Vladimir Aleksi c Phd

8/15/2019 Vladimir Aleksi c Phd

http://slidepdf.com/reader/full/vladimir-aleksi-c-phd 1/156

Methods for Handling EqualityBased on Proof Transformations

Vladimir Aleksic

2007

A thesis submitted to the University of London

for the degree of Doctor of Philosophy

King’s College London

Department of Computer Science



Abstract

This thesis investigates methods for theorem proving in first-order logic with equal-

ity, whose refutational completeness is shown by transformations of derivation trees.

These methods are built around refinements of the the paramodulation calculus,

which, in terms of modern results in paramodulation-based theorem proving, can

be labelled as non-standard.

All recent results in paramodulation-based theorem proving are closely related

to the model generation technique for proving its refutational completeness. This

technique has proved to be powerful enough to show refutational completeness of nu-

merous refinements of paramodulation, including superposition with tautology elim-

ination, subsumption and simplification. Moreover, model generation can be used to

show that these refinements are complete even if they implement special strategies of

derivations, such as the basic strategy, and the classical literal selection strategies.However, certain applications require derivations by paramodulation-based cal-

culi, which implement strategies that can not be shown complete using the model

generation method. An example of such strategies is the regular strategy of deriva-

tions (needed for equality elimination), which is a specialization of derivations that

seems not to have a model generation completeness proof. A more general example

is the arbitrary literal selection strategy, which is a super-set of the classical literal

selection strategies, and also seems not to fit into the model generation framework.

The focus of this thesis is in developing an alternative framework for showing

refutational completeness of methods built around these non-standard refinements of

paramodulation. This framework is based on special transformations of derivation

trees, which work even where the model generation technique fails. The key strength

of the transformation-based methods developed in this thesis is that they, opposite

to other modification methods, provide as much as the model generation method. In

particular, they allow for elimination of classes of redundant clauses, and are powerful

enough to show completeness even for methods based around refinements of the basic

superposition calculus.

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work. Their numerous suggestions on how to improve the content of this thesis have

contributed to its quality to the extent that is not measurable.

It has been a joyful and most pleasant experience to be part of the PhD group at

King’s. Many thanks to the staff of the Poetry Place. I would also like to thank my

family for their constant support and presence, and my friends for making the past

four years a truly enjoyable time. Thank you Chiara, Helen, Adam, Alex and Ozan.

Finally, thank you Maja, for being so very special.



Contents

1 Introduction 7

2 Preliminaries 12

2.1 Terms, term orderings and rewrite relations . . . . . . . . . . . . . . . 12

2.2 Equations, atoms and clauses . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.1 Constrained clauses . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Closures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.3 Equational satisfiability . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4 Inference systems and derivations . . . . . . . . . . . . . . . . . . . . . 20

3 Paramodulation

and basic superposition 233.1 Resolution-based reasoning with equality . . . . . . . . . . . . . . . . . 24

3.2 Paramodulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.3 Brand’s modification method . . . . . . . . . . . . . . . . . . . . . . . 27

3.3.1 Flattening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.3.2 Symmetry elimination . . . . . . . . . . . . . . . . . . . . . . . 29

3.3.3 Transitivity elimination . . . . . . . . . . . . . . . . . . . . . . 30

3.3.4 Brand’s proof of completeness of paramodulation . . . . . . . . 31

3.3.5 Importance and limitations of Brand’s results . . . . . . . . . . 34

3.4 State-of-the-art refinements of paramodulation . . . . . . . . . . . . . 36

3.4.1 Ordered paramodulation . . . . . . . . . . . . . . . . . . . . . . 37

3.4.2 Basic strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4.3 The model generation method . . . . . . . . . . . . . . . . . . 43

3.4.4 Practical redundancy criteria and deletion rules . . . . . . . . . 52

3.4.5 Selection strategies . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 Examples of basic superposition calculi . . . . . . . . . . . . . . . . . . 59

3.5.1 Calculus BS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

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CONTENTS 6

3.6 Calculus EBFP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4 Ordered paramodulation without term orderings 63

4.1 Non-lengthening superposition of ground unit clauses . . . . . . . . . . 64

4.2 Non-lengthening superposition of ground Horn clauses . . . . . . . . . 67

4.3 Non-lengthening superposition of non-ground clauses . . . . . . . . . . 71

5 Regular derivations in basic superposition 73

5.1 Permutation rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2 A proof by transformation . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Regular derivations on closures . . . . . . . . . . . . . . . . . . . . . . 87

6 Basic superposition with arbitrary selection strategies 90

6.1 Completeness for superposition with ordering constraint inheritance . 93

6.2 Arbitrary selection for superposition on closures . . . . . . . . . . . . . 100

6.3 Arbitrary selection and tautology elimination . . . . . . . . . . . . . . 101

7 Basic folding revisited 103

7.1 The basic folding calculus . . . . . . . . . . . . . . . . . . . . . . . . . 105

7.2 Completeness of basic folding of Horn clauses . . . . . . . . . . . . . . 108

7.2.1 The “a folding descendant or a proper subsumer of” relation . 111

7.2.2 Relations m

−→ and c

−→ . . . . . . . . . . . . . . . . . . . . . . . 1167.2.3 Proof transformations involving superposition and

m−→ . . . . . 120

7.2.4 Proof transformations involving c−→ . . . . . . . . . . . . . . . 126

7.2.5 The completeness statement . . . . . . . . . . . . . . . . . . . . 129

7.3 Completeness of basic folding of general clauses . . . . . . . . . . . . . 131

7.3.1 Calculus EBMP : a new approach to factoring . . . . . . . . . 136

7.3.2 Proof of completeness . . . . . . . . . . . . . . . . . . . . . . . 138

8 Conclusions and future work 144



Chapter 1

Introduction

Automated reasoning for first-order logic (FOL) with equality has always been of

great importance in mathematics and logic. In recent years, through development

and implementation of practically efficient reasoning formalisms, theorem proving in

FOL has become a critical component of applications of formal methods in computer

science. These applications include integrated circuit design and verification, security

protocol verification, logic and functional programming, machine learning, and, most

recently, the semantic web.

The most promising results to date for reasoning in FOL with equality are based

around the paramodulation rule, originally introduced by Robinson and Wos in[RW69]. Discouraged by the approach that uses resolution with congruence axioms,

they introduced a rule that handles the equality predicate directly, treating it as a

part of the language. Named paramodulation, this rule produces fewer clauses by

combining a number of resolution steps involving the congruence axioms, and ne-

glecting the intermediately generated consequences. Confident in efficiency of their

approach, Robinson and Wos joined paramodulation with the reflexivity resolution

and positive factoring rules, and proved refutational completeness of thus obtained

paramodulation calculus. This result, however, held true only in presence of func-

tional reflexivity axioms (clauses of the form f (x1

, . . . , xn

) ≈ f (x1

, . . . , xn

) for all

n-ary functional symbols f ), which was its serious shortcoming. Although it was im-

mediately clear that the functional reflexivity axioms made paramodulation a more

prolific calculus than resolution in presence of congruence axioms, Robinson and Wos

could not remove the former because the technique they used to show refutational

completeness strongly relied on their presence.

It was only with Brand’s results (see [Bra75]) that paramodulation was proved

refutationally complete without the functional reflexivity axioms. Despite failing to

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remove them, Brand more importantly introduced his famous modification method

(for proving completeness of paramodulation), which pointed out some redundanciesthat appear as a result of applying paramodulation in its original, unrestricted form.

Namely, Brand’s completeness proof strongly hinted that paramodulation inferences

into variables and into terms introduced by unification in previous inferences 1 were

not necessary: this immediately initiated prolific research. The productiveness of

this research measures in numerous refinements of paramodulation that use term

rewriting techniques, which were pioneered by Knuth and Bendix in [KB70] and

which provided important clues toward eliminating inferences into variables. These

refinements came with a bonus, embodied in a number of techniques for proving their

refutational completeness. They are the transfinite semantic tree method by Hsiang

and Rusinowich (see [HR91]), the proof ordering method by Bachmair (see [Bac89]),

and forcing techniques of Zhang [Zha88] and Pais and Peterson [PP91].

The most distinguished forcing technique, the model generation method of Bach-

mair and Ganzinger [BG90], has turned out to be powerful enough to prove com-

pleteness of many refinements of paramodulation, some of them much stronger then

ordered paramodulation. The model generation method is a way of proving complete-

ness of a calculus, name it IC , by constructing an equality Herbrand interpretation

for a set of clauses S , which is closed under IC and does not contain the empty clause.

Informally, the interpretation built by the method is a congruence R∗ induced by a

set of ground rewrite rules R. Each rule in R is generated by some clause of S . The

process of generating R is defined by induction on an ordering on clauses (which is

an extension of a given term ordering). A clause C generates a rule and therefore

contributes to the model if it does not hold in the model generated by the clauses

that are smaller than C (in the given ordering). Otherwise, the clause C holds in

this partial model, and thus follows from the clauses that are smaller, which makes it

redundant in the process of model generation.

The key feature of the model generation method is that it automatically provides

a powerful redundancy criterion (for clauses and inferences) in the framework of every

calculus that has a model generation completeness proof. This includes calculi thatimplement the basic strategy and simultaneously employ a selection of literals which

are allowed to be involved in inferences (selection functions). Consequently, the model

generation method has a huge advantage over all other known techniques, and has

been established as the method of choice for proving refutational completeness of

paramodulation-based calculi.

1Blocking paramodulation into terms introduced by unification in previous inferences is known as

the basic strategy of derivations.



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Another distinctive aspect of the model generation method is that it is parametrized

by an ordering on terms. Almost every proof by model generation requires this order-ing to be a reduction ordering on ground terms, i.e. to be well-founded, monotonic

and to satisfy the sub-term property. These restrictions, however, prove to be the

main weakness of the method, because for some applications they are simply too

strong. Assume, for example, that an application necessitates derivations by or-

dered paramodulation from a set of first-order clauses with equality and non-equality

(predicate) literals, in which all inferences on equality literals occur before all other

inferences.

This specialization of derivations2 can be formulated as a refinement of ordered

paramodulation where the ordering does not satisfy the sub-term property and is

therefore a non-reduction ordering. Unfortunately, if the underlying term ordering

is not a reduction ordering, the model generation technique will at least give up a

lot of its power, while in some cases it even fails to produce a completeness proof.

Likewise, as shown by Bofill and Rubio in [BR02], for a non-monotonic ordering which

does not satisfy the sub-term property, model generation can provide completeness

only of ordered paramodulation (it does not work for superposition) and only for the

calculus of Horn clauses. For the full first-order case a model generation proof seems

to not exist at all. Furthermore, the result for the Horn case no longer extends to the

basic version of the calculus, and even ceases to be compatible with subsumption and

simplification.

An alternative way of obtaining the above described specialization of derivations

is to use a paramodulation calculus which employs a suitable selection function but

keeps a reduction ordering. By such a selection function only equality literals are

selected, and all of them, which implies that a positive equation s ≈ t can be selected

in presence of P (t). This, however, does not fit in the class of selection functions

that have a model generation completeness proof. By these selection functions, a

positive literal is selected only if it is maximal, which clearly contradicts (because

of the sub-term property of reduction orderings) the selection of s ≈ t in presence

of P (t). Subsequently, the proposed derivation strategy by paramodulation with thespecial selection function does not have a model generation proof either.

To conclude, there exist refinements of paramodulation which can not be proved

complete by using model generation, nor by any other known technique. This thesis

2The strategy of derivations in which inferences on equality literals take place before all other

inferences is known as the regular strategy of derivations. The proof of refutational completeness

of paramodulation in [RW69] by Robinson and Wos used the regular strategy to transform an e-

unsatisfiable set of clauses with congruence axioms into an unsatisfiable set of clauses.



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would like to contribute to the realm of paramodulation-based theorem proving by an-

alyzing some of these refinements of paramodulation (especially those based on specialselection strategies and term ordering requirements), building methods for handling

equality around them, and developing techniques for proving their completeness. The

main quality of all results presented in this thesis is that they strengthen related re-

sults and answer questions that they open, while not compromising the state-of-art

requirements of paramodulation calculi like basicness and redundancy elimination.

Summary of main results

The following is a summary of the main contributions of this thesis to the knowledge

in the field of paramodulation-based reasoning with equality.

1. Refutational completeness of so-called non-lengthening superposition of ground

Horn clauses with tautology elimination. This result is built on an idea of

Kanger [Kan63], who was the first one to suggests the use of ordering restrictions

in equational inference rules. From the point of view of modern paramodulation-

based theorem proving, the result explores the area beyond the mark set by

Bofill and Rubio in [BR02], by dropping even well-foundedness from the re-

quirements of term orderings used to orient paramodulation.

This result is proved using a transformation method, and presented in Chap-

ter 4, in Lemma 4.6.

2. Refutational completeness of the regular strategy of derivations for basic super-

position calculus with tautology elimination and subsumption. By the regular

strategy (cite Kanger [Kan63]) all equality inferences take place before all other

steps in the proof. The completeness proof is given using an unusually powerful

transformation technique, which, unlike other transformation techniques, nei-

ther compromises basicness of the calculus nor elimination of redundant clauses.

The result is given in Chapter 5, Theorem 5.14, and has been published at

LPAR 2005 under the title “Regular Derivations in Basic Superposition-BasedCalculi” (see [AD05]).

3. Refutational completeness of arbitrary selection strategies for derivations by ba-

sic supersposition, provided that there exists a refutation that does not contain

factorisation. The result is an answer to questions posed in [dN95] and [BG01b].

The result can not be further strengthened with redundancy elimination, as il-

lustrated by Example 5.4.



11

This result is presented in Chapter 6, Theorem 6.6, and has been published

at JELIA 2006 under the title “On Arbitrary Selection Strategies for BasicSuperposition” (see [AD06]).

In case of resolution, this result can be strengthened by tautology elimination,

which is shown in Theorem 6.9.

4. Refutational completeness of basic folding for Horn clauses with tautology elimi-

nation and subsumption. This result is a continuation of the work of Degtyarev,

and Voronkov presented in [DV96], who give a completeness result without any

redundancy notions. The method used to show this result is a transformation

method, which simultaneously builds on the method used to show completeness

of regular derivations, and the method used in the original paper by Degtyarev,

Koval and Voronkov [DKV95].

This result is presented in Chapter 7, in Theorem 7.24.

5. Refutational completeness of a basic superposition-based calculus BMP of gen-

eral first-order clauses, which takes a novel approach to factoring of positive

literals. This result is proved by the model generation method and therefore

provides powerful redundancy criteria for clauses and inferences. The result is

formulated in Chapter 7, in Lemma 7.29.

6. Extension of the completeness result of basic folding from Horn case to general

first-order case. The result is obtained by using a transformation method, which

proves powerful enough to retain all qualities of the corresponding result for

Horn clauses.

A detailed sketch of the proof is given in Chapter 7, in Theorem 7.38.



Chapter 2

Preliminaries

This chapter gives basic definitions on terms and term orderings, equations, clauses,

inference systems and derivations.

2.1 Terms, term orderings and rewrite relations

Let X be a denumerable set of variable symbols x , y , z , . . . and F be a fixed finite

function signature f , g , h , . . ., so that F ∩X = ∅. Let arity : F → N be a function on

function symbols, which maps them to the number of their arguments. If arity(f ) = 0

for some f ∈ F , then f is a constant symbol. Constants are denoted by a , b, c , . . ..It is assumed that every function symbol has a unique arity, i.e. if F i = f ∈ F |

arity(f ) = i, then the sets F i are disjoint. The set T F (X ) of free, first-order term is

a minimal set such that

• X ∈ T F (X ),

• if t1, . . . , tn ∈ T F (X ) for some n, and arity(f ) = n then f (t1, . . . , tn) ∈ T F (X ).

The set of ground, or variable-free, terms T F is defined as T F (∅). From this definition,

it follows that whenever F 0 = ∅ then also T F = ∅. Therefore, it is always assumed

that F always contains at least one constant symbol. The fact that terms t1 and t2

are syntactically equivalent is denoted by t1 ≡ t2. Note that the symbols = and ≡

do not belong to the formal language used throughout the thesis. They are a part of

the meta-language.

A term t ∈ T F (X ) can be viewed as a finite ordered tree formed such that

• the term t is the root of the tree,

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2.1 Terms, term orderings and rewrite relations 13

• if t = f (t1, . . . , tn), it is the parent of finite ordered trees formed from the terms

t1, . . . , tn; the edge between t and ti is labelled by i, for all i ∈ 1, . . . , n,

• the leaves of the tree are variable symbols and constants.

A position p within a term t is a sequence of positive integers that corresponds to

the sequence of labels of edges of the tree-representation of t, on the path from the

root to some node. A term at position p in a term t is denoted by t | p. For example,

the position of f (x) in f (g(h(a), f (x))) is 1.2 (denoted as t |1.2= f (x)), whereas the

position of x in the same term is 1.2.1 (t |1.2.1= x). The top position, denoted by ε,

is an empty sequence of integers, and therefore t |ε= t. It is said that a term s occurs

at a position p in a term t if t | p≡ s. A term s occurs below a position p in a term t,if there exists a positive integer q such that t | p.q≡ s. Similarly, a term s occurs above

a position p in a term t, if there exists a position r and a positive integer q such that

t |r≡ s and p = r.q . If it is important to emphasize that if a term t contains s as a

subterm (at position p), then t can be written as t[s] (t[s] p). In the same fashion, the

result of replacing s by r (at p) in t is denoted by t[r] (t[r] p).

A substitution σ is a mapping X → T F (X ). Substitutions are usually represented

by their non-identity part, i.e. if σ = [x1 → t1, . . . , xn → tn], it means that

• σ(xi) = ti, for i ∈ 1, . . . , n,

• σ(x) = x, for all other variables.

The domain of a substitution σ, dom(σ), is the set x | σ(x) = x. Similarly, a

range of σ, denoted by ran(σ), is the set t | t = σ(x), where x ∈ dom(σ). If

range(σ) ⊆ T F , σ is a ground substitution. The set of variables of a substitution σ,

denoted by var(σ) is the union of variables of dom(σ) and ran(σ).

The application of a substitution σ to a term t, denoted as tσ is defined as

tσ =

σ(t) if t ∈ X

f (t1σ , . . . , tnσ) if t = f (t1, . . . , tn) and n > 0t if t is a constant symbol

For example, g(x, y)[x → f (y), y → h(a, a)] = g(f (y), h(a, a)).

The composition of substitutions σ and θ, denoted σθ, is a function from X to

T F (X ), defined as a composition of functions, i.e. x(σθ) = (xσ)θ for all x ∈ X . For

example, [x → f (a, y)][y → a] = [x → f (a, a), y → a], since g(x, y)[x → f (a, y)][y → a] =

g(f (a, a), a).



2.1 Terms, term orderings and rewrite relations 14

The union of two substitutions σ and θ , denoted by σ ∪ θ and defined if dom(σ) ∩

dom(θ) = ∅, is a mapping from dom(σ ∪ θ) (= dom(σ) ∪ dom(θ)) such that

x(σ ∪ θ) =

xσ if x ∈ dom(σ)

xθ if x ∈ dom(θ)

A term s is a (ground) instance of a term t, if there exists a (ground) substitution σ

such that tσ ≡ s. Two terms are variants if they are instances of each other.

Two terms s and t are unifiable if there exists a substitution (unifier) σ such that

tσ ≡ sσ . A substitution is a most general unifier of terms s and t, denoted mgu(s, t),

if it is a unifier of s and t and sσθ ≡ tσθ for every other substitution θ.

A (strict partial) ordering is an irreflexive and transitive relation. An ordering

on T F (X ) is

• total on T F if either s = t, s t or t s,

• well-founded if there are no infinite descending chains t1 . . . tn . . .,

• stable under substitutions if s t implies sσ tσ for all terms s, t and every

substitution σ.

• monotonic if l r implies t[l] p t[r] p for all terms l ,r, t and positions p,

• compatible with the sub-term property if t[s] p s for all terms t, s and positions

p = ε.

A rewrite ordering is a monotonic ordering that is stable under substitutions. A

reduction ordering is a well-founded rewrite ordering. A simplification ordering is

a reduction ordering with the subterm property. The following is easy to check –

reduction ordering that is total on T F is a simplification ordering on T F .

Let be a term ordering and ∼ be an equivalence relation. Then is said to be

compatible with ∼ if s ∼ s t ∼ t implies s t.

A rewrite rule is an ordered pair of terms (s, t), written s ⇒ t. A set of rewrite

rules, usually denoted by R is a rewrite system . A rewrite relation with R on T F (X ),

denoted →R is the smallest monotonic relation such that lσ →R rσ for all l → r ∈ R

and all σ.

The relation →R is a binary relation, with ←R being its inverse, ↔R its symmetric

closure, →+ its transitive closure and →∗ its reflexive closure. The relation →R is



2.2 Equations, atoms and clauses 16

Note that equality literals are also identified as multisets: positive equations s ≈ t are

denoted by s, t, while the negation of the same equation has the multiset notations,s,t,t. To distinguish positive and negative literals, especially in examples, clauses

are sometimes represented as ¬A1, . . . , ¬Ak, B1, . . . Bl or equivalently, as sequents

A1, . . . , Ak → B1, . . . , Bl.

A multiset extension of a congruence ∼ is defined as the smallest relation ∼mul

such that

• ∅ ∼mul ∅, and

• S ∪ s ∼mul S ∪ s if S ∼mul S and s ∼ s .

Similarly, a multiset extension of a term ordering (which is compatible with a

congruence ∼) is the smallest relation mul compatible with ∼mul such that

S ∪ s mul S ∪ s1, . . . , sn if S ∼mul S and s s i for all i ∈ 1, . . . , n.

Ordering on equality literals is defined as the multiset extension of a term ordering .

Bearing in mind the multiset definition of (positive and negative) literals, it follows

that for terms s, t and u, if s t u, then s ≈ u s ≈ t s ≈ u. Also, for all

equations e, ¬e e.

Ordering on clauses is defined as a two-fold multiset extension of a given ordering

on terms. The two-fold multiset extensions of a given well-founded and total relation

is a well-founded and total relation.

Clauses that contain no more than one positive literal are called Horn clauses .

Analogously, clauses that contain at most one positive equality literal are Horn with

respect to equality literals . The empty clause is the clause that contains no literals.

It is also known as a contradiction. The set of variables of a clause C is denoted as

var(C ).

A clause C is an instance of a clause D if there exists a substitution C = Dσ. It

is a ground instance of D if σ is a ground substitution. Two clauses are variants if they are instances of each other.

Assume that C is an instance of D. Then there exists a bijection ϕ that maps the

literals of C to the literals of D, such that for every literal L of C , ϕ(L)σ is a literal

of D. A position p in a literal ϕ(L)σ of the clause D is a variable position if L p is a

variable. The set of variable positions of a clause C (which is an instance of D) refers

to the set of all variable positions of all literals in C .



2.2 Equations, atoms and clauses 18

can be considered as equation P (x1, . . . , xn) ≈ true, where true is a new constant

symbol and P is considered as a function symbol rather than a predicate symbol. Inorder for this encoding to be sound, and to avoid meaningless expressions in which

predicate symbols occur as proper subterms, one should use a two-sorted logic, in

which the sort of the symbol true and the predicate symbols is different from the sort

of the function symbols. After this satisfiability preserving transformation, resolution

inferences of the formΓ1, P Γ2, ¬P

Γ1, Γ2(res)

can be looked as a sequence of two inferences, where paramodulation

Γ1, P ≈ true Γ2, P ≈ trueΓ1, Γ2,true ≈ true ( par)

is followed by an equality solution that removes the trivial literal true ≈ true.

If predicate symbols are considered as functional symbols of another sort, it is

possible to use ordering on terms to compare atoms with other atoms and terms,

and also predicate literals with equality literals. For example, if the given ordering

of ground terms is a reduction ordering, then P (t) t for all ground terms t. Also,

P (t) t ≈ s. This approach can be taken in the opposite direction, too. Namely, the

result of comparing two predicate literals can be defined as the result of comparingtheir corresponding equational representations.

2.2.1 Constrained clauses

A constraint is a possibly empty conjunction of atomic equality constraints s = t or

atomic ordering constraints s t or s t. The empty constraint is denoted by .

A constrained clause is a pair consisting of a clause C and a constraint T , written as

C | T . The part C will be referred to as the clause part and T the constraint part of

C | T . A constrained clause C | will be identified with the unconstrained clause

C . A constrained clause is called a Horn (general first-order) constrained clause if itsclause part is a Horn (general first-order) clause.

A substitution σ is said to be a solution of an atomic equality constraint s = t,

if sσ and tσ are syntactically equivalent. It is a solution of an ordering constraint

s t (with respect to a reduction ordering > which is total on ground terms), if

sσ > tσ, and a solution of s t if it is a solution of s t or s = t. Generally,

a substitution σ is a solution of a constraint T , if it is a simultaneous solution to

all its atomic constraints. If T contains atomic ordering constraints, σ is required



2.3 Equational satisfiability 19

to be a ground substitution. Note that a purely equality ordering constraint has a

unique solution, while a constraint which contains atomic ordering constraints mayhave more solutions. A constraint is satisfiable if it has a solution.

A ground instance of a constrained clause C | T is any ground clause Cσ, such

that σ is a ground substitution and σ is a solution to T . The notation irredR(C | T )

stands for a set of ground instances of C | T which are irreducible by R, for every

rewrite system R contained in a given ordering (that is, l r for all rewrite rules

l ⇒ r ∈ R). Because C | T represents a set of ground clauses, than a set S of

constrained clauses represents a set of ground clauses, too. Therefore, irreducibility

property can be extended to sets of constrained clauses.

A contradiction is a constrained clause | T , with the empty clause part such that

the constraint T is satisfiable. A constrained clause is called void if its constraint is

unsatisfiable. Void clauses have no ground instances and therefore are redundant. A

set of constrained clauses is satisfiable if the set of ground instances of all the clauses

is satisfiable.

2.2.2 Closures

A closure is a pair C ·σ, where C is a clause, σ is a substitution and dom(σ) = var(C ).

A closure is referred to as a Horn (general first-order) closure if its clause part is a

Horn (general first-order) clause. The set of variables of a closure C · σ, denoted by

var(C ) is the set var(C ) ∪ var(σ). An instance of a closure C · σ is any clause Cσρ

for some substitution ρ. The notion of irreducible (with respect to rewrite systems)

instances of closures and sets of closures is defined in the same way as for constrained

clauses.

If the clause part of the closure is empty, the closure is identified with the empty

clause – it follows from the definition of closures that the substitution part is empty,

too. Closures whose substitution part is the empty substitution are called ε-closures.

Two closures C 1 ·σ1 and C 2 ·σ2 are variants iff the clauses C 1 and C 2 are instances

and C 1σ and C 2σ are instances, too.

2.3 Equational satisfiability

In this thesis the main focus is on equational clauses. The equality predicate, as it is

considered in first-order logic with equality, is a predicate that satisfies the following



2.4 Inference systems and derivations 21

Inference rules can also be defined on sets of clauses, constrained clauses or clo-

sures. For example, the rule on sets of clauses

S S ∪ C

where S is a set of clauses, denotes a set-representation of the rule

C 1 · · · C nC

if there exist clauses C 1, . . . , C n from S . If a set S 2 is obtained from S 1 by applying

an inference rule, it will be denoted by S 1 ⇒ S 2. Inference rules defined on sets are

particularly handy in presence of so-called deletion rules, which can also be definedon clauses, constrained clauses or closures. A deletion rule is a rule of the form

S ∪ C

S if condition

and denotes that the clause C can be eliminated from a set S ∪ C , provided that

condition holds.

A derivation is a possibly infinite ordered sequence of sets of clauses S 0 ⇒ S 1 ⇒

· · · , where each set S i+1 is obtained from the previous one, the set S i, either by

adding a conclusion of an inference or by deleting a clause by using a deletion rule.

A refutation is a common name for a derivation of the empty clause. A sequence

S 1 ⇒ · · · ⇒ S n is a derivation of a clause C ∈ S n, if S n−1 ⇒ S n by applying an

inference whose premises are from S n−1 and whose conclusion is C .

A derivation of an e-empty clause from a set of clauses which contain both predi-

cate and equality literals is called e-refutation . An e-refutation that ends with equal-

ity inferences is called s-e-refutation (from short e-refutation). Note that the empty

clause is also e-empty. Similarly, every refutation is also an e-refutation.

The limit S ω of a derivation S 0 ⇒ S 1 ⇒ · · · is the set ∪ j ∩k≥ j S k of clauses. The

derivation is called fair iff for any application of in inference rule of the form

S ωS ω ∪ C

there exists i such that C ∈ S i. A derivation is called saturated iff it is finite and fair.

A derivation tree is a tree-like representation of a derivation of a clause. Assume

Ω is a derivation S 1 ⇒ · · · ⇒ S n of a clause C . It can be transformed into a derivation

tree T Ω recursively, as follows:



Chapter 3

Paramodulation

and basic superposition

This chapter presents the history of paramodulation-based theorem proving. Since

the introduction of the paramodulation rule and calculus in [RW69], this history can

be looked at as a simultaneous development in two directions. On one hand, there has

been a lot of research into inefficiencies of paramodulation and ways of fixing them.

This has resulted in numerous refinements of paramodulation. On the other hand,

there have been many results introducing methods for proving refutational complete-

ness of the refinements of paramodulation. Very often, existing methods would not

be powerful enough to show completeness of some novel refinements. This would

trigger research into novel techniques for proving completeness of paramodulation,

which would then, though not always, motivate further refinements of paramodula-

tion, starting the cycle all over again.

The emphasis of the chapter, in the way it connects to the new results presented

in this thesis, is on the state-of-art refinements of the paramodulation rule. Methods

that are proved complete in the thesis are claimed to be satisfying the state-of-art

requirements that are described in this chapter. In the process of developing anddefining these modern requirements, the most important were Brand’s modification

(see [Bra75]), and the model generation method (see [BG90]) for proving refutational

completeness of paramodulation and its refinements.

Brand’s modification method and, generally, his results, are closely related to all

new contributions of this thesis. All results except for those presented in Chapter 4

address the basic strategy of derivations, origins of which can be traced back to

Brand’s method. Additionally, Brand’s method is an attempt of equality elimination,

23



3.1 Resolution-based reasoning with equality 24

which is a topic addressed in Chapter 5 and Chapter 7. Finally, Brand’s method

of proving refutational completeness of paramodulation is by proof transformation,which is the approach used to prove every contribution of the thesis.

Although the model generation method is not directly connected to the results

given in the thesis, it is important to carefully analyse it for a number of reasons.

Firstly, the model generation method gives the most powerful redundancy criterion

to date, which is in the heart of all state-of-are theorem provers. The redundancy

criteria used in Chapter 5 and Chapter 7 are required to fit into this redundancy

criterion. Secondly, understanding the model generation method gives a straight for-

ward answer to why is it the case that every result given in this thesis is proved using

transformations of derivation trees.

For the reasons given in the last two paragraphs, a big part of this chapter fo-

cuses on these the Brand’s modification and model generation methods. Before going

further, note that the term “paramodulation” will often be used to mean both a

refinement of the paramodulation rule and the corresponding calculus.

3.1 Resolution-based reasoning with equality

The first approach to reasoning on clauses with equality was using the resolution

calculus (see [Rob65]):

Resolution Γ1, A Γ2, ¬B

(Γ1, Γ2)mgu(A, B) (res)

Positive factoring Γ, A , B

(Γ, A)mgu(A, B) (f ac)

As far as the resolution calculus is concerned, the equality predicate is treated as

any other predicate. Therefore, in order to use resolution to check equational unsat-

isfiability of a set of clauses in first-order logic with equality, it is necessary to add

congruence axioms CA (see Section 2.3) to the theory.

It turns out, however, that this approach is very prolific, since it is possible to make

infinitely many resolution inferences from a set of clauses that include an arbitrary

clause with a positive literal and a congruence axiom.



3.2 Paramodulation 25

Example 3.1. Consider the functional monotonicity axiom x ≈ y → f (x) ≈ f (y)

and a clause → a ≈ b. These two clauses generate the following infinite chain of resolution inferences:

a ≈ b x ≈ y, f (x) ≈ f (y)

f (a) ≈ f (b) (res)

x ≈ y, f (x) ≈ f (y)

f (f (a)) ≈ f (f (b)) (res)

x ≈ y, f (x) ≈ f (y).... (res)

3.2 Paramodulation

Attempting to eliminate the congruence axioms, Robinson and Wos, in [RW69], sug-gested an approach to reasoning with equality, which treats the equality predicate as

a part of logical language. Motivated by the Leibniz law for the replacement of equals

by equals, they proposed its clausal form, and named it paramodulation:

Γ1, l ≈ r Γ2, L[l]

(Γ1, Γ2, L[r])mgu(l, l) ( par)

where expression L[l] denotes that a term l appears as a subterm of L, and L[r]

denotes the literal obtained by replacing the particular occurrence of l by r.

Similarly to resolution, the paramodulation rule is defined using unification. Its

conclusion is a result of replacing an indicated occurrence of l mgu(l, l) (denoted by

lσ) from the “into” premise by rσ , and adding side literals Γ1σ of the “from” premise.

Note 3.1. By analyzing the definition of paramodulation, it is easy to spot the rule’s

first deficiency. Paramodulation is symmetric with respect to the equation l ≈ r.

The rule identifies l ≈ r and r ≈ l , so that the conclusion could also be obtained by

replacing an occurrence of rσ by lσ.

A possibility of constraining this symmetry, though for sequent calculi, was first

suggested by Kanger in [Kan63]. He allowed only non-increasing applications of equal-

ity rules, i.e. those in which the complexity of l is greater than the complexity of r.Guided by this idea, many authors suggested refinements of paramodulation, in which

the paramodulation rule is oriented using term orderings. Some of them are analyzed

in detail in Section 3.4.1.

To obtain refutational completeness, Robinson and Wos joined paramodulation

with the resolution and factoring rules, thus forming the paramodulation calculus P .

The following is the formulation of their completeness result.



3.2 Paramodulation 26

Theorem 3.2 (Robinson and Wos, [RW69]). If a functionally reflexive set of clauses

S is closed under paramodulation and factoring, and if S is equationally unsatisfiable,then S is unsatisfiable.

This result is important for the findings presented in this thesis for two differ-

ent reasons. Firstly, the completeness proof is derived using the regular strategy of

derivations , which is further discussed and proved complete in the framework of basic

superposition in Chapter 5. Secondly, the formulation of the above theorem suggests

that the authors, in order to prove completeness of paramodulation, first eliminate

the equality predicate. This way, they reduce the problem of equational unsatisfia-

bility (of a set of clauses) to the problem of unsatisfiability. This idea underlines the

ideas of the regular strategy of derivations and the basic folding calculus, which areproved complete in the framework of basic superposition in Chapter 5 and Chapter 7.

Unfortunately, the method (for proving refutational completeness of paramodula-

tion) that Robinson and Wos used was limited in that it could provide completeness

of paramodulation only in presence of the functional reflexivity axiom (one instance

of the axiom is added to the set of clauses for every function symbol f )

f (x1, . . . , xn) ≈ f (x1, . . . , xn),

which makes paramodulation a very prolific rule.

Note 3.2. Paramodulation in presence of the functional reflexivity axioms is even

more prolific than resolution in presence of congruence axioms. This observation,

which follows from the proof of Theorem 1 of [BG98a] is very counter-intuitive, since

paramodulation is a more direct rule (in the way it handles equality) than resolution.

Therefore, many authors conjectured that the functional reflexivity axioms are not

needed for completeness of the calculus.

This conjecture, however, could only be proved with the appearance of novel meth-

ods for proving refutational completeness of paramodulation. Most notable among

them are the methods developed by Brand [Bra75] and Peterson [Pet83].

Careful analysis of the use of functional reflexivity in the completeness proof of

[RW69] points out another imperfection in the definition of paramodulation. However,

to get to that point, it is necessary to start from the fact that the functional reflexivity

axioms are used in [RW69] only to provide lifting. The lifting argument is the main

step in proving refutational completeness of calculi. Completeness is first shown for

a ground version of the calculus. The calculus has the lifting property if for each

ground inferenceD1 . . . Dn

D



3.3 Brand’s modification method 27

there is an inference from arbitrary generalizations C 1, . . . , C n of D1, . . . , Dn with a

conclusion which is a generalization of D. Refutational completeness then follows,since lifting implies that each derivation of the empty clause (denoted by ) from a

set of ground clauses S has a derivation of the generalization of (which is itself)

from a set of generalizations of the clauses from S .

Unlike resolution and factoring, the paramodulation rule is not liftable. As it has

already been pointed out, the presence of functional reflexivity makes paramodulation

liftable. For example, the non-liftable inference

a ≈ b P (f a , f a)

P (f a , f b) ( par)

can be lifted using the axiom f x ≈ f x:

a ≈ b f x ≈ f x

f a ≈ f b ( par)

P (y, y)

P (f a , f b) ( par)

Note 3.3. Analyzing the way functional reflexivity is used for lifting, it becomes

evident that its main function is to provide the possibility of substituting an arbitrary

term for a variable. In other words, functional reflexivity axioms are always used for

inferences into variable positions. In the previous example, x is substituted for a by

first paramodulating a ≈ b into f x ≈ f x.Paramodulation into variables is always possible, because every term unifies with

a variable. Therefore, many authors looked at refining paramodulation so that infer-

ences into variables could not be possible.

3.3 Brand’s modification method

Trying to refine the extensively prolific paramodulation rule in presence of the func-

tional reflexivity axioms (see Note 3.2), Brand [Bra75] proved that elimination of the

functional reflexivity axioms does not affect completeness of the paramodulation cal-

culus 1. This result has become a landmark in the history of paramodulation-based

theorem proving.

Dealing with the problem of checking the equational unsatisfiability of a set of

clauses, Brand took an approach that combined the above described resolution and

paramodulation-based dealing with equality. On the one hand, Brand suggests using

1A similar result was independently obtained by Peterson [Pet83]




resolution, which is applied to a set S of clauses (without equality) obtained after

applying some “efficient” transformations to a given set S of first-order clauses withequality. As a result of these transformations, assuming that S is equationally unsat-

isfiable, Brand showed that there exists a refutation of S by resolution in which the

only congruence axiom that is needed is reflexivity. On the other hand, Brand took

the paramodulation-based approach, by showing that there exists a refutation of S

by resolution, that can be mapped into a refutation of S ∪ x ≈ x by paramodula-

tion. It immediately follows that neither the congruence (except reflexivity) nor the

functional reflexivity axioms are necessary for refutational completeness of paramod-

ulation.2

The transformations applied to the clauses from S are performed in three stages:

flattening (elimination of the monotonicity axioms), elimination of the symmetry

axiom, and elimination of the transitivity axiom. Intuitively, Brand suggested that

the equality predicate is removed by repeated applications of resolution with the

congruence axioms and the clauses in S . The congruence axioms are disposed of at

the end of the process, and reasoning continues using resolution.

This section will go over the stages of Brand’s modification process, exemplifying

each step on the equationally unsatisfiable set of clauses T :

→ a ≈ b

→ P (f (a))P (x) → Q(x, x)

Q(f (a), f (b)) →

The final part of the section contains an exemplified description of the strategy which

Brand uses to construct a refutation from S ∪ x ≈ x which is always possible to

map into a refutation of S ∪ x ≈ x by paramodulation.

3.3.1 Flattening

As defined in Section 2.2, a literal is flat if non-variable terms appear only at the top

position, and a clause is flat if it contains only flat literals.

2Precisely, if congruence axioms are not necessary for completeness, neither are the functional

reflexivity axioms, since they follow from the monotonicity axioms: f (x) ≈ f (x) follows from x ≈

y → f (x) ≈ f (y). However, it is worth emphasizing that Brand’s result is stronger than the results

addressed previously in this chapter: completeness of paramodulation in presence of the congruence

axioms and in presence of the functional reflexivity axioms.




is replaced by a clause

Γ, t ≈ z, s ≈ z

where z is a fresh variable. The latter clause, which replaces Γ , s ≈ t, is obtained by

applying resolution with Γ, s ≈ t and the transitivity axiom x ≈ y, y ≈ z → x ≈ z.

Example 3.9. Clauses

→ a ≈ b

→ b ≈ a

are replaced by

a ≈ x → b ≈ x

b ≈ x → a ≈ x

The set obtained from T by eliminating the transitivity axiom is, for further reference,

denoted by T .

Elimination of the transitivity axiom is best described by the following theorem.

Theorem 3.10 (Brand, see [Bra75]). Let S be a set of flat clauses and let S be

obtained from S by eliminating the symmetry and transitivity axioms. The set S is

equationally satisfiable if and only if S ∪ x ≈ x is satisfiable.

To summarise, the transitivity axiom is redundant in checking whether a set of

flat clauses, obtained after the elimination of symmetry, is equationally satisfiable.

3.3.4 Brand’s proof of completeness of paramodulation

Definition 3.11 (companions). Let L be a literal of a clause C . Assume that L is

of the form L[x], and that there exists a negative literal of the form s ≈ x such that

the only two occurrences of x are in L and s ≈ x. Then s ≈ is a companion of L.

Example 3.12. In the clause

a ≈ x, f (x) ≈ z, f (y) ≈ u, b ≈ y, Q(z, u) →

the literal a ≈ x is a companion of f (x) ≈ z, which is a companion of ¬Q(z, u).

The following theorem formulates Brand’s result about refutational completeness

of the paramodulation calculus P . The given proof of the theorem is not complete.

Provided below is rather a detailed sketch of the proof, the integral version of which

can be found in [Bra75].




Theorem 3.13 (Brand, [Bra75]). Let S be a set of clauses that is equationally un-

satisfiable. Then there exists a refutation of S ∪ x ≈ x by the rules of unrestricted paramodulation.

Proof. By Theorem 3.10, a set S is equationally unsatisfiable if and only if the set

S ∪ x ≈ x is unsatisfiable, where S is obtained from S by flattening its clauses and

applying symmetry and transitivity elimination. Therefore, there exists a refutation

S ∪x ≈ x by resolution. Out of all refutation by resolution, there exists a refutation

Ω that implements a special strategy which is possible to map into a refutation Ω of

S ∪ x ≈ x by paramodulation. This strategy, which Brand proves to be complete,

is defined by restrictions imposed to applications of resolution and positive factoring

rules. Namely, positive factoring can be applied only on clauses that contain nocompanions. Similarly, a resolution inference

Γ1, A Γ2, ¬B


is allowed if

• one of the premises contains no companions, and

• the literal of the other premise which is resolved upon has no companions.

These restrictions make it possible to map every inference of Ω to an inference of Ω .

This mapping is defined as follows, with a correctness proof available in [Bra75]:

• factoring inferences are mapped into the same factoring inferences.

• resolution inferences

Γ1, s ≈ t Γ2, s ≈ x, L[x]

(Γ1, Γ2, L[t])mgu(s, s) (res)

where s ≈ x is a companion of L[x] is mapped into a paramodulation inference

Γ1, s ≈ t Γ2, L[s](Γ1, Γ2, L[t])mgu(s, s)

( par)

where the premise Γ2, L[s] is a clause of the original set S , obtained by appli-

cation of resolution with the reflexivity axiom and all the companions of the

clause Γ, s ≈ x, L[x].

From the definition of the allowed resolution inferences, it follows that the literal

s ≈ x has no companions. Therefore, this is a mapping in which the refutation is




associated a paramodulation refutation that corresponds to innermost rewriting.

It is also worth noting that the resolution inferences always take place on non-variable terms that exist in the original set of clauses. This very observation is

the origin of the basic strategy of derivations .

• resolution inferencesΓ1, A Γ2, ¬B


where neither A nor B have companions map to the same inference.

The proof of the previous theorem, including finding of a resolution refutationwhich is possible to map into a pramodulation proof, and the resulting paramodula-

tion refutation, can best be depicted by the following example.

Example 3.14. Consider the sets of clauses T from the beginning of the section and

T from Example 3.9. Here is a refutation of T , which applies the restrictions, from

the previous proof, to performing the rules of the resolution calculus.

1. a ≈ x → b ≈ x

2. b ≈ x → a ≈ x

3. a ≈ x, f (x) ≈ y → P (y)4. a ≈ x, b ≈ y, f (x) ≈ u, f (y) ≈ v, Q(u, v) →

5. P (x) → Q(x, x)

6. → x ≈ x

7. → b ≈ a [res 1, 6]

8. → a ≈ b [res 2, 6]

9. f (b) ≈ y → P (y) [res 8, 3]

10. → P (f (b)) [res 9, 6]

11. b ≈ y, f (b) ≈ u, f (y) ≈ v, Q(u, v) → [res 8, 4]

12. f (b) ≈ u, f (b) ≈ v, Q(u, v) → [res 11, 6]

13. f (b) ≈ v, Q(f (b), v) → [res 12, 6]

14. Q(f (b), f (b)) → [res 13, 6]

15. → Q(f (b), f (b)) [res 5, 10]

16. [res 14, 15]

This refutation is mapped, inference by inference, to the following refutation by




paramodulation.

1. → a ≈ b

2. → P (f (a))

3. Q(f (a), f (b)) →

4. P (x) → Q(x, x)

5. → x ≈ x

6. → a ≈ b [par 1, 5]

7. → a ≈ b [par 1, 5]

8. → P (f (b)) [par 7, 2]

9. → P (f (b)) [par 8, 5]

10. Q(f (b), f (b)) → [par 7, 3]

11. Q(f (b), f (b)) → [par 10, 5]

12. Q(f (b), f (b)) → [par 11, 5]

13. Q(f (b), f (b)) → [par 12, 5]

14. → Q(f (b), f (b)) [res 4, 9]

15. [res 13, 14]

Note that inferences of the original refutation which derive clauses numbered from 7

onwards, correspond to inferences numbered form 6 onwards in the resulting refutation

by paramodulation.

Analyzing the previous proof, it can be seen that Brand shows that a refutation of

a set of clauses, obtained after his transformations, can be mapped to a refutation by

paramodulation of the original set of clauses. Moreover, the existence of this mapping

is shown only in case when a resolution derivation is obtained by applying a special

derivation strategy, as described in the previous proof.

3.3.5 Importance and limitations of Brand’s results

With respect to the history of equational reasoning in first-order logic, Brand’s results

are, without a doubt, amongst the most important. As shown by Theorem 3.13,Brand proved that the functional reflexivity axioms are not needed for refutational

completeness of the paramodulation calculus P . Furthermore, as pointed out in

the proof of the same theorem, Brand’s modification method played a big role in

motivating the definition of the basic strategy of derivations, by which paramodulation

inferences are forbidden into terms introduced by unification in previous steps (see

[Deg79, BGLS92, NR92b]).




As a side effect, Brand’s results strongly hint that paramodulation into variables

(see Note 3.3) is not needed for completeness3.

In the context of this thesis, Brand’s method is important for numerous reasons.

Firstly, Brand’s method provides, similarly to the proof of Theorem 3.2, a way of

translating logic with equality into logic without equality. As stated in Section 3.2,

the idea of “eliminating” the equality predicate from the logic also underlines the ideas

of the regular derivation strategy and the basic folding calculus, which are proven

complete (in the framework of superposition-based theorem proving) in Chapter 5

and Chapter 7. Also, Brand’s method is a transformation method , since it proves

completeness of paramodulation by transforming a refutation by another calculus

(resolution) into a refutation by paramodulation. Transformation of a refutation by

some complete calculi is the main method for proving refutational completeness of

the refinements of paramodulation presented in this thesis.

However important, Brand’s modification method has its deficiencies. Consider

the following example.

Example 3.15. Applying Brand’s modification method, the following set of flat

clauses→ a ≈ x

→ b ≈ c

can be transformed into a set of clauses that contains the following two clauses

x ≈ y → a ≈ y

c ≈ z → b ≈ z

The resolution inference on these new clauses

c ≈ z → b ≈ z x ≈ y → a ≈ yc ≈ z → a ≈ z (res)

corresponds to the following paramodulation inference

→ b ≈ c → a ≈ x→ a ≈ c ( par)

which is a paramodulation inference between the two original clauses that takes place

into a variable position. Therefore, it is redundant from the point of view of the

state-of-art results, which render the inferences into variable positions redundant.

3Neither Brand nor Peterson could entirely prove paramodulation into variables redundant. This

problem was completely solved later by Bachmair et.al. [BGLS92] and Nieuwenhuis and Rubio

[NR92b].



3.4 State-of-the-art refinements of paramodulation 36

Another deficiency of Brand’s method is that the number of clauses obtained

after the modification method results in a potentially exponential number of clausesderived from the original set. Finally, Brand’s modification method does not address

elimination of classes of redundant clauses from derivations by paramodulation.

A way of using Brand’s ideas in a more efficient way, and hence generating more

efficient methods based on equality elimination, would be based on direct dealing

with the equality predicate. In this light, Degtyarev and Voronkov in [DKV95] sug-

gest a way of eliminating the equality predicate using a form of flattening and basic

superposition, and define the basic folding calculus. Chapter 7 contains a definition

of basic folding and a completeness proof for its first-order version.

3.4 State-of-the-art refinements of paramodulation

Although Brand managed to resolve the issue highlighted in Note 3.2, he did not

give answers to the problems given in Note 3.1 and Note 3.3. These hard, and for

that time unresolved questions, did not terminate research in paramodulation-based

approach to reasoning in first-order logic with equality. To the contrary, they moti-

vated researchers in the field to try to overcome them by refining the paramodulation

rule and/or discovering new methods for proving refutational completeness of the

paramodulation calculus. Detailed description of some of these efforts can be found

in the works of Degtyarev [Deg79, Deg82], Peterson [Pet83], Degtyarev and Voronkov

[DV86], Hsiang and Rusinowitch [HR86, HR91], Pais and Peterson [PP91], Zhang and

Kapur [ZK88], Bachmair and Ganzinger [BG90, BG94, BG98b], Rusinowitch [Rus91],

Bachmair et.al. [BGLS92, BGLS95], Nieuwenhuis and Rubio [NR92b, NR95] and

Lynch [Lyn97].

The most successful approach to refining paramodulation, as noticed by Peterson

in [Pet83], would be aiming at developing “a refutationally complete set of inference

rules for first-order logic with equality which reduces to the Knuth-Bendix procedure

when restricted to equality units”. From the point of view of Peterson, every such

refutationally complete set of inference rules defines a refinement of paramodulation.In modern terms though, a refinement of paramodulation can also be considered a

special strategy of derivations by a paramodulation-based inference system. For ex-

ample, a refinement of paramodulation is a calculus which contains a paramodulation

rule parametrized by term orderings (ordered paramodulation). Other refinements of

paramodulation include: paramodulation-based calculi enriched with deletion rules

for removing redundant clauses, paramodulation-based calculi that implement the

basic strategy of derivations to avoid inferences into variables, and calculi that em-




ploy selection strategies that guarantee finding a refutation in, on average, an efficient

manner.The above refinements define the features that state-of-art refinements of paramodulation-

based calculi are required to have. More detailed descriptions of all of these follow in

the rest of the chapter.

3.4.1 Ordered paramodulation

In its original form, paramodulation allows for building infinitely large terms by re-

peatedly using the same clause as the “from” premise of subsequent paramodulation

inferences (see Note 3.1 from the beginning of this chapter).

Example 3.16. The equation x ≈ f x can repeatedly be applied as in the derivation

below:

x ≈ f x

x ≈ f x P a

P (f a) ( par)

P (f f a) ( par)

...P (f . . . f a)

obtaining longer and longer literals P (f na).

This is where term orderings come into play. They implement the idea of non-

increasing applications of equality rules, originally introduced by Kanger in [Kan63]for sequent calculi. Term orderings constrain the way premises can be used in the

paramodulation rule, by allowing replacement of terms only with terms which are

smaller in the given term ordering. This is the main idea of ordered paramodula-

tion, that was originally considered by Peterson [Pet83] and Hsiang and Rusinowitch

[HR86].

Orderings that are commonly used in the literature to define the relation “simpler”

on terms, are reduction orderings. An ordering is a reduction ordering on non-

ground terms if it is well-founded and if, for all terms s,t, r and substitutions σ, if

s t then r[sσ] r[tσ]. Although reduction orderings are not always total on non-

ground terms (one can not compare the terms f (x, y) and f (y, x), where x and y

are variables), they are total or totalizable on ground terms, which usually suffices in

proofs of refutational completeness.

It is easily verifiable that ordered paramodulation, as defined below, disallows in-

ferences between P (a) and x ≈ f x, and therefore increasing applications of paramod-

ulation from Example 3.16:Γ1, l ≈ r Γ2, L[l]

(Γ1, Γ2, L[r])σ




where

• σ = mgu(l, l),

• term l is not a variable, and

• rσ lσ .

Note that when lσ and rσ are ground terms and is linear on ground terms, the last

condition is equivalent to lσ rσ.

There are further refinements of ordered paramodulation, implemented by a more

extensive use of reduction orderings. For example, Pais and Peterson [PP91] further

strengthen the ordered paramodulation rule by adding the following literal ordering

conditions:

• (l ≈ r)σ is maximal w.r.t. in (Γ1, l ≈ r)σ, and

• L[l]σ is maximal w.r.t. in (Γ2, L[l])σ,

This refinement is known as maximal paramodulation .

An even stronger refinement in terms of ordering restrictions is superposition .

The idea of superposition is taken from Knuth and Bendix [KB70], who define it as

a rule on equations (positive unit clauses). Its generalization to first-order clauses

is therefore defined on purely equational clauses, and is applied only with maximalterms of maximal equations:

Γ1, l ≈ r Γ2, s[l] ≈ t

(Γ1, Γ2, s[r] ≈ t)σ

such that

• σ = mgu(l, l),

• term l is not a variable,

• rσ lσ ,

• (s[l] ≈ t)σ is maximal w.r.t. in (Γ2, s[l] ≈ t)σ,

• (l ≈ r)σ is maximal w.r.t. in (Γ1, l ≈ r)σ, and

• tσ s[l]σ.

Finally, the strongest version of superposition, proposed by Bachmair and Ganzinger

[BG98b] as strict superposition , is obtained from superposition by adding the condi-

tion (l ≈ r)σ (s[l] ≈ t)σ.




3.4.2 Basic strategy

The idea of the basic strategy is motivated by the idea of improving efficiency of

paramodulation by disallowing inferences into arbitrarily chosen terms. The first

insight into how to restrict paramodulation inferences to only certain terms was pro-

vided by Brand in [Bra75] (see Section 3.3). By its modern definition, the basic

strategy of derivations disallows inferences into variables and terms obtained by uni-

fication in previous steps of the derivation. In order to implement the basic strategy,

it is therefore best to avoid unification. However, unification is needed for the calculus

to be sound and complete. This problem of opposite requirements has been solved by

introducing an additional piece of syntax, which replaces the standard clause by a pair

consisting of the clause part and the part that contains unification conditions. Thereexist two such notations, known as constrained clauses, described in Section 2.2.1,

and closures, which are defined in Section 2.2.2.

Although they were defined at about the same time (closures in [BGLS92] and

constrained clauses in [NR92b, NR92a]), and despite they are both used to implement

the basic strategy of paramodulation, closures and constrained clauses are slightly

different formalisms. This will be explained in Section 3.4.2 in more detail.

With constrained clause in play, the paramodulation rule is then changed so that

unification is not applied to the conclusion. Rather, the unification condition is stored

in the constraint part of the conclusion. The effect of unification is then extendedto the rest of the derivation by constraint inheritance, meaning that the constraint

of the conclusion contains constraints of the premises. An inference is then valid if

there is a simultaneous solution of all unification conditions in the constraint part of

the conclusion.

The basic strategy (without ordering restrictions) was originally introduced in

Degtyarev [Deg79, Deg82] by the name of monotone paramodulation , and was later

defined in [DV86] as a rule on conditional clauses . The ideas of the basic strategy

(described in the previous paragraph) are best seen in the following two forms of

monotone paramodulation introduced in [DV86]:

Γ1, l ≈ r | T 1 Γ2, L[l] | T 2

Γ1, Γ2, L[r] | T 1 ∪ T 2 ∪ l ≈ l

andΓ1, l ≈ r | T 1 Γ2, L[l] | T 2

Γ1, Γ2, L[z] | T 1 ∪ T 2 ∪ l ≈ l, z ≈ r

where in both rules l is not a variable and in the second rule z is a new variable. Note

that the rules of monotone paramodulation do not contain any ordering restrictions




for their application.

Independently, the basic strategy with ordering restrictions was also introducedby Nieuwenhuis and Rubio in [NR92b, NR92a] and Bachmair et.al. [BGLS92].

Basic superposition of closures

The following is the basic superposition calculus of Horn closures, presented in [BGLS92].

Basic (positive and negative) superposition

Γ1, l ≈ r · σ1 Γ2, s[l] t · σ2

Γ1, Γ2, s[r] t · (σ1 ∪ σ2)ρ (sup)

where

• ∈ ≈, ≈,

• l is not a variable,

• ρ is the most general unifier of lσ1 and lσ2,

• rσ1ρ lσ1ρ,

• lσ1ρ ≈ rσ1ρ is strictly maximal in Γ1, l ≈ r · σ1ρ,

• tσ2ρ sσ2ρ,

• sσ2ρ ≈ tσ2ρ is strictly maximal in Γ2, s[l] t · σ2ρ, and

• lσ1ρ ≈ rσ1ρ sσ2ρ tσ2ρ in case of positive superposition.

The basic strategy is implemented by applying most general unifiers only to the

substitution part of closures, which prevents introduction of new variable positions

to clauses.

Basic superposition of constrained clauses

The authors of [NR92b] came up with another variant of basic superposition, which

proved to be even more restrictive than basic superposition of closures. This isachieved by moving the ordering conditions, which were previously checked prior

to applying inference rules as side conditions, into the constraint part of the clause.

Simultaneously, the inference rules of the calculus are changed to implement the or-

dering inheritance strategy. The following is the basic superposition calculus of Horn

constrained clauses from [NR92b] with equality and ordering constraint inheritance.





Γ1, l ≈ r | T 1 Γ2, s[l] t | T 2

Γ1, Γ2, s[r] t | T 1 ∧ T 2 ∧ l = l ∧ δ ∧ γ (sup)

where

• ∈ ≈, ≈,

• l is not a variable,

• δ stands for l r ∧ s t ∧ smax(l ≈ r, C 1) ∧ smax(s t, C 2), where C 1

and C 2 are the left and right premises, and smax(l, C ) expresses that the

literal l is strictly greater than any literal in C , and• γ is empty in case of negative superposition, or stands for s t l ≈ r

otherwise.

Basic superposition of closures vs.

basic superposition of constrained clauses

The previous subsection gives a definition of basic superposition with equality and

ordering constraint inheritance. In this section, it is argued that if ordering constraints

are inherited, it is (theoretically) the most restrictive variant of basic superposition.

Before proceeding further, consider the following basic superposition rule on con-straints, where only equality constraints are inherited.


Γ1, l ≈ r | T 1 Γ2, s[l] t | T 2

Γ1, Γ2, s[r] t | T 1 ∧ T 2 ∧ l = l (sup)

where

• ∈ ≈, ≈,

• l

is not a variable,

• rσ lσ , where σ is a solution to T 1 ∧ T 2 ∧ l = l,

• lσ ≈ rσ is strictly maximal in Γ1σ,lσ ≈ rσ,

• tσ sσ,

• sσ ≈ tσ is strictly maximal in Γ2σ, s[l]σ tσ, and

• lσ ≈ rσ sσ tσ in case of positive superposition.




Basic superposition of constrained clauses with equality constraint inheritance,

like the rule of closures, is applied only if the ordering conditions are satisfied. Theequality constraints propagate the unification information in the same way closures

do. As a consequence, if a calculus appears in literature as a calculus of closures, it

is always possible to formulate it as a calculus on constrained clauses with equality

constraint inheritance. The converse holds, too.

Consider now the following example, that distinguishes the ordering constraint

inheritance the most restrictive.

Example 3.17. Consider the following derivation on closures, and assume an order-

ing on terms by which f g and f (a, b) f (b, a).

f (x, y) ≈ g(x) · ε f (u, v) ≈ f (v, u) · ε

f (v, u) ≈ g(x) · [u → x, v → y] (sup)

f (a, b) ≈ c

g(b) ≈ c (sup)

This derivation is valid, because each application of superposition satisfies the neces-

sary conditions, i.e. f (x, y) f (y, x) is true for any ordering on ground terms.

Consider now a derivation from the same set of clauses, which this time uses the

calculus of constrained clauses with ordering constraint inheritance.

f (x, y) ≈ g(x) f (u, v) ≈ f (v, u)f (v, u) ≈ g(x) | δ

(sup1)f (a, b) ≈ c

g(b) ≈ c | f (v, u) = f (a, b) ∧ δ (sup2)

where δ replaces f (x, y) = f (u, v) ∧ f (u, v) > f (v, u) ∧ f (v, u) > g(x) ∧ f (u, v) ≈

f (v, u) > f (x, y) ≈ g(x).

The constraint of the conclusion of sup1 has a solution, for example [u → a, v →

b, x → a, y → b]. However, the constraint of the conclusion of sup2 contains an extra

ordering constraint, by which the variable v would have to be substituted by a and v

by b. Therefore, the whole constraint has no solution.

To summarize, a derivation with ordering constraint inheritance is correct if con-

straints of all clauses have solutions. Since the constraints are inherited, it follows

that a derivation is correct if the constraint in the conclusion of a derivation has a so-

lution. On the other hand, derivations by calculi that use closures (or implement only

equality constraint inheritance) check for ordering conditions at every step, without

recording the conditions under which the conditions hold. As a result, it is possible

that contradictory conditions are satisfied at different steps of a derivation, while




the derivation is still correct. Consequently, as shown by the previous example, a

derivation by a calculus of closures may exist even if it cannot be instantiated. Suchredundant derivations do not exist by basic calculi that employ ordering constraint

inheritance.

Finally, the following can be inferred. Assume that there is a refutation of a set

of unsatisfiable (empty-constrained) clauses S by basic superposition with ordering

constraint inheritance. Then, there exists a refutation by basic superposition without

ordering constraint inheritance (the ordering conditions are checked as side conditions

every time an inference rule is applied). This fact is used further in this chapter and

in Chapter 7.

3.4.3 The model generation method

Before introducing other state-of-the-art features of paramodulation-based calculi (re-

dundancy notions and selection strategies), it is necessary to describe the model gen-

eration method for proving their refutational completeness. The model generation

method is used as the standard technique for establishing the completeness of or-

dered paramodulation-based calculi, with almost all modern results in the field being

derived using this method. The main strength and a reason that the method is the

folklore way of proving refutational completeness of refinements of paramodulation is

that it provides a powerful redundancy criteria for inferences and clauses. Some of

the known redundancy classes for clauses that fit into this framework are tautology

elimination, subsumption and simplification. This redundancy criteria, however, only

applies in case paramodulation is refined by reduction orderings (see Section 3.4.1):

if the ordering requirements are weakened, the model elimination method does not

provide this powerful redundancy criteria anymore (see [BG01b, BR02]).

Historically, the model generation method appeared after the forcing techique, pre-

sented by Zhang in [Zha88] and Pais and Peterson [PP91]. Precisely, it is the most

distinguished forcing technique, and was first presented by Bachmair and Ganzinger

[BG90] as a method for proving refutational completeness of ordered paramodula-

tion with simplification. Later, many authors used model generation for their re-

sults, which addressed refutational completeness of various refinements of paramod-

ulation (and superposition), including basic paramodulation and superposition (see

[BGLS92, NR92a, NR92b, BGLS95, NR95, BG98b, Lyn97, BG01b, BR02]).

The following will describe the idea underlying the model generation method. As-

sume that S is a set of purely equational clauses, which is closed with respect to




applications of inference rules of a given calculus. Soundness of the given calculus

guaranteed, the calculus is then complete if, from the assumption that the emptyclause is not in S , it follows that S is satisfiable. In order to satisfy the set S , one

then first generates an equality Herbrand interpretation and then proves that the

interpretation is a model of S . The interpretation is built as a congruence R∗ gener-

ated by a set of ground rewrite rules R. Each rule in R is generated by some clause

from S . The process of generating the system R is defined by induction on ordering

on clauses, which is the clause extension of a given (usually reduction) ordering on

ground terms. Namely, a clause C from S generates a rule or not, depending on the

set RC of rules generated by clauses smaller than C .

To formally show how the model generation method works for a given calculus,

it is necessary to first define a calculus. For the sake of simplicity, the choice of the

calculus is a ground version of superposition of Horn clauses, denoted S .


Γ1, l ≈ r Γ2, s[l] t

Γ1, Γ2, s[r] t (sup)

where

• l r and s t,

• l ≈ r is maximal in Γ1, l ≈ r , and

• s t is maximal in Γ2, s t.

Equality solution Γ, s ≈ s

Γ (eq sol)

where s ≈ s is maximal in its premise.

Definition 3.18 (model generation, see [NR01]). Let C be a ground clause Γ, l ≈ r,

and let be a given reduction ordering of ground terms. The equation l ≈ r is called

productive, and E C = l ⇒ r if

• C is not true in R∗C ,

• l r and l ≈ r is greater then all literals in Γ, and

• l is irreducible by RC .




The set RC denotes the union of all E D for D ≺ C , while R∗C denotes the congruence

defined by RC .

The first condition of the definition states that a clause contributes to the model if

it does not hold in the partial model built using smaller clauses. Therefore, extension

of the inductively built model is forced , whenever needed for a productive clause to

be true in it. However, a new equation (rewrite rule) can only be added to the rewrite

system (which induces the model) if it is a maximal literal in the clause and if its

largest term is irreducible by the rules of the rewrite system generated by smaller

clauses. The last two conditions are crucial in proving the following lemma which is

used in proving refutational completeness of the above calculus.

Lemma 3.19. Let S be a set of ground clauses. For every clause C in S , if C is true

in R∗C then C is true in R∗. Furthermore, the system R is convergent, i.e. confluent

and terminating.

Theorem 3.20 (Model generation proof of completeness, see [NR01]). The calculus

S is refutationally complete for ground Horn clauses.

Proof. The goal is to show that either ∈ S or there is a model for S . Assume now

the opposite, that S does not have a model and that ∈ S . Let R be the union of

E D for all D ∈ S , and let R∗ be the corresponding congruence. Since, by assumption,

R∗ is not a model of S , there must exist the smallest counterexample C ∈ S (with

respect to the clause extension of a given term ordering ) for R∗. It then follows

that C is not a productive clause.

Assume that C is not productive because its maximal literal is negative. It can

be assumed that C is of the form Γ, s ≈ t. Then, either s and t are syntactically

equivalent, and therefore equality solution on C yields a smaller counterexample then

C (the clause Γ), or s t (w.l.o.g.) and s is order-reducible by some rule in R C . This

means that there exists a clause C , smaller than C , which contributes to R C with an

equation l ≈ r. Assume that C is of the form Γ, l ≈ r. Then there is an inference

from C into C with the conclusion D:

Γ, l ≈ r Γ, s[l] ≈ t

Γ, Γ, s[r] ≈ t (sup)

Γ and Γ are false in, respectively, R∗C and R∗

C . The goal is to show that Γ and Γ

are also false in R∗. The rules in R \ RC , taken as equations, are all greater than the

rules in RC . Therefore, they can not be used in rewriting the literals from Γ, which

is therefore false in R∗. As for Γ, the same argument applies, only the rules from




R \ RC are considered. Additionally, the literals from Γ are not reducible by l ⇒ r

either, because l is the maximal literal in C , since the clause is productive. So Γ isalso false in R∗. The literal s[l] ≈ t is false for R∗ by assumption. Since l ≈ r is true

for R∗, the literal s[r] ≈ is also false in R∗. To conclude, D is false in R∗.

Provided that D ≺ C , D is a smaller counterexample than C (with respect to

R∗), so the initial assumption that C is the smallest counterexample for R ∗ is wrong.

Finally, assume that C is a minimal counterexample for R∗ and that it is of

the form Γ, s ≈ t. Assume also that s ≈ t is a maximal literal in Γ. C is then

not productive because s is reducible by some rule in R. Similar to the previous

case, there exists a superposition inference into C , which yields a clause that is a

counterexample for R∗ and also smaller than C .

The following example shows how the model generation method works. The given

set of clauses is not closed with respect to S . The intention of the example is to

show how saturation and model generation can be done simultaneously. Note that

the process of model generation is not monotonic. Namely, as the closure with respect

to the calculus is being computed, the model changes.

Example 3.21. Let ≺ be a lexicographic path ordering generated by the precedence

c ≺ b ≺ a ≺ f . Let S be the following set of clauses, where the underlined literals are

maximal.

→ f (a) ≈ bf (a) ≈ d → a ≈ c

The model generated for this set is f (a) ⇒ b∗. By saturating this set with respect

to S , the following clause is added:

b ≈ d → a ≈ c

which changes the model of the set: it is now a ⇒ c∗. After making another step

in the saturation process, these clauses are added:

b ≈ d, f (c) ≈ d → a ≈ c

b ≈ d → f (c) ≈ b

f (a) ≈ d, b ≈ d → c ≈ c

As in the previous step, the model changes again, and is now a ⇒ c, f (c) ⇒ b∗.

The set can be further saturated, and the saturation process never stops. However,

the clauses that are being added are all of the form

b ≈ d, b ≈ d, . . . , b ≈ d, . . . → a ≈ c

b ≈ d, b ≈ d, . . . , b ≈ d, . . . , f (c) ≈ d → a ≈ c

b ≈ d, b ≈ d, . . . , b ≈ d, . . . → f (c) ≈ b




and therefore the model will not change.

Redundancy notions induced by the model generation method

Paramodulation-based theorem proving is based on the concept of computing a closure

of a given set of clauses with respect to a given calculus. Computing a closure of a

given set S means repeatedly applying inferences of the given system to clauses from

S , adding to S conclusions of these inferences until there are no new clauses to add.

Even in case of the most refined inference system, the number of derived clauses

grow rapidly, with a great likelihood of performing inferences and deriving clauses

that do not contribute to the proof search. A solution to this problem is identify-

ing such inferences and clauses, describing them in the form of redundancy criteriaand augmenting the procedure of computing a closure accordingly. The result is a

procedure called saturation up to redundant inferences and/or clauses.

This is where the model generation method shows its strength. Namely, the

model generation method provides strong clues about which clauses and inferences

are redundant, and are therefore not necessary in the process of closing a given set

of clauses. If model generation can be used to show refutational completeness of a

calculus, then the completeness result comes with a bonus, embodied in powerful re-

dundancy criteria for inferences and clauses.

By analyzing the proof of Theorem 3.20, it is easy to see that the only inferences

that are necessary for showing the refutational completeness of S are the ones that

reduce the minimal, amongst the clauses from S , counterexample C for the generated

candidate model R∗. This motivates the following definition.

Definition 3.22 (redundant inferences, see [BG98a, NR01]). An inference with max-

imal premise C and conclusion D is redundant with respect to a set of ground clauses

S if there exist ground clauses C 1, . . . , C n, such that C 1, . . . , C n |= D and C C i for

i ∈ 1, . . . , n.

Definition 3.23 (redundant clauses, see [BG98a, NR01]). A ground clause C is called

redundant with respect to a set of ground clauses S if there exists clauses C 1, . . . , C n

such that C 1, . . . , C n |= C and C C i for i ∈ 1, . . . , n.

It is easy to see that the previous two definitions are tightly coupled. Namely, an

inference is redundant if its maximal premise is a redundant clause. These definitions

make a base for defining an efficient saturation procedure.




Definition 3.24 (saturation up to redundant inferences). A set S is saturated (up

to redundancy) with respect to an inference system if every inference with premisesfrom S , by a rule from the given inference system, is redundant with respect to S .

Saturation up to redundant inferences is sufficient for computing a refutation of

a given unsatisfiable set of clauses. This is proved by the following theorem which,

without a loss of generality, addresses the previously defined calculus S .

Theorem 3.25 (completeness of superposition of Horn clauses with redundancy

elimination, seenieuwenhuis01paramodulationbased). Let S be a set of ground Horn

clauses which is saturated up to redundant inferences by the calculus S . Then S is

unsatisfiable if and only if

∈ S .

Proof. The proof is similar to the proof of Theorem 3.20, although a contradiction is

derived using the fact that S is saturated up to redundant inferences, not using the

assumption about the minimality of a counterexample.

Assume the opposite, that S is unsatisfiable and that the empty clause is not in

S . Let R∗ be the model generated the same way as in the proof of Theorem 3.20.

Then, there exists a clause C which is a minimal counterexample for R∗ in S .

Case analysis is very similar to the one given in the proof of Theorem 3.20. Since

C is a counterexample, this means that it is not a productive clause. Assume that C

is of the form Γ, s ≈ t, and that the reason it is not productive is because s is reducibleby a rule l ⇒ r from R. Since C is a counterexample for R∗, by Lemma 3.19, it is also

a counterexample for R∗C . This means that there exists a productive clause Γ 1, l ≈ r,

which is smaller than C , such that the superposition inference

Γ1, l ≈ r Γ, s[l] ≈ t

Γ, Γ1, s[r] ≈ t (sup)

induces the conclusion D. In the proof of Theorem 3.20, it is shown that D is also a

counterexample for R∗. On the other hand, is is assumed that the set S is saturated

up to redundant inferences. By definition, the clause D is then true in R∗C , and

therefore in R∗, which is a contradiction.

Note that the redundancy criterion given by Definition 3.23 contains all known

redundancy notions. For example, tautologies, which are clauses that are true in any

model, are trivially covered. It is also trivial to se that a subsumed clause is redundant,

with respect to Definition 3.23, in presence of its subsumer. As a reminder, a ground

clause C subsumes a ground clause C if C is a non-trivial sub-multiset of C . Consider

now the definition of simplification.




from top clause above. However, the obtained set is then not unsatisfiable anymore:

x ≈ a → f (x) ≈ y

→ f (x) ≈ y | x = a

f (b) ≈ b →

→ a ≈ b

To conclude, the usual definitions of redundancies do not apply in the basic setting.

The model generation method in the basic setting implies the following redun-

dancy criteria. The definition is given for constrained clauses, but can trivially be

changed to hold for closures (it has been shown earlier in this section that the closures

representation of the basic strategy is equivalent to the constraints representationwithout ordering constraints).

Definition 3.28 (redundant inferences with constrained clauses). An inference is

redundant with respect to a set S of constrained clauses if for every rewrite system

R compatible with a given reduction ordering , and every ground instance of the

inference, with premises C 1, . . . , C n, maximal premise C and conclusion D, either

R ∪ irredR(S )≺C i |= C i for some i ∈ 1, . . . n, or R ∪ irredR(S )≺C |= D.

Definition 3.29 (redundant constrained clauses). A constrained clause C | T is

redundant with respect to a set of constrained clauses S , if for every rewrite system

R compatible with a reduction ordering , R ∪ irredR(S )Cσ |= C σ, for every Cσ ∈

irredR(C | T ).

The above redundancy criteria is, opposite to the redundancy criteria induced

by the model generation method in the ground case, of limited practical use. The

problem of deciding whether a clause is redundant, using the previous definition, is

generally undecidable. Since redundancy elimination is the most important feature

in modern provers, it is very important to formulate some practical instances of the

above abstract redundancy criterion. These new criteria should be decidable and,

ideally, computationally cheap to check. A more thorough analysis of this issue is

given in Section 3.4.4.

Limitations of the model generation method

A closer look at Definition 3.18 reveals the inductive nature of the model generation

method, where the induction is defined on the clause extension of a given ordering of

terms. The given ordering, in order for the model to work for a given calculus, has

to be the same as the ordering which the rules of the calculus are parametrized by.




The fact is: a vast majority of refinements of paramodulation that use term or-

derings are parametrized by reduction orderings. The reason for this lies in the factthat the model generation method, to its full power, works only for reduction or-

derings. Although there are results, derived by model generation, about refutational

completeness of paramodulation calculi which are parametrized by non-reduction or-

derings (see [BG01b, BR02]), they are limited in the way that the redundancy criteria

given in Definition 3.23 and Definition 3.22 do not apply.

There also exist applications that require existence of refinements of paramodu-

lation that use term orderings that are weaker than reduction orderings.

Example 3.30. Consider the unsatisfiable set of clauses:

→ f (a) ≈ c

f (a) ≈ c → a ≈ b

f (b) ≈ c →

and let be a lexicographic path ordering given by f a b c. By definition,

is a reduction ordering. Look, for a moment, at this set of clauses as an equational

logic program, and assume that an application requires that inferences are performed

either between a head of a program clause and a goal clause or between heads of

program clauses (see [Fri84]). This strategy does not fit into the model generation

method parametrized by a reduction ordering. To show this, consider performing thenecessary inferences (that reduce minimal counterexamples with respect to partial

models), as explained in the proof of Theorem 3.21. The smallest productive clause

is → f (a) ≈ c, which generates the rule f (a) ⇒ c. The minimal counterexample

with respect to rewrite system f (a) ⇒ c and the given ordering is the clause

f (a) ≈ c → a ≈ b. This necessitates the inference

→ f (a) ≈ c f (a) ≈ c → a ≈ b

c ≈ c → a ≈ b ( par)

This inference clearly does not fit into the required strategy, since the negative literal

of the right premise, with which the inference tales place, is the head of the respective

clause. If it was possible to continue the search for a refutation using the model

generation method, the clause f (a) ≈ c → a ≈ b would have to be a productive

clause, in which case the literal a ≈ b would have to be greater than f (a) ≈ c, which

is only possible if the ordering is not a reduction ordering.

There are some results, like [BG01b, BR02], which explore the possibility of fine-

tuning the model generation method so that it works for non-reduction orderings.




The latter of the two results addresses a basic paramodulation calculus parametrized

by non-monotonic orderings that do not satisfy the sub-term property. Even thoughthe model generation is successfully used to prove refutational completeness of the

calculus, in this case it fails in providing a powerful redundancy criteria (similar to

the criteria given by Definition 3.29) together with the completeness result.

After the results of [BR02], an open problem was to answer whether well-foundedness

was necessary for refutational completeness of paramodulation. Regardless the answer

to this question, which is given in Chapter 4, it is certain that the model generation

method can not be extended for (paramodulation parametrized by) orderings that are

not well-founded. The definition of model generation (Defintion 3.18) is inductive on

the clause extension of a given ordering of terms. In case this ordering was not well-

founded, the induction definition would become not well-founded, since the smallest

clause, the starting point of the model generation process, would not be defined.

Like Example 3.30, Chapter 5, Chapter 6 and Chapter 7 address refinements

paramodulation-based methods that do not fit into the framework of model generation

with reduction orderings. To show refutational completeness of the corresponding

calculi in presence of redundancy criteria, it is necessary to take an approach different

to the model generation method.

3.4.4 Practical redundancy criteria and deletion rules

Definition 3.28 and Definition 3.29, although very powerful, give only abstract notions

of redundancy for inferences and constrained clauses in the framework of basic su-

perposition. In order to implement redundancy elimination in provers, it is necessary

to define some practical redundancy criteria, define corresponding deletion rules , and

add them to the calculus. Here is general form of a deletion rule:

S ∪ C

S \ C

where C is redundant.

Early examples of deletion rules, inherited from resolution, are tautology deletion

and subsumption , the latter being originally introduced by Robinson [Rob65] for the

general resolution. Simplification is another deletion rule, and perhaps the most

widely used (by reasoners). It can be formalized in the following way, as a rule that

transforms multisets of clauses:

S ∪ Γ, L[l] ∪ l ≈ r

S ∪ Γ, L[rσ] ∪ l ≈ r




where lσ = l , lσ rσ and L[lσ] (lσ ≈ rσ) (Peterson [Pet83]).

The meaning of simplification is that l can be replaced by rσ, thus discardingthe clause Γ, L[l]. Simplification has been recognized as a very strong strategy for

equational reasoning, and is a primary mode of computation in state-of-art provers,

with the other inferences being used only sparingly.

The idea of simplification was introduced in [Kan63] by Kanger for equational

reasoning in sequent calculi. However, the first complete equality reasoning procedure

that incorporates simplification appeared much later, in the famous paper by Knuth

and Bendix [KB70].

Other discussion of various aspects of practical redundancy criteria can be found

in Wos et.al. [WRCS67] Knuth and Bendix [KB70], Slagle [Sla74], Lankford [Lan75],

Loveland [Lov78], Peterson [Pet83], Wos, Overbeek and Lusk [WOL91], Rusinowitch

[Rus91], Lusk [Lus92], Voronkov [Vor92], Bachmair and Ganzinger [BG94, BG98b],

Bachmair et.al. [BGLS95], Nieuwenhuis and Rubio [NR95], Tammet [Tam96], Mints,

Orevkov and Tammet [MOT96], Lynch [Lyn97].

Especially relevant for this thesis are redundancy criteria in the framework of the

basic strategy of derivations by superposition. In particular, the results in the thesis

address the definitions of subsumption and tautologies in the basic setting.

Although very abstract, provers like Saturate (see [GN94, NN93]) instantiate this

abstract notion in a few practical simplification and deletion rules.

There are situations when deletion rules are not explicitly given in a definition of

a calculus. In this case, if the calculus is said to be complete and compatible with

elimination of a certain redundancy class, it means that the calculus is complete with

the addition of the appropriate deletion rule. For example, if a calculus I is said

to be refutationally complete and compatible with tautology elimination, it means

that the calculus obtained by adding the tautology deletion rule to I is refutationally

complete.

Practical redundancy notions for constrained clauses

Definition 3.31 (tautologies, subsumed clauses). A clause C → D | T is a tautology

if for every solution σ of T , there exists a literal L in D, such that Cσ |= Lσ.

A constrained clause C | T is a subsumer of a constrained clause C | T , denoted

C | T C | T , if there exists an injective mapping ϕ of the literals from C to the

literals from C , such that for every σ that is a solution of T there exists σ , a solution




of T , for which:

• Lσ = ϕ(L)σ, for every literal L from C , and

• all non-variable positions in ϕ(L) are also non-variable positions in L.

Before analyzing the previous definition, it is necessary to show that the redun-

dancies it defines fit into the abstract redundancy criterion of Definition 3.29.

Lemma 3.32. Subsumption, as defined in Definition 3.31, fits into the abstract def-

inition of redundant clauses.

Proof. Let C | T C | T and R be a rewrite system compatible with . Choose

ϕ, σ and σ as an injection and solutions to, respectively, T and T , so that they

match the conditions from the subsumption definition above. For every literal L

from C , L σ = ϕ(L)σ. Since all non-variable positions in ϕ(L) are also non-

variable positions in L, it follows that the variables in L can only occur below the

positions of the variables in L. Therefore, if the substitution σ is reduced with respect

to R, so is the substitution σ. We prove that irredR(C | T )Cσ |= Cσ. If ϕ is not

a bijection, the clause C σ ∈ irredR(C | T )≺Cσ , because σ is reduced and C σ is

smaller in the multiset extension of the ordering then Cσ. From C σ |= Cσ, it

follows that C | T is redundant in presence of C | T .

Otherwise, if ϕ is a bijection, the analysis is the same as in the previous case, with

the only difference being that now Cσ = C σ and therefore C σ ∈ irredR(C | T )Cσ .

Lemma 3.33. Tautologies, as defined in Definition 3.31, fit into the abstract redun-

dancy criterion.

Proof. For a clause C | T and any solution σ to T , the clause C σ is a ground tautology,

and therefore |= C σ.

The practical value of the given definitions of tautologies and subsumed clauses

varies, depending on the implementation of basic strategy. For superposition with

(only) equality constraint inheritance, they are certainly useful. This is because theconstraint solutions can be given in the form of most general unifiers, which reduces

the problem of checking whether a clause is redundant to checking for syntactic equal-

ity of terms. However, for basic superposition with ordering constraint inheritance,

there does not seem to exist an efficient method of checking if clauses are redundant.

The presence of ordering constraints makes the number of different constraint solu-

tions potentially infinite, which makes the problems of determining whether a clause

is a tautology or subsumed by another clause undecidable.




Practical redundancy notions for closures

A closure C · σ is a subsumer of a closure C · σ, denoted C · σ C · σ, if there exists

an injection ϕ that maps the literals from C to the literals from C and a substitution

η, such that

• Lση = ϕ(L)σ, for all literals L from C , and

• all non-variable positions in ϕ(L) are also non-variable positions in L.

Closures are proper subsumers if they are not variants of each other.

A closure C → D · σ is a tautology if there exists a literal L in D such that

Cσ |= Lσ.

The above definitions are certainly suitable for practical use. Checking whether a

given closure is a tautology or a subsumed clause are decidable problems.

3.4.5 Selection strategies

Another way of improving efficiency of reasoning with paramodulation-based calculi

is by allowing only some literals to take part in inferences. This is where selection

strategies come into play: they are used to restrict inferences so that they are possible

only on selected literals. This section gives an overview of widely used selection

strategies. A vast majority of selection strategies that are used in theoretical results

on paramodulation-based theorem proving are inherited from works on resolution-

based theorem proving. Some of definitions are therefore going to be given in the

context of resolution.

In practice, one implements a selection strategy by embedding it into definitions of

inference rules. A trivial example is the definition of the resolution rule. For example,

resolution implements a selection strategy by which one of the literals resolved upon

is always positive, while the other is negative.

Selection strategies are defined as selection functions. There are many definitionsof selection functions given by different authors. For example, authors of [BG01a]

define a selection function as a mapping which assigns a possibly empty set of literals

to each clause C . The literals in this set are called selected literals. If the set is

non-empty, all the selected atoms (the atoms upon which the selected literals are

built) have to be true in every interpretation in which C is false. This implies that all

negative literals may be selected and that any selection that contains only negative

literals is a correct one.




An alternative definition for selection functions can be found in [BGLS92]. In

the framework of their model generation method, selection functions are defined asmappings from clauses to sets of literals that contain some negative literals, or all

maximal literals. Such selection functions are referred to as classical throughout this

thesis.

A similar, yet different, definition of selection functions can be found in [DVN00].

Here, a selection function maps a clause to a set of literals that either contains exactly

one negative literal, or only positive literals.

These general definitions of selection functions are often refined to more specific

ones. Consider, for example, an instance of a general selection function as defined

by [BG01a], by which one, and only one, negative literal is selected in a clause (if

the clause contains any negative literals). This defines the positive strategy , or syn-

onymously the strategy of eager selection of negative literals , which implies that one

premise of a paramodulation or resolution inference always contains only positive

literals.

Alternatively, let a selection function map a clause C to all its negative literals.

This function defines the maximal selection strategy, which is implemented in the

ordered resolution rule with maximal selection, see [BG01a]:

Γ1

, A1

Γ2

, A2

. . . Γn, An ΓN , ΓP

Γ1, . . . , Γn, ΓP (ord. res)

where

• Γi, Ai are positive clauses for i ∈ 1, . . . , n, and so is ΓP ,

• no clause Γi contains the literal A j, for i, j ∈ 1, . . . , n,

• the clause ΓN contains all the literals ¬Ai, for i ∈ 1, . . . , n, and only them,

and

• each Ai is maximal in Ai, Γi, for i ∈ 1, . . . , n.

Resolution with maximal selection is closely related to the hyper-resolution rule of

[Rob65]:Γ1, A1 Γ2, A2 . . . Γn, An D0

Dn+1(hyp res)

where

• Γi, Ai are positive clauses for i ∈ 1, . . . , n,




• Ai is maximal in Γ1, Ai for i ∈ 1, . . . , n, and

• Dn+1 is a positive clause, obtained by resolving D0 with, successively, Γ1, A1,

. . . , Γn, An.

The hyper-resolution rule is trivially less restrictive: it does not implement the second

condition in the definition of the ordered resolution with maximal selection.

Finally, [BG01a] gives a definition of free selection , which is of particular interest

for the result presented in Chapter 6. A free selection function is a mapping that

selects only one (positive or negative) literal from each non-empty clause. Free selec-

tion was addressed in [dN95] and [Lyn97], where the authors prove completeness of,

respectively, resolution and superposition with free selection.Although the free selection strategy is very general, it is possible to define an even

less restrictive way of selecting literals which take part in inferences. In [BG01b], the

authors define arbitrary selection as a function that maps a non-empty clause to a

non-empty set of literals. Completeness of basic superposition with arbitrary selec-

tion is addressed in Chapter 6 of this thesis.

The free selection can be put in connection with the lock resolution rule, proposed

in [Boy71]. Lock resolution is defined in the following way. Assume that each literal

in each clause is assigned a unique index, so that different literals in different clauses

have different indices assigned to it, except for, optionally, different occurrences of

the same literal. The lock inference rule can take place only with the literal with

the maximal index. The assignment of the indices is performed only once, at the

beginning of the derivation, to the literals of the clauses of the initial set. The indices

in derived clauses are inherited from premises of each inference to the conclusion.

It can therefore be said that the lock selection function is free for the clauses from

the original set of clauses, since the indices can be randomly assigned to literals in a

clause. However, for the derived clauses it is not free, since the selection is determined

by the inheritance of the indices. This is an important difference between the lock

selection and free selection functions. While free selection is not complete for non-Horn clauses, even in the propositional case, the lock resolution is complete, although

without tautology elimination.

The definition of lock resolution rule implies that it is possible to derive instances

of the same clause that contain different selected literals.

Example 3.34. Consider the following set of propositional clauses, where each literal

is assigned an index. In each clause the literal with the maximal index is selected




(and underlined).

1¬ p ∧ 2r 3¬q ∧ 4¬r 5¬q ∧ 6¬ p

Resolution over the first two clauses yields the clause 1¬ p ∧ 3¬q , which then has to

be considered separately to 5¬q ∧ 6¬ p.

This implies that, in lock resolution, two instances of a clause are considered dif-

ferent if their indices are different.

For further reading on selection strategies, one should refer to Dershowitz [Der91],

Nieuwenhuis and Nivela [NN91], Bachmair and Ganzinger [BG94], de Nivelle [dN95],

Lynch [Lyn97], Bofill and Godoy [BG01b], and Aleksic and Degtyarev [AD06].

Discussing selection strategies further, the following question arises: Is a deriva-

tion which applies a certain selection strategy fair? Fairness with respect to a se-

lection strategy can be defined in the following way. A derivation which implements

a selection strategy is fair, if it is guaranteed that all possible inferences with every

selected literal (especially if more than one literal is selected, e.g. classical selection)

will eventually be made. This fairness condition is satisfied in all saturation-based

procedures.

Another interesting question regarding selection functions would be: Is it possible

to select, using one selection function, different (sets of) literals in different occur-

rences of a clause in a derivation? Generally, different occurrences of a clause are

considered the same clause. Since every selection function is, by definition, a function

on a set of clauses, the selected set of literals is uniquely determined. This principle

is assumed to hold throughout the thesis, particularly in Chapter 6.

However, a few methods for proving the refutational completeness (e.g. model

generation) seem to allow for selection of different sets of literals for different oc-

currences of a clause. This is fine as long as the selection is made according to the

definition (e.g. for classical selection, as long as either some negative or all maximal

literals are selected). Also, the previous example shows a situation in which thereappear two instances of the same clause with different “selected” literals. Strictly

speaking, however, lock resolution does not implement any selection strategy. Selec-

tion of literals in derived clauses is not defined as a selection function, it is rather

determined by index assignment to the literals from the original set of clauses.

Finally, the authors of [SW02] pose the following question: Does the strong com-

pleteness take place if a chosen selection strategy is fair? Strong completeness in

the context of selection means that any derivation that implements the chosen fair



3.5 Examples of basic superposition calculi 59

selection strategy can be continued to the empty clause (for an unsatisfiable set of

clauses). The same result claims that strong completeness takes place in case of levelsaturation procedures. All the above defined selection strategies, and therefore the

strategies considered throughout the thesis, are considered only in level-saturation

procedures, and should therefore imply strong completeness.

3.5 Examples of basic superposition calculi

This section presents examples of complete paramodulation-based calculi that imple-

ment the state-of-art features described in the previous sections. More precisely, the

section contains definitions of basic superposition calculi that are addressed in the

original work presented in the thesis.

In further chapters, when the context is unambiguous, the main paramodulation-

based rules of the below calculi will be referred to as superposition.

3.5.1 Calculus BS

Calculus BS is a variant of the basic superposition calculus of constrained clauses with

ordering constraint inheritance given in [NR95]. Unlike in [NR95], the positive and

negative superposition and equality factoring rules of BS do not contain the literal

ordering requirements.

Positive and negative superposition

Γ1, l ≈ r | T 1 Γ2, s[l] t | T 2

Γ1, Γ2, s[r] t | T 1 ∧ T 2 ∧ l = l ∧ l r ∧ s t (sup)

where l is not a variable and ∈ ≈, ≈.

Equality solution Γ, s ≈ t | T

Γ | T ∧ s = t (es)

Equality factoring Γ, s1 ≈ t1, s2 ≈ t2 | T

Γ, s1 ≈ t1 | T ∧ s1 = s2 ∧ t1 = t2(f ac)



3.6 Calculus EBFP 60

BS is used in Chapter 6, for proving completeness of basic superposition that imple-

ments arbitrary selection strategy. Since literal ordering conditions in the definitionof the calculus from [NR95] implement classical selection strategies, these conditions

are dropped in the definition of BS .

Refutational completeness of the calculus BS follows from the completeness of the

original calculus from [NR95]: BS is complete since it is a less restrictive calculus.

3.6 Calculus EBFP

Calculus EBFP , defined below, is a version of the calculus which was introduced and

proved refutationally complete in [MLS95]. The difference is in that the rules of EBFP

do not contain the literal ordering conditions of the calculus from [MLS95]. The literal

ordering conditions, in the original calculus, implement the classical selection strategy.

However, EBFP is used, in Chapter 5, to show refutational completeness of a strategy

of derivations that does not fit into the classical selection strategy, hence the omission

of the literal ordering rules.

Factored (positive and negative) overlap

l1 ≈ r1, . . . , ln ≈ rn, Γ1 | T 1 s[l] t, Γ2 | T 2

s[r1] t , . . . , s[rn] t, Γ1, Γ2 | T 1 ∧ T 2 ∧ δ (sup)

where

• δ stands for (l1 r1 ∧ . . . ∧ ln rn ∧ s t ∧ l1 = l ∧ . . . ∧ ln = l), and

• ∈ ≈, ≈.

Equality solution 4

s ≈ t, Γ | T

Γ | T ∧ s = t (es)

Refutational completeness of EBFP follows from the completeness of the calculus

from [MLS95], since EBFP is a less restricted calculus.

The calculus EBFP has a model generation proof parametrized by reduction or-

derings. Therefore, it is compatible with the abstract redundancy criterion given by

4In [MLS95] that introduced the calculus BFP , this inference is called “reflection”.




Definition 3.29. Specially, it is complete with basic tautology elimination and ba-

sic subsumption. Therefore, completeness of EBFP is not affected by adding thesubsumption and tautology elimination deletion rules to it.

Finally, EBFP also contains the following resolution and (positive and negative)

predicate factoring rules.

Relational resolutionΓ1, P | T 1 Γ2, ¬Q | T 2

Γ1, Γ2 | T 1 ∧ T 2 ∧ P = Q (res)

Relational factoring (positive and negative)

Γ, L1, L2 | T

Γ, L1 | T ∧ L1 = L2(f ac)

where

• L1 and L2 are either both positive or both negative literals;

• L1 and L2 are identical up to variable renaming.

The reason for adding these rules is that Chapter 5 where EBFP is used, it is preferred

to distinguish between predicate and equality literals. Although the relational rules

(resolution and positive factoring) can be treated as equational rules in a two-sortedlogic (see Section 2.2), their presence improves readability of the completeness proof

in Chapter 5. As for negative factoring, it is needed in the proof of the result of

Chapter 5 for purely technical reasons.

Since both the resolution and relational factoring rules are sound, their inclusion

does not affect completeness of the calculus EBFP .

Interestingly, the calculus EBFP is defined on general first-order clauses, and yet

it does not contain a factoring rule. The reason for this is that it actually postpones

factoring till the end of a derivation. In case or a refutation, these postponed factoring

inferences are never performed.

From refutational completeness of EBFP and the discussion from Section 3.4.2 of

this chapter, it follows that the closures version of EBFP is a complete calculus. The

closures version of EBFP , the rules of which are stated below, is used in Chapter 7.

Factored (positive and negative) overlap

l1 ≈ r1, . . . , ln ≈ rn, Γ1 · σ1 s[l] t, Γ2 · σ2

s[r1] t , . . . , s[rn] t, Γ1, Γ2 · (σ1 ∪ σ2)ρ (sup)




where

• ρ is the simultaneous most general unifier of l1σ1 = lσ2, . . . , lnσ1 = lσ2),

• ∈ ≈, ≈, and

• r1σ1ρ l1σ1ρ, . . . , rnσ1ρ lnσ1ρ, and tσ2ρ sσ2ρ.

Equality solution s ≈ t, Γ · σ

Γ · σρ (es)

where

• ρ is the most general unifier of sσ and tσ.

Deletion rules are defined with respect to the redundancy notions for closures, given

in Chapter 3.4.4.



Chapter 4

Ordered paramodulation

without term orderings

As discussed and described by Example 3.30, certain applications of theorem proving

in FOL with equality require derivations by refinements of paramodulation that do

not fit in the model generation framework of the state-of-art paramodulation-based

calculi. Such refinements can be looked at from a couple of different but connected

perspectives. Firstly, they can be defined as derivations by standard paramodulation-

calculi (parametrized by reduction orderings), that implement non-classical selec-

tion strategies. Secondly, they can be defined as derivations by calculi which are

parametrized by non-reduction orderings. While the former kind of refinements are

discussed in Chapter 5 and Chapter 6 in more detail, this chapter focuses to refine-

ments of the latter type.

The usual way of constraining the symmetry (see Note 3.1) of the paramodulation

rule is by using term orderings for orienting the paramodulation rule. These orderings

are required to be reduction orderings on ground terms, which often proves to be too

strong a restriction (see Example 3.30). Trying to weaken these restrictions, Bofill

and Rubio in [BR02] go the furthest by showing that well-foundedness is a sufficientproperty of a term ordering for completeness of ordered paramodulation (of Horn

clauses). It has since been an open research problem to investigate whether the

ordering conditions can be weakened even further, i.e. whether well-foundedness is

also a necessary condition.

This takes the consideration back to an idea of Kanger [Kan63, DV01b], who was

the first to restrict the symmetry of the paramodulation rule by ordering the positive

equational literals. To be more precise, Kanger allowed only replacement of terms by

63



4.1 Non-lengthening superposition of ground unit clauses 64

“less complex” terms, where complexity was defined as term depth. Although Kanger

worked with sequent calculi, his idea can be transferred to paramodulation-basedcalculi. The pioneering attempt was made by Orevkov in [Ore69], who generalized

Kanger’s approach of orienting terms depending on their depth, by using an arbirtary

relation instead. Such a relation, as required by Orevkov, has to be asymmetric and

total on ground terms. In [DV01b], Orevkov gives a rough sketch of a completeness

proof of unit ground paramodulation oriented by an asymmetric and total relation –

a statement that is fully proved in Lemma 4.3.

This is where the model generation method completely fails and where transfor-

mation methods come into play. By studying the model generation method (see, for

example, [BG90], and the proof of Lemma 7.29), it can be seen that it is defined by

induction on a given term ordering, or more precisely, on the clause extension of the

given term ordering. However, in order for this induction to be well-founded, the

ordering on clauses has to be a well-founded relation, which is not the case.

The rest of the chapter shows the advantage that transformation methods hold

over the model generation method in proving non-standard refinements of paramod-

ulation. This further develops in Chapter 6, Chapter 5 and Chapter 7.

4.1 Non-lengthening superposition of ground unit clauses

Let be a total and asymmetric relation on ground terms and E be a set of ground

equations. Every equation l ≈ r in E can be oriented with respect to . An oriented

equation is written as l ⇒ r or l ⇐ r, depending on whether ¬(r l) or ¬(l r).

The set E is defined as the set of oriented equations in E . Note that only positive

equations are oriented. Negative equations are going to be written in the usual way.

An equational proof of a ground equation s ≈ t, with respect to a set of ground

equations E , is a sequence

s ≡ s0 ≈ s1 ≈ . . . ≈ sn−1 ≈ sn ≡ t

where for each i such that 0 < i < (n − 1), si ≡ si[l] and si+1 ≡ si[r] for some l ≈ r

in E . Equational proofs can also be considered over the set of oriented equations E ,

when they are referred to as oriented equational proofs. In this case the step s i ≈ si+1

is written as si ⇒ si+1 if l ⇒ r , or si ⇐ si+1 if l ⇐ r . An oriented equational proof

of the form

s ≡ s0 ⇒ s1 . . . ⇒ sm ⇐ . . . ⇐ sn ≡ t




is called a rewrite proof.

Consider the following definition of a superposition rule for ground equationsoriented with respect to a given total and asymmetric relation . Using this definition,

any rewrite proof (of an equation) can be represented as a sequence of inferences by

non-lengthening superposition.

Non-lengthening positive superposition

l ⇒ r s[l] ⇒ t

s[r] ≈ t

It is easy to see that a rewrite proof consists of a (possibly empty) sequence of

⇒-steps followed by a (possibly empty) sequence of ⇐-steps. However, an oriented

proof generally contains more than one sequence of both ⇐-steps and ⇒-steps.

Definition 4.1 (wrong ⇐ and ⇒-step). Every ⇐-step in an oriented equational proof

which occurs to the left of the rightmost sequence of ⇒-steps is referred to as a wrong

⇐-step. Similarly, every ⇒-step, which occurs on the right of the rightmost wrong

⇐-step is called a wrong ⇒-step.

A wrong steps pair (m, n) is a pair assigned to an oriented equational proof, where

m and n are counts of, respectively, wrong ⇐-steps and wrong ⇒-steps in the proof.Example 4.2. Consider the oriented equational proof

t1 ⇐ t2 ⇒ t3 ⇒ t4 ⇐ t5 ⇒ t6.

The number of wrong ⇐-steps in the proof is 2: the steps t1 ⇐ t2 and t4 ⇐ t5. The

number of wrong ⇒-steps is 1: it is t5 ⇒ t6. Consequently, for the same oriented

equational proof, the value of the wrong steps pair is (2, 1).

Consider now the following sub-proof of the above proof:

t2 ⇒ t3 ⇒ t4 ⇐ t5.

This is a rewrite proof, in which the number of wrong ⇐-steps and wrong ⇒-steps

both equal to zero. Therefore, the wrong steps pair of the proof is (0 , 0).

Note that Definition 4.1 implies that if the number of wrong ⇒-steps, in an ori-

ented equational proof, is positive, then the number of wrong ⇐-steps has to be

positive, too. The converse also holds. This means there does not exist oriented

equational proof that can be assigned a wrong steps pairs of the form (0 , k) or (k, 0),

for a positive integer k.




Lemma 4.3. Let E be a set of ground equations and E be the corresponding set

of equations in E oriented with respect to a total and asymmetric relation . Every consequence s ≈ t of E has a rewrite proof in E .

Proof. Let > be a relation on pairs (m, n), such that

(m1, n1) > (m2, n2) iff m1 = m2 and n1 > n2 or

m1 > m2

The relation is obviously well-founded, with the minimal element (0 , 0).

Consider an ordered equational proof in E of s ≈ t. The proof is by induction on

the wrong step pair of the the equational proof. In case the pair is (0, 0), the proof is

already a rewrite proof, and the lemma holds.Otherwise, there is at least one wrong ⇐-step in the proof. Let v be the term

at which the rightmost such step occurs. There exist terms u and w, such that

u ⇐ v ⇒ w, where v ≡ v[l1] p ⇒ v[r1] p ≡ w and u ≡ v[r2]q ⇐ v[l2]q ≡ v, for some

l1 ⇒ r1, and l2 ⇒ r2 in E . Depending on whether the positions p and q overlap or

not, there are two cases.

Assume that p and q are disjoint. In this case, v ≡ v[l1] p[l2]q , and the sequence u ≡

v[l1] p[r2]q ⇐ v[l1] p[l2]q ⇒ v[r1] p[l2]q can be modified to u ≡ v [l1] p[r2]q ⇒ v[r1] p[r2]q ⇐

v[r1] p[l2]q. This modification is a result of swapping the wrong ⇐ and the wrong

⇒-steps. Consequently, both the numbers of wrong ⇐-steps and wrong ⇒-steps

decrement, and therefore the induction hypothesis applies.

Assume that positions p and q overlap. It can be assumed, w.l.o.g. that q is below

p, i.e. q = p.p. Therefore, the term v is equivalent to v[l1[l2] p] p, and the sequence

u ⇐ v ⇒ w can be modified to u ≡ v[l1[r2] p ] p ⇐ v[l1[l2] p ] p ⇒ v[r1] p ≡ w. This

effectively means that there exists the inference

l1[l2] ⇒ r1 l2 ⇒ r2

l1[r2] ≈ r1

The inferred equation can be oriented with respect to in two ways.

• Let l1[r2] ⇐ r1, which is in case ¬(l1[r2] r1). By applying this oriented equa-

tion, one can obtain u ≡ v[l1[r2] p] p ⇐ v[r1] p ≡ w. Thus modified oriented

proof contains one wrong ⇒-step less, and the lemma holds by the induction

hypothesis.

• l1[r2] ⇒ r1, in case ¬(l1[r2] r1). This case is symmetric to the previous, with

the only difference being that the induction step is applied after eliminating a

wrong ⇐-step.



4.2 Non-lengthening superposition of ground Horn clauses 67

The following example illustrates the cases analysed in the previous lemma.

Example 4.4. Let E be the set of equations a ≈ b, b ≈ g(c, b), b ≈ g(a, b), a ≈ c.

The equation g(c, b) ≈ g(b, a) is a consequence of the equations in E , with the following

equational proof:

g(c, b) ≈ b ≈ g(a, b) ≈ g(a, a) ≈ g(b, a).

Let be a total and asymmetric relation such that the set E is a ⇐ b, b ⇒

g(c, b), b ⇒ g(a, b), a ⇒ c. The above equational proof can therefore be rewritten as

g(c, b) ⇐ b ⇒ g(a, b) ⇒ g(a, a) ⇐ g(b, a).

which can be assigned the wrong steps pair (1, 2).

The rightmost wrong ⇐-step occurs at term b. There are two steps that take place

with b, which make the following sub-proof of the above oriented proof: g(c, b) ⇐ b ⇒

g(a, b). By performing the inference

b ⇒ g(c, b) b ⇒ g(a, b)

g(c, b) ≈ g(a, b)

the above oriented proof can, having assumed that c ⇐ a, be transformed into

g(c, b) ⇐ g(a, b) ⇒ g(a, a) ⇐ g(b, a)

with the wrong steps pair (1, 1). The next term of the proof, upon which the rightmost

wrong ⇐-step takes place, is g(a, b). The adjacent steps at this term are g(c, b) ⇐

g(a, b) ⇒ g(a, a). Their sequence can easily be changed by applying the rule b ⇒ a,

which rewrites g(a, b) into g(a, a), on the term g(c, b). Simultaneously, the rule a ⇒ c

can be applied to g(a, a), which results in the sequence g(c, b) ⇒ g(c, a) ⇐ g(a, a).

By this step, the whole oriented proof is therefore transformed into

g(c, b) ⇒ g(c, a) ⇐ g(a, a) ⇐ g(b, a)

which is a rewrite proof and has the wrong steps pair (0, 0).

4.2 Non-lengthening superposition of ground Horn clauses

The result of the previous section can be straightforwardly extended to Horn clauses.

The following are rules of the non-lengthening superposition of Horn clauses , defined

using the notation given for the case of unit equations.




Non-lengthening negative superposition

l ⇒ r Γ2, s[l] ≈ t

Γ1, Γ2, s[r] ≈ t (sup)

where ¬(t s[r]).

Equality solution s ≈ s, Γ

Γ (es)

The rule of non-lengthening positive superposition is the rule defined for the unit case.Note that the above calculus implements a selection strategy by which a negative

literal is selected every time a clause contains one.

Definition 4.5 (block-shaped derivations). A block-shaped derivation is a special form

of derivations by a calculus that implements a selection strategy by which a negative

literal is selected in each clause that contains one. A block-shaped derivation Ω is

recursively defined in the following way.

• Ω consists of only one clause

• Ω is a derivation of a clause L(m1)1 , . . . , L

(mp)

p (which can be empty) from anegative clause L1, . . . , Lk, and is of the following form:

Ωkmk

lkmk ≈ rkmk

Ω1m1

l1m1 ≈ r1m1

Ω11l11 ≈ r11 L1, . . . , Lk

L1, . . . , Lk

(sup).... (sup)

L(m1−1)1 , . . . , Lk

L(m1)1 , . . . , L

k

(sup)

.... (sup)

L

(m1)

1 , . . . , L

(mk−1)

k

L(m1)1 , . . . , L

(mk)k

(sup)

L(m1)1 , . . . , L

(mk−1)k−1

(es)

.... (es)

L(m1)1 , . . . , L

(mp) p

where p ∈ 1, . . . , k, and each Ωij, for i ∈ 1, . . . , k and j ∈ 1, . . . , ki is a

block-shaped derivation.




• Ω is a derivation of a positive literal s(mk+1) ≈ t(mk+1) from a clause L1, . . . , Lk, s ≈

t, and of the form:

Ωkmk

lkmk ≈ rkmk

Ω1m1

l1m1 ≈ r1m1

Ω11l11 ≈ r11 s ≈ t, L1, . . . , Lk

s ≈ t, L1, . . . , Lk

(sup).... (sup)

s ≈ t, L(m1−1)1 , . . . , Lk

s ≈ t, L(m1)1 , . . . , L

k

(sup)

.... (sup)

s ≈ t, L(m1)1 , . . . , L

(mk−1)k

s ≈ t, L(m1)1 , . . . , L

(mk

)k

(sup)

L(m1)1 , . . . , L

(mk−1)k−1

(es)

.... (es)s ≈ t.... (sup)

s(mk+1) ≈ t(mk+1)

where the derivation of s(mk+1) ≈ t(mk+1) from s ≈ t is of the form

Ω(k+1)mk+1

l(k+1)mk+1 ≈ r(k+1)mk+1

Ω(k+1)1

l(k+1)1 ≈ r(k+1)1 s ≈ t

s ≈ t (sup)

.... (sup)

s(mk+1−1) ≈ t(mk+1−1)

s(mk+1) ≈ t(mk+1) (sup)

and where each Ωij, for i ∈ 1, . . . , k + 1 and j ∈ 1, . . . , ki is a block-shaped

derivation.

Informally, in a block-shaped derivation, all supersposition inferences into one literal

take place consecutively, in a block. Because of the selection strategy that every block-

shaped derivation implements, equality solution inferences into L

(m1)

1 , . . . , L

(mk)

k takeplace only at the end of the derivation, immediately before superposition inferences

into s ≈ t.

Lemma 4.6. Non-lengthening superposition of Horn clauses is refutationally com-

plete and compatible with tautology elimination.

Proof. Let S be an unsatisfiable set of ground Horn clauses. Since S is unsatisfiable,

there exists a refutation of S by any complete paramodulation calculus I , which does




not contain literal ordering conditions in the definitions of its rules. Let Ω be such a

refutation, and assume that it employs eager selection of negative literals. It is easyto show, following the same procedure like in Theorem 6.6, that Ω can be transformed

into a block-shaped derivation. Assume therefore that Ω is already block-shaped.

The proof is by induction on the number of sub-derivations of Ω that are also

block-shaped derivations. Let Ω contain only one sub-derivation that is block-shaped.

In that case Ω is of the form

Ω1l ≈ r

Ω2l ≈ r

r ≈ r (sup)

(es)

The proof of this case is by another induction, on the length of Ω1. The base case is

when Ω1 contains of only one clause, the unit clause l ≈ r. The statement trivially

holds, since there always exists an inference by non-lengthening superposition between

l ≈ r and l ≈ r .

Let Ω1, with a conclusion l ≈ r, contain more block-shaped derivations. By

definition, Ω1 is then of the form

L1, . . . , Ln, u ≈ v.... (sup)

L1, . . . , L

n, u ≈ v

.... (es)u ≈ v.... (sup)l ≈ r

where l ≈ r is derived from u ≈ v by applying a sequence of paramodulation

inferences from the calculus I . The left premises of these inferences are clauses

l1 ≈ r1, . . . , lk ≈ rk, which are conclusions of block-shaped definitions Ω11, . . . , Ω1k.

The literal u ≈ v follows from the equations l1 ≈ r1, . . . , lk ≈ rk, and therefore has

a rewrite proof. Equivalently, the clause l ≈ r has a proof by non-lengthening su-

perposition from the clauses l1 ≈ r1, . . . , lk ≈ rk. By the induction hypothesis, sinceΩ11, . . . , Ω1k are shorter than Ω1, they all have derivations by non-lengthening super-

position.

Let now Ω contain more than one block-shaped sub-derivation. It is then of the



4.3 Non-lengthening superposition of non-ground clauses 71

formL1, . . . , Ln

.... (sup)

L1, . . . , L

n.... (es)

Each derivation of the literals L 1, . . . , L

n from L1, . . . , Ln is a block-shaped derivation

shorter than Ω. By the induction hypothesis, they can be transformed into derivations

by non-lengthening superposition of Horn clauses. The claim of the lemma follows.

The claim about the compatibility with tautology elimination can be proved by

following the ideas given in the proof of Theorem 6.9.

4.3 Non-lengthening superposition of non-ground clauses

Although the results presented in Lemma 4.3 and Lemma 4.6 show advantages of

transformation methods over the model-generation method, the fact is that trans-

formation methods have a lot of disadvantages. For example, consider the problem

of lifting. In order to show refutational completeness of a superposition calculus of

non-ground clauses, using the model generation technique, is to show that the cal-

culus is complete for ground clauses. Lifting comes for free, provided that the term

ordering which orients the superposition rule is suitable (normally, it is a reduction

ordering on ground terms). Moreover, the completeness result for non-ground clauses

keeps all properties of the corresponding result for ground clauses. However, the ex-

istence of a transformation-based completeness proof for a superposition calculus on

ground clauses says nothing about completeness of the same calculus on non-ground

clauses. Furthermore, it is possible that there exists a transformation-based com-

pleteness proof of a calculus on ground clauses, even if the calculus is not complete

in the non-ground case.

Example 4.7. Consider the following unsatisfiable set of unit clauses

h(u, g(u)) ≈ c, h(g(x), g(x)) ≈ c, a ≈ g(a)

and let ¬(c h(g(x), g(x))), ¬(g(a) a), and ¬(g(g(a)) g(a)), i.e. h(g(x), g(x)) ⇒ c,

a ⇒ g(a), and g(a) ⇒ g(g(a)).

There is a tautology-free refutation of this set by non-lenghtening superposition

of unit clauses (a unit clause is a tautology if all its ground instances are of the form



4.3 Non-lengthening superposition of non-ground clauses 72

t ≈ t):

h(g(x), g(x)) ⇒ ca ⇒ g(a) h(u, g(u)) ≈ c

h(g(a), g(a)) ≈ c (sup)

c ≈ c (sup)

(es)

This derivation, however, allows for performing inferences into variable positions. If

this was disallowed, the only other refutation is

h(g(x), g(x)) ⇒ c

a ⇒ g(a)

a ⇒ g(a) a ⇒ g(a)

g(a) ≈ g(a) (sup)

g(a) ⇒ g(g(a)) (sup)

h(u, g(u)) ≈ c

h(g(g(a)), g(g(a))) ≈ c (sup)

c ≈ c (sup)

(es)

It follows that, in the non-ground case, either tautologies or inferences into variable

positions have to be allowed. consequently, a version of non-lengthening superposi-

tion for non-ground clauses with state-or-art features, like basicness and tautology

elimination, is not complete at all.



Chapter 5

Regular derivations in basic

superposition

The main contribution of this chapter is in a proof of refutational completeness of

the regular strategy of derivations by the calculus EBFP (see Section 3.6), which is

an implementation of basic superposition with ordering constraint inheritance. The

proof, from which it follows that tautology deletion and subsumption do not affect

completeness of the regular strategy, is given by transformations of derivation trees.

The transformation technique is based on so-called permutation rules, which are used

to change the order in which inferences appear in a derivation.

The regular strategy is a specialization of derivations in (sequent or clause) calculi

with equality, by which all equality inferences take place before all other steps in the

derivation. The strategy, for derivations by sequent calculi, was originally introduced

by Kanger in [Kan63] (see also [DV01b]) and was later used by Maslov in [Mas71] to

generalize the inverse method to predicate calculi with equality.

In the case of paramodulation-based clause calculi, regular derivations can be

defined in the following way.

Definition 5.1 (regular derivations). A derivation by a paramodulation-based calcu-

lus (which also contains the equality solution, factoring and resolution rules) is regular

if all applications of superposition, equality solution and factoring (on equational lit-

erals) precede all other inferences.

In this framework regular derivations were first used by Robinson and Wos in

[RW69] for their proof of refutational completeness of paramodulation. They used

paramodulation and factoring (which are equality inference rules) to obtain an un-

satisfiable set of clauses from an E-unsatisfiable set of clauses.

73



75

Example 5.4. Consider the following set of clauses

→ P (c,b,b)

P (c,c,b), P (c,b,c) → b ≈ c

P (x , y , y) → P (x , y , x)

P (x , y , y) → P (x,x,y)

P (c,c,c) →

and assume an ordering such that b c. It is possible to make exactly two legal

superposition inferences with the above premises, so that they are not preceded by

other inferences. By legal superposition inferences, it is referred to the inferences that

do not involve two instances of the same clause, do not take place into variable posi-

tions, and involve the only genuine equality literal of the above set: b ≈ c. Therefore,

both these inferences are between the first two clauses of the above set, deriving the

tautologies:

P (c,c,b), P (c,b,c) → P (c,c,b)

P (c,c,b), P (c,b,c) → P (c,b,c)

This set of clauses can be transformed to a logically equivalent set of e-flat clauses,

which have the property that all arguments of predicate literals are variables.

1. x ≈ c, y ≈ b → P (x , y , y)

2. x ≈ c, y ≈ b, P (x,x,y), P (x , y , x) → b ≈ c

3. P (x , y , y) → P (x , y , x)

4. P (x , y , y) → P (x,x,y)

5. x ≈ c, P (x,x,x) →

It is apparent that, since superposition into variables is forbidden, the inferences that

previously led to tautologies can not be preformed anymore. As a result, there exists

a regular derivation without tautologies.

6. y ≈ b → P (c,y,y) [es 1]

7. y ≈ b, P (c,c,y), P (c,y,c) → b ≈ c [es 2]

8. P (c,c,b), P (c,b,c) → b ≈ c [es 7]

9. y ≈ c, P (c,c,b), P (c,b,c) → P (c , y , y) [s 8, 6]

10. P (c,c,b), P (c,b,c) → P (c,c,c) [es 9]

11. P (c,c,c) → [es 5]

12. → P (c,b,b) [es 6]

The Horn subset consisting of the “relational” clauses 3, 4, 10, 11 and 12 is un-

satisfiable, i.e. is refuted by resolution without tautologies under arbitrary selection

function.



76

In order to prove our result, we introduce a method based on transformations

of derivation trees. The introduction of this method is necessary because the regu-lar strategy of derivations simply does not fit into the model generation framework

(another such example is given in Section 3.4.3 in Example 3.30). In a purely equa-

tional logic (Section 2.2 explains how to treat predicate literals as equational literals),

regular derivations can be executed by a basic superposition calculus which:

• employs a special selection strategy by which all equality literals and only they

are selected, and

• uses an ordinary selection strategy, but is parametrized by orderings which do

not satisfy the subterm property. In particular, these orderings allow t P (t)for every term t and predicate symbol P , thus ensuring that the equality literals

are always greater than predicate literals.

On the other hand, as discussed in Section 3.4.3, almost all completeness results using

the model generation method address calculi which:

• use the classical selection strategies (see Section 3.4.5). By them, if a positive

literal is selected, then it has to be maximal in the literal extension of a given

(reduction) term ordering. Since, because of the subterm property, P (t) s ≈ t

holds true in every reduction ordering where t s, it follows that the classical

selection strategy does not encompass the regular strategy.

• are parametrized by reduction orderings.

There have been attempts to use model generation for proving completeness of calculi

that go beyond the restrictions described in the previous two items. However, these

results are not sufficient to prove completeness of the regular strategy of derivations.

Regarding arbitrary selection strategies, the latest achievement is in [AD06] (pre-

sented in Chapter 6). It represents an extension of the results of Bofill and Godoy

given in [BG01b] about completeness of arbitrary selection strategies for basic paramod-

ulation on Horn clauses. Although it addresses basic superposition on general clauses,it can be applied only on derivations that contain no factoring inferences, and there-

fore can not be used to cover the claim of Conjecture 5.3. The result has another

weakness: it does not contain any redundancy notions.

Trying to weaken term ordering constraints, Bofill and Rubio (see [BR02]) prove

completeness of ordered paramodulation for Horn clauses that is based on orderings

without the sub-term property. This result, however, is not suitable to cover the claim

of Conjecture 5.3. It addresses only paramodulation which does not implement basic



5.1 Permutation rules 77

strategies, and can not be extended to the case of basic superposition. Moreover, it

is not certain whether this result can be strengthened with redundancy notions likesubsumption and tautology elimination.

For the above reasons, our result can be looked at as a theorem about completeness

of a non-standard state-of-art refinement of a paramodulation calculus.

5.1 Permutation rules

Permutation rules are applied to derivation trees (of derivations by EBFP , see Sec-

tion 3.6) with the effect of inverting the order of two consecutive inferences, whenever

a relational inference precedes an equality inference. Because the equational rules

take place strictly with equality literals while the relational inferences can only be

applied with predicate literals, this inversion is always possible to make, as shown in

the below definition of the permutation rules.

Before proceeding to the definitions of the rules, please note that the superposition

rules of the underlying calculus EBFP do not contain an explicitly stated factoring

rule. Hence there are no permutation rules that involve factoring of equality literals.

Secondly, in order to make the rules more readable, the literals that take part in the

inferences are underlined.

res-es rule – Resolution precedes equality solutionΓ1, s ≈ t, ¬Q | T 1 Γ2, P | T 2

Γ1, Γ2, s ≈ t | T 1 ∧ T 2 ∧ P = Q (res)

Γ1, Γ2 | T 1 ∧ T 2 ∧ s = t ∧ P = Q (es)

This sequence transforms to:

Γ1, ¬Q, s ≈ t | T 1

Γ1, ¬Q | T 1 ∧ s = t (es)

Γ2, P | T 2

Γ1, Γ2 | T 1 ∧ T 2 ∧ s = t ∧ P = Q (res)

fac-es rule – Relational factoring precedes equality solution

Γ, s ≈ t, L1, L2 | T

Γ, s ≈ t, L1 | T ∧ L1 = L2(f ac)

Γ, L1 | T ∧ s = t ∧ L1 = L2(es)

where L1 and L2 are either both positive or both negative predicate literals.



5.1 Permutation rules 78

Similarly to the previous rule, this sequence transforms to:

Γ, L1, L2, s ≈ t | T

Γ, L1, L2 | T ∧ s = t (es)

Γ, L1 | T ∧ s = t ∧ L1 = L2(f ac)

This permutation, as well as the previous one, is always possible to make, since

predicate inferences always take place on predicate literals, while equality solu-

tions are always preformed on equality literals.

res-sup rule – Resolution followed by superposition

Γ11, l1 ≈ r1, . . . , ln ≈ rn, ¬Q | T 1 Γ12, P | T 2

Γ11, Γ12, l1 ≈ r1, . . . , ln ≈ rn | T 1 ∧ T 2 ∧ P = Q (res) Γ2, u[l] v | T

Γ11, Γ12, Γ2, u[r1] v , . . . , u[rn] v | T 1 ∧ T 2 ∧ T ∧ P = Q ∧ T 4(sup)

where T 4 stands for (l1 r1 ∧ . . . ∧ ln rn ∧ u v ∧ l1 = l ∧ . . . ∧ ln = l), and

∈ ≈, ≈. In this case, the sequence transforms to:

Γ11, l1 ≈ r1, . . . , ln ≈ rn, ¬Q | T 1 Γ2, u[l] v | T

Γ11, Γ2, u[r1] v , . . . , u[rn] v, ¬Q | T ∧ T 1 ∧ T 4(sup)

Γ12, P | T 2

Γ11, Γ12, Γ2, u[r1] v , . . . , u[rn] v | T ∧ T 1 ∧ T 2 ∧ P = Q ∧ T 4(res)

fac-sup rule – Relational factoring followed by superposition

Γ1, l1 ≈ r1, . . . , ln ≈ rn, L1, L2 | T 1

Γ1, l1 ≈ r1, . . . , ln ≈ rn | T 1 ∧ L1 = L2(f ac)

Γ2, u[l] v | T

Γ1, Γ2, u[r1] v , . . . , u[rn] v | T 1 ∧ T 2 ∧ T ∧ L1 = L2 ∧ T 3(sup)

where T 3 stands for (l1 r1 ∧ . . . ∧ ln rn ∧ u v ∧ l1 = l ∧ . . . ∧ ln = l) and

∈ ≈, ≈. L1 and L2 are either both positive or both negative literals. In this

case, the sequence transforms to:

Γ1, l1 ≈ r1, . . . , ln ≈ rn, L1, L2 | T 1 Γ2, u[l] v | T

Γ1, Γ2, u[r1] v , . . . , u[rn] v, L1, L2 | T 1 ∧ T ∧ T 3(sup)

Γ1, Γ2, u[r1] v , . . . , u[rn] v | T 1 ∧ T 2 ∧ T ∧ L1 = L2 ∧ T 3(f ac)

By analyzing the permutation rules res-sup and fac-sup, it can be noticed that,

once applied to derivation trees, they can introduce some tautologies. Importantly,

these tautologies are exclusively tautologies with respect to predicate literals, i.e. the

predicate parts of these clauses are tautologies. Consider and example that illustrates

this.



5.2 A proof by transformation 79

Example 5.5. Assume that a derivation contains the following sequence of inferences,

in which resolution takes place before a superposition inference.

P (a) → a ≈ b, a ≈ c → P (a)

→ a ≈ b, a ≈ c (res)

f (a) ≈ b → P (a)

f (b) ≈ b, f (c) ≈ c → P (a) (sup)

Applying the res-sup rule, this sequence is changed into

P (a) → a ≈ b, a ≈ c f (a) ≈ b → P (a)

P (a), f (b) ≈ b, f (c) ≈ b → P (a) (sup)

→ P (a)

f (b) ≈ b, f (c) ≈ c → P (a) (res)

In the latter derivation a tautology appears as a conclusion of the superposition

inference. This is due to the fact that the permutation rule causes literals, which do

not appear together in any clause of the original derivation, to appear in the same

clause of the transformed derivation. Looking at the general description of the res-

sup rule, it results in the literal ¬Q appearing in the same clause as Γ2 in the modified

derivation, which does not happen in the original derivation.

Note however, that this can only happen with respect to predicate literals, which

implies that permutation rules can result only in the appearance of tautologies with

respect to predicate literals.

Lemma 5.6. The above permutation rules modify EBFP derivations into EBFP

derivations.

Proof. Every permutation rule defines a way of inverting the order of two adjacent

inference rules in a derivation tree. After changing positions, the inferences still take

place with the same literals at the same positions in terms as it was in the original

derivation. Also, all ordering constraints are kept. Therefore, the resulting derivation

is a valid EBFP derivation.

5.2 A proof by transformation

In order to prove the completeness of the regular strategy for basic superposition, we

start with a refutation by EBFP of an unsatisfiable set of clauses S . Assume that the

root of the refutation is | T , where T is a satisfiable constraint. Since the calculus

employs constraint inheritance, we can find a solution of T , and apply it to the whole

refutation. Having that our transformations do not introduce inferences ”from” and

”to” some fresh literals, and that they they do not change the positions at which




the inferences take place, we can consider only ground instances of the refutation.

Further in this work, all the transformations will be assumed to take place on groundderivations.

Lemma 5.7. Any EBFP derivation Ω of the form:

Π1¬P, C 1

Π2P, C 2

C 1, C 2(res) Π3

D

E (sup)

.... (eq infs)F

where the inferences that follow res are all equality inferences, can be split into two

derivations Ω1 and Ω2 with conclusions F 1 and F 2 for which:

• The clause F 1 contains the literal P (can be written as F ∗1 , P ) and F 2 contains

the literal ¬P (can be written as F ∗2 , ¬P ).

• The union of the literals from F ∗1 and F ∗2 contains all the literals that appear in

F and only those literals, with possible duplicates.

Proof. The proof is by induction on the number of (equality) inferences in Ω that take

place after the inference res. In the base case the number of equality inferences after

res is 0. The statement of the lemma then holds, with F 1 and F 2 being, respectively,

clauses ¬P, C 1 and P, C 2, and Ω1 and Ω2 being of the form

Πi

¬P, C i

for i ∈ 1, 2.

Otherwise, let Ω with a conclusion F be a derivation that is obtained from Ω by

cutting off its last inference. By the induction hypothesis, Ω can be split into Ω1 and

Ω2, rooted by F 1 and F 2 respectively.

Focus to the final inference of Ω. It involves one or more literals from the clause F

.Let the final inference of Ω, without a loss of generality, be a positive superposition

inference with F as the ”from” clause. Note that the conclusion of this inference is

in fact the clause F .Γ1, l ≈ r1, . . . , l ≈ rm Γ2, u[l] ≈ v

Γ1, Γ2, u[r1] ≈ v, . . . , u[rm] ≈ v

In case all the literals l ≈ r1, . . . , l ≈ rm belong to (w.l.o.g.) F 1, we add the following

derivation to Ω1, thus defining the final form of Ω1. The added inference has F 1 as




the ”from” premise:

F

∗

1 , l ≈ r1, . . . , l ≈ rm Γ2, u[l] ≈ vΓ2, F ∗1 , u[r1] ≈ v , . . . , u[rm] ≈ v

There are no added inferences to Ω2, which is then the same as Ω2. By the induction

hypothesis, the clauses F 1 and F 2 contain all the literals form Γ1. Besides, F 1 contains

the literal P and F 2 the literal ¬P . Therefore, the conclusion F 1 of Ω1 inherits the

literal P from the F 1, and similarly F 2 inherits ¬P from F 2, and the union of the

literals from F 1 and F 2 contains only (and all of them) the literals from Γ 1, Γ2.

Otherwise, assume that the literals l ≈ r1, . . . , l ≈ rk appear in F 1, while the

literals l ≈ rk+1, . . . l ≈ rm appear in F 2. It is easy to see that, in order to obtain all

the literals that appear in Ω, both F

1 and F

2 should paramodulate into the negative

premise of the last inference of Ω. We therefore produce Ω1 and Ω2 by adding an

inference to both Ω1 and Ω

2. These inferences have the clauses F 1 and F 2 as positive

premises.F ∗1 , l ≈ r1, . . . , l ≈ rk Γ2, u[l] ≈ v

Γ2, F ∗1 , u[r1] ≈ v, . . . , u[rk] ≈ v

andF ∗2 , l ≈ rk+1, . . . , l ≈ rm Γ2, u[l] ≈ v

Γ2, F ∗2 , u[rk+1] ≈ v, . . . , u[rm] ≈ v

Similarly to the previous case, the statement of the lemma holds. It is worth pointing

out that this case produces duplicate literals in the union of the literals from theclauses F 1 and F 2. It due to the fact that the ”to” clause of the final inference of Ω

appears as the ”to” clause of the final inferences of both Ω 1 and Ω2, and therefore

the literals from Γ2 are inherited to both F 1 and F 2.

Note that the same reasoning applies when the last inference of Ω is equality

solution. The consideration then forks in two sub-cases, determined by whether the

literal inferenced upon in Ω appears in both F 1 and F 2 or just in one of them.

The previous lemma can be best illustrated by an example.

Example 5.8. Consider a derivation Ω which is of the form as assumed at the

beginning of the previous lemma.

→ b ≈ c

P (a) → a ≈ b → P (a), a ≈ c

→ a ≈ b, a ≈ c (res)

f (a) ≈ b → P (a)

f (b) ≈ b, f (c) ≈ c → P (a) (sup1)

f (c) ≈ b, f (c) ≈ c → P (a) (sup2)

The transformation of this derivation is performed by induction down the derivation

tree. Consider first the sub-derivation Ω with the induction parameter 1, which is




the sub-derivation rooted by the inference sup1. Following the procedure described

in the proof of the previous lemma, this Ω can be split in two derivations Ω1 and Ω2.Since the “from” literals of the inference sup1 come from both premisses of res, the

derivations Ω1 and Ω

2 contain one of these premisses each. These derivations are

P (a) → a ≈ b f (a) ≈ b → P (a)

P (a), f (b) ≈ b → P (a) (sup)

and→ P (a), a ≈ c f (a) ≈ b → P (a)

f (c) ≈ b → P (a), P (a) (sup)

By induction, these two derivations can be used to compute the derivations Ω 1 and

Ω2, which Ω can be split into. Indeed, consider the inference sup2. The “into” literal

of this inference is f (b) ≈ b, which belongs to the conclusion of Ω 1 and does not

exist in Ω2. Therefore, Ω2 is the same as Ω

2, while Ω1 is modified by adding another

superposition inference to Ω1:

→ b ≈ c

P (a) → a ≈ b f (a) ≈ b → P (a)

P (a), f (b) ≈ b → P (a) (sup)

P (a), f (c) ≈ c → P (a) (sup)

The conclusions of the derivations Ω1 and Ω2, named F 1 and F 2 are such that

• they contain the literals resolved upon in res, i.e. can be written as F ∗1 , P (a)

and F ∗2 , ¬P (a), and

• the union of the literals from F ∗1 and F ∗2 contains all the literals of the conclusion

of Ω, and only them, with possible duplicates.

The above example show that it is possible, by applying the previous lemma, to

end up with two proofs which may contain tautologies. However, these tautologies

are a result of combining the predicates, which are resolved upon in the original proof,

with literals they do not appear with clauses of the original proof. However, it follows

from the proof of the previous lemma, that these tautologies are only with respect to

predicate literals.

Nevertheless, the clauses F 1 and F 2 are such that there exists a derivation, with

them as premises, of the clause F (the conclusion of the original derivation) by ap-

plying predicate factoring and resolution.

Lemma 5.9. An e-refutation by EBFP can be transformed into a regular EBFP

e-refutation with the same conclusion.




Proof. In a derivation tree, a predicate inference for which there is an equality in-

ference following it is called a non-terminating predicate inference. Let Ω be ane-refutation by EBFP with a conclusion R. Without a loss of generality, we assume

that the final inference of Ω is an equality inference. Otherwise, we can always ne-

glect the predicate inferences at the end of the derivation tree, and apply the lemma

on the sub-derivation obtained this way. Let n be the number of non-terminating

predicate inferences in Ω. Among all the predicate inferences in the derivation that

are not followed by other predicate inferences, pick the one that is followed by the

least number of inferences and call it inf . If the number of the inferences that follow

inf is m, the induction is on the regularity pair (n, m), where:

(n1, m1) > (n2, m2) if n1 > n2 or

n1 = n2 and m1 > m2

A regular derivation is assigned the pair (0, 0).

Assume that inf is a resolution inference. In case of factoring, the discussion

is similar (the difference is in the permutation rules applied) and less complex. The

inference that follows inf can be equality solution. In this case, the rule res–es applies,

which modifies the Ω to a derivation Ω, which at least has the second member of the

regularity pair lesser than m. This transformation does not change the conclusion of

Ω. The induction hypothesis applies to the sub-derivation of Ω without the trailing

resolution inferences.

Alternatively, the derivation Ω is of the form:

Π1C 1

Π2C 2

Γ1, l ≈ r1, . . . l ≈ rk(inf ) Π3

Γ3, u[l] ≈ v

Γ1, Γ2, u[r1] ≈ v, u[r2] ≈ v, . . . u[rk] ≈ v

If all the literals l ≈ r1, . . . , l ≈ rk belong to either C 1 or C 2, then similarly to

the previous case, the permutation res–sup can be applied, which also results in

obtaining a derivation with a smaller regularity pair and the same conclusion R.

If neither of the previous two scenarios apply, then some of the literals l ≈

r1, . . . , l ≈ rk appear in C 1, while the others are inherited from C 2. In other words,

C 1 = P, Γ1, l ≈ r1, . . . , l ≈ rl and C 2 = ¬P, Γ2, l ≈ rl+1, . . . l , ≈ rm.

By Lemma 5.7, the derivation can be split into two derivations Ω 1 and Ω2. They can

be transformed, by the induction hypothesis, to regular e-refutations Ω 1 and Ω

2 with

the conclusions F 1 and F 2. The Lemma 5.7 states that the clauses F 1 and F 2 contain




the literals P and ¬P . By the same lemma, as described following Example 5.8, it is

possible to construct a derivation, by predicate inference rules, from F 1 and F 2 withconclusion R.

The base of the induction is a derivation with the regularity pair (1, k) where

k ≥ 1. More precisely, in case Lemma 5.7 applies to a derivation with only one non-

terminating predicate inference, k is allowed to be greater than 1. This is because

the Lemma 5.7 makes it possible to push all predicate inferences down, below all

equality inferences that follow. Otherwise, the base of the induction is any derivation

which can be assigned the pair (1, 1). By applying a suitable permutation rule, such

derivation can be made regular.

Lemma 5.10. Any unsatisfiable set of e-flat clauses has a regular refutation in which

tautologies are redundant.

Proof. Because of its completeness property and compatibility with tautology elimi-

nation, there is always a tautology-free BF P refutation from an unsatisfiable set of

e-flat clauses. Every such refutation is also an e-refutation, and by the Lemma 5.9, it

can be transformed to a regular EBFP refutation. As it has been already stated, the

preformed transformation does not cause the appearance of tautologies w.r.t. equality

literals. Having a regular derivation means that there can be derived a set of purely

predicate clauses from which the empty clause can be derived by resolution. Each

of those purely predicate clauses is actually the root of a regular derivation. If there

are tautologies w.r.t. predicate clauses in such regular derivations, the corresponding

root will be a tautology, too. As such, it is not needed in the further refutation by

resolution, and can be discarded. By discarding this clause, we discard the whole

sub-derivation where tautologies appeared. This proves that even tautologies w.r.t.

predicate literals can be eliminated.

Tautologies are not the only class of redundant clauses that can be removed from

regular derivations. The following shows that the calculus EBFP is complete with

subsumption. As a direct consequence, we have the completeness of the regular

strategy with subsumption.

Lemma 5.11. Let C 1 C 2

C (sup)

be an application of factored overlap and let C 1 C 1 and C 2 C 2. Then either C 1

or C 2 is a subsumer of C , or there exists a subsumer C of C , obtained by applying

superposition to C 1 and C 2.




Proof. Assume that the clauses C 1 and C 2 are, respectively, of the form Γ1, l1 ≈

r1, . . . , ln ≈ rn | T 1 and Γ2, s[u] ≈ t | T 2, C of the form Γ1, Γ2, s[r] ≈ t | T 1 ∧ T 2 ∧ l >r ∧ s[u] > t ∧ l = u, and that ϕ1 and ϕ2 are injections that map the literals from

C 1 and C 2 to, respectively, C 1 and C 2. Due to the fact that the clauses C 1 and C 2

have no variables in common and that a constraint solution is a ground substitution

by definition, any σ which is a solution of the constraint of C can be represented as

a composition σ1 σ2, such that

• σ1 is a solution of the constraint T 1 ∧ l > r,

• σ2 is a solution of T 2 ∧ s[u] > t,

• lσ1 = uσ2.

If there are no literals in C 1 that map into the literals l1 ≈ r1, . . . , ln ≈ rn, we

prove that C 1 C . It is obvious that the injection ϕ1 maps the literals from C 1 to the

literals from C . Since C 1 C 1, there exists σ 1 which is a solution of the constraint of

C 1, such that for every literal L from C 1, ϕ(L)σ1 = L σ1. Since ϕ(L) also belongs

to C , it follows that C 1 C .

Otherwise, if there is no literal in C 2 that maps into s[u] ≈ t, following the steps

from the previous paragraph, it is easy to show that C 2 C .

Assume now that the clauses C 1 and C 2 are, respectively, of the form Γ1, l1 ≈

r1, . . . , lm ≈ rm | T 1 and Γ2, s[u] ≈ t | T 2. Let ϕ1 be an injection that maps the

literals from Γ1 to the literals from Γ1 and the literals l1 ≈ r1, . . . , lm ≈ rm to a

subset of l1 ≈ r1, . . . , ln ≈ rn (without a loss of generality we can assume that they

map to l1 ≈ r1, . . . , lm ≈ rm respectively). Similarly, let ϕ2 be an injection that maps

the literals from Γ2 to the literals from Γ2 and s[u] ≈ t to s[u] ≈ t. Consider the

following inference between C 1 and C 2:

Γ1, l1 ≈ r 1, . . . , lm ≈ r m | T 1 Γ2, s[u] ≈ t | T 2

Γ1, Γ2, s[r1] ≈ t, . . . , s[rm] ≈ t | T 1 ∧ T 2 ∧ s[u] > t ∧ δ (sup)

where δ represents the constraint l1 = u

∧ . . . ∧ lm = u

∧ l1 > r

1 ∧ . . . ∧ l

m > r

m.

To make the rest of the proof more readable, we name the conclusion of the previous

inference C .

Since σ1 is a solution of T 1, σ2 a solution of T 2 and having that C 1 C 1 and C 2 C 2,

there exist substitutions σ 1 and σ 2 which are solutions to T 1 and T 2. From liσ1 = li σ1,

riσ1 = ri σ1 and liσ1 > riσ1 for i ∈ 1, . . . m, it follows that li σ1 > ri σ1 for

i ∈ 1, . . . m, and subsequently that σ1 is a solution of T 1 ∧ l1 > r1, . . . , lm > rm.

Similarly, it can be concluded that σ2 is a solution of T 2 ∧ s[u] > t. Also, from




liσ1 = li σ1 for i ∈ 1, . . . m and uσ2 = u σ2, it can be deduced that liσ1 = u σ2

for i ∈ 1, . . . m, and because the domains of σ1 and σ2 do not intersect, that thecomposition σ1 σ2 is a solution to the constraint of the clause C . Let ϕ be defined

as the union of ϕ1 and ϕ2, added that ϕ(s[u] ≈ t) = s[u] ≈ t. Clearly, ϕ maps

the literals from C to the literals from C , and ϕ(L)σ = L σ1 σ2, for all L from C .

Therefore, C C .

Lemma 5.12. Let C 1C

(es)

be an application of equality solution and let C 1 C 1. Then either C 1 C or there

is a subsumer C

of C , obtained by applying equality solution of C 1.

Proof. Assume that the clauses C 1 is of the form Γ1, s ≈ t | T 1, C of the form

Γ1 | T 1 ∧ s = t, and that ϕ is an injection that maps the literals from C 1 to the literals

from C 1.

Let C 1 be of the form Γ1, s ≈ t | T 1, ϕ(s ≈ t) = s ≈ t, and assume C is obtained

by equality solution from C 1 and of the form Γ1 | T 1 ∧ s = t. Name σ a solution of

T 1 ∧ s = t (also a solution of T 1). Since C 1 C 1, there is a substitution σ that is

a solution of T 1, such that s σ = sσ and t σ = tσ. It follows that σ is a solution

of T 1 ∧ s = t. The injection ϕ maps the literals from C to the literals from C and

ϕ(L)σ = Lσ, for all literals L from C . It follows that C C .The case when C 1 is such that it does not contain a literal that maps to s ≈ t is

very similar and it straightforwardly follows that C C .

Lemma 5.13. Let C i be subsumers of C i for i ∈ 1, . . . , n. If there exists a regular

refutation C 1, . . . , C n, then there exists a refutation from C 1, . . . , C n.

Proof. The statement follows from the previous two lemmas and the fact that reso-

lution is compatible with subsumption.

By incorporating the statement of the previous lemma with Lemma 5.10, the

following holds.

Theorem 5.14. Any unsatisfiable set of e-flat clauses has a regular EBFP -refutation.

The following statement is an instance of Conjecture 5.3, and is a straightforward

consequence of the previous consideration.

Corollary 5.15. Let S be a set of Horn with respect to equality literals with the

following property: the arguments of every non-equality atom in S are variables. Then



5.3 Regular derivations on closures 87

there exists a refutation of S with tautology elimination and subsumption in which

applications of superposition precede applications of all other rules (resolution, equality solution and factoring).

5.3 Regular derivations on closures

The result presented in Chapter 7 later in this thesis, relies on completeness of the

regular strategy of derivations by basic superposition of closures. It is therefore

necessary to show that the regular strategy is complete for the calculus EBFP of

closures (see Section 3.6).

Although the completeness of EBFP of constrained clauses given in Section 3.6

implies completeness of EBFP of closures, this implication does not extend to the

regular strategy of derivations. The reason for this is that the proof of completeness

of the regular strategy of derivations by basic superposition with ordering constraint

inheritance, given in Theorem 5.14, is derived by transformations of proof trees.

Transformation methods may not work for basic superposition of closures, as it is

explained in Example 6.8, later in this thesis. It is therefore necessary to show that it

is possible to create a set of permutation rules, similar to the ones given in Section 5.1,

which can be applied on closures. Such set of permutation rules should transform

correct EBFP derivations (on closures) to correct EBFP derivations (see Lemma 5.6).

Permutation rules for basic superposition of closures are constructed in the same

way as the rules for basic superposition with ordering constraints. For example,

consider the following sequence of inferences by EBFP of closures:

Γ11, l1 ≈ r1, . . . , ln ≈ rn, ¬Q · σ1 Γ12, P · σ2

Γ11, Γ12, l1 ≈ r1, . . . , ln ≈ rn · (σ1 ∪ σ2)mgu(P σ2, Qσ1) (res)

Γ2, u[l] v · σ

Γ11, Γ12, Γ2, u[r1] v , . . . , u[rn] v · ((σ1 ∪ σ2)mgu(P σ2, Qσ1) ∪ σ)ρ (sup)

where

• η is the substitution part in the conclusion of the resolution inference, i.e η =

(σ1 ∪ σ2)mgu(P σ2, Qσ1),

• ρ is the composition of the substitutions mgu(liη,lσ), for i ∈ 1, . . . , n, and

∈ ≈, ≈, and

• the condition for applying the superposition inference is r1ηρ l1ηρ, . . . , rnηρ

lnηρ, and vσρ uσρ.




Following the idea of the permutation rules from Section 5.1, the res-sup rule on

closures should transform the previous sequence of inferences into

Γ11, l1 ≈ r1, . . . , ln ≈ rn, ¬Q · σ1 Γ2, u[l] v · σ

Γ11, Γ2, u[r1] v , . . . , u[rn] v, ¬Q · (σ1 ∪ σ)ρ (sup)

Γ12, P · σ2

Γ11, Γ12, Γ2, u[r1] v , . . . , u[rn] v · ((σ1 ∪ σ)ρ ∪ σ2)mgu(Pσ,Q(σ1 ∪ σ2)ρ) (res)

where

• ρ stands for the composition of the substitutions mgu(lσ,liσ1) for i ∈ 1, . . . , n,

and

• the condition for applying the superposition inference is r1σ1ρ l1σ1ρ, . . . ,

rnσ1ρ lnσ1ρ and vσρ uσρ.

Other permutation rules for closures can be defined in a similar way, from the per-

mutation rules for constrained clauses, given in Section 5.1.

Lemma 5.16. Permutation rules modify EBFP derivations to EBFP derivations.

Proof. A permutation rule transforms a sequence of inferences to another sequence of

inferences. Assume that the conditions for applying inferences in the original sequence

of inferences hold. It is sufficient to show that the conditions for applying inferences

in the modified sequence also hold. This proof addresses the res-sup in detail. The

same argument can be used to analyze the rule fac-sup, while for all other rules thelemma trivially follows.

Consider the definition of res-sup rule for closures, given above. In the original

sequence, there is one condition that has to be satisfied, which is the condition for

applying the superposition rule. Assume that this condition holds, i.e that the formula

ϕ : r1ηρ l1ηρ ∧ rnηρ lnηρ ∧ vσρ uσρ

holds true. It should be shown that, under this assumption, the formula

φ : r1σ1ρ l1σ1ρ ∧ rnσ1ρ lnσ1ρ ∧ vσρ uσρ

which represents the condition for applying the superposition inference after applying

the res-sup rule, also holds. From

η : (σ1 ∪ σ2)mgu(P σ2, Qσ1)

ρ : mgu(l1η,lσ) · · · mgu(lnη,lσ)

ρ : mgu(lσ,l1σ1) · · · mgu(lσ,lnσ1)

it is easy to see that σ1ρ and σρ are, respectively, more general then ηρ and that

σρ. It follows that if ϕ holds then φ also holds.




From the above consideration, the following is easy to show.

Lemma 5.17. Any unsatisfiable set of flat clauses has a regular refutation by EBFP

of closures, in which tautologies and subsumed closures are redundant.



Chapter 6

Basic superposition with

arbitrary selection strategies

The results presented in this chapter aim at extending the results given in Chapter 5

by investigating strategies for derivation by basic superposition which are more general

than the regular strategy. As a reminder, the regular strategy of derivations can be

encoded as a selection function that selects only equality literals, and all of them,

whenever there are any in the clause. This chapter investigates more general selection

strategies. More precisely, it addresses the most general selection strategy, known as

the arbitrary selection strategy (see [BG01b]), by which a non-empty set of literals

is selected in each non-empty clause. Arbitrary selection is a generalization of free

selection, defined in [BG01a], by which a single (positive or negative) literal is selected

in each non-empty clause.

Selection strategies provide a way of controlling the search space by restricting all

inferences between clauses to only those that involve selected literals. Using selection

strategies is far more desirable for paramodulation than for resolution – resolution,

by definition, always takes place with one positive and one negative literal, while

paramodulation can involve two positive literals, which makes it a less restrictive

rule.

Definition 6.1 (arbitrary selection). A selection strategy is a function from a set of

clauses, that maps each clause to one of its sub-multisets. If a clause is non-empty,

then the selected sub-multiset is non-empty too. A derivation is compatible with a

selection strategy if all the inferences are performed on the selected literals, i.e. all

the literals involved in the inferences are selected. A selection strategy is complete if

any unsatisfiable set of clauses has a refutation which is compatible with the strategy.

90



91

All modern results of refutational completeness of paramodulation-based calculi

(which have a model generation proof) also show compatibility of these calculi witha class of so-called classical selection strategies. By these strategies, either some

negative or all maximal positive literals are selected. One such selection strategy is

the maximal strategy (see [BG01a]), by which all maximal positive literals in a given

ordering are selected. For instance, the inference

→ a ≈ b, b ≈ c → a ≈ c, c ≈ d

→ b ≈ c, b ≈ c, c ≈ d (sup)

is compatible with the maximal strategy only if a ≈ b and a ≈ c are maximal literals,

in a given ordering, in their corresponding clauses. Another selection strategy that can

be described as classical is the strategy of eager selection of negative literals, by which

a single negative literal is selected if the clause contains any. Since negative literals

can only be paramodulated into, it follows that “from” premises can not contain

any negative literals and must therefore be positive clauses. Hence, this strategy is

often called the positive selection strategy. For more on selection strategies refer to

Section 3.4.5.

Although classical selection strategies impose restrictions on how to apply in-

ference rules, and therefore lead to a reduced number of derived clauses, they are

not sufficient for practically efficient theorem proving. In particular, Dershowitz in

[Der91] explains that the maximal strategy is not very efficient and, more generally,Voronkov (during JELIA 2006) points out that their Vampire prover certainly uses

selection strategies that can not be described as classical. Added that certain appli-

cations require completeness of strategies that can not be described as classical, (as

pointed out in [Lyn97] and in Chapter 5 on the example of regular derivations) it is

undeniable that investigation of non-classical selection strategies, for both resolution

and paramodulation, is rather necessary.

In consequence, de Nivelle in [dN95] (Theorem 6.7.4) proves that resolution is com-

plete with arbitrary selection strategies for Horn clauses (without equality). More

precisely, de Nivelle shows completeness of arbitrary selection for general clausesprovided that there exists a refutation without factorization. In the framework of

paramodulation-based theorem proving, it is Lynch in [Lyn97] who comprehensively

discusses and tries to prove compatibility of arbitrary selection strategies for basic

superposition on Horn closures (with equality). In [BG01b], Bofill and Godoy point

out some severe flaws in Lynch’s proof, and give their own proof of a similar result,

with addition of the requirement that the calculus is compatible with the positive

selection strategy.



92

Our result is a step further in this direction, as we generalize de Nivelle’s result tothe case of basic superposition (with ordering constraint inheritance), and strengthen

the result by Bofill and Godoy, firstly by eliminating the condition that the calculus

is complete with the positive strategy, secondly by showing their result for a more

restrictive implementation of basic superposition, and thirdly by considering the case

of general first-order clauses.

The latter generalisation, to calculi of general first-order clauses, comes with a

pinch of salt. Namely, basic superposition of general clauses is complete with arbitrary

selection only in case of unsatisfiable sets of clauses which have refutations without

applications of a factoring rule. Otherwise, for unsatisfiable sets of clauses factoring is

needed, it is not possible to find a refutation with applying arbitrary selection. This

is best shown by the following example, originally given in [Lyn97].

Example 6.2. Consider a selection rule on general clauses which selects all positive

literals in a clause, regardless their maximality. In the following set of clauses, the

selected literals are underlined.

p → r p, q → r → q

q, r → q → p r, p → → p, q , r

This set is unsatisfiable. The conclusion of every inference is subsumed by the originalset, and therefore the empty clause can not be generated.

Note that the given selection strategy can not be described as classical. If this was

possible, a selected positive literal would have to be maximal in the corresponding

clause, which would result in r p, p q and q r. It follows that r r, which is

impossible.

Our result does not say anything about redundancy elimination. In general, Ex-

ample 5.4 from the previous chapter shows that arbitrary selection strategies for

superposition on Horn clauses are not complete with tautology elimination. The

fact that they are not complete with tautology removal is a good indicator thatarbitrary selection strategies may not have a completeness proof using the model

generation technique. As it has already been pointed out, the model generation tech-

nique, roughly speaking, allows for elimination of clauses that follow from smaller

clauses. Since tautologies follow from the empty set of clauses, they would certainly

be redundant should a model generation completeness proof exist.

Bearing this in mind, it is not a surprise that proofs of all the above mentioned

results about completeness of arbitrary selection strategies are by transformations



6.1 Completeness for superposition with ordering constraint inheritance 93

of derivation trees. Our proof of Theorem 6.6 is no exception. Unlike de Nivelle’s

method, we transform derivations by superposition, while unlike the technique usedby Bofill and Godoy the application of our technique is not conditioned by whether

the calculus is compatible with the positive selection strategy.

This work has been presented at JELIA 2006 under the title “On Arbitrary Se-

lection Strategies for Basic Superposition” (see [AD06]).

6.1 Completeness for superposition with ordering con-

straint inheritance

The main contribution of this section is a proof that shows that basic superposition

with ordering constraint inheritance is complete with arbitrary selection strategies,

provided that there exists a refutation without factoring inferences.

The result is given for the calculus BS from Section 3.5.1. From completeness of

BS it follows that there is a refutation from any inconsistent set of clauses. This,

however, does not give any information about the existence of a refutation that is

compatible with a given selection function. One way of answering this (affirmatively)

is by transforming a given refutation to a refutation that is compatible with the given

selection function.

This transformation can be done in the following way. Assume there is an inference

inf 1 with clauses C 1 and C 2 as premises and C as the conclusion. Assume that C 1 is of

the form Γ1, L1, L2 and that inf 1 uses a literal L2, which is not selected and different

to a selected literal L1. In every superposition and equality solution inference L1

would be a literal of the conclusion C , as an inherited literal from the premise C 1.

Assume now that the clause C is a premise of another inference inf 2, which uses L1 –

this can be assumed, as it is shown in the proof below. The idea is to swap the order

in which L1 and L2 are used, by making permutations of the inferences inf 1 and inf 2,

which results in using the selected literal L1 in inf 1. As it is shown below, this can

be applied to all inferences that use literals that are not selected, which results in a

refutation that is compatible with any given selection function.

Analyzing the idea of permutations of inference rules, it becomes obvious that

swapping two inferences only makes sense if the literals L1 and L2 are different (in

the multiset way). This observation is used below in the definition of permutation

rules.




To summarize, permutation rules that swap the order of two consecutive inferences

fall into three categories, and apply whenever inf 1 and inf 2 are:

• two consecutive superposition inferences,

• two consecutive equality solutions, and

• a superposition inference followed or preceded by equality solution.

Formally, definitions of the permutation rules are (the symbol represents either

≈ or ≈):

sup-es rule – Superposition followed by equality solution

Γ1, l1 ≈ r1 | T 1 Γ2, s ≈ t, l2[l

] r2 | T 2Γ1, Γ2, s ≈ t, l2[r1] r2 | T 3

(sup)

Γ1, Γ2, l2[r1] r2 | T 3 ∧ s = t (es)

where T 3 stands for T 1 ∧ T 2 ∧ l = l1 ∧ l1 r1 ∧ l2 r2. This sequence of

applications of inference rules permutes into:

Γ1, l1 ≈ r1 | T 1

Γ2, s ≈ t, l2[l] r2 | T 2

Γ2, l2[l] r2 | T 2 ∧ s = t (es)

Γ1, Γ2, l2[r1] r2 | T 1 ∧ T 2 ∧ s = t ∧ l = l1 ∧ l1 r1 ∧ l2 r2(sup)

Note that, in order for the permutation to be possible, it is essential that theliterals s ≈ t and l2 r2 are distinct in the multiset context. In case they

were not, equality solution in the original derivation would be possible only

after superposition, and therefore the two inferences would never be possible to

swap.

es-sup rule – Equality solution followed by superposition. This rule is defined as

the converse of supup-es, and its application is always possible.

es-es rule – Two equality solution inferences occur immediately after one another

Γ, s1 ≈ t1, s2 ≈ t2 | T

Γ, s1 ≈ t1 | T ∧ s2 = t2 (es)

Γ | T ∧ s2 = t2 ∧ s1 = t1(es)

Since they take place on different literals, they trivially swap.

Γ, s1 ≈ t1, s2 ≈ t2 | T

Γ, s2 ≈ t2 | T ∧ s1 = t1(es)

Γ | T ∧ s1 = t1 ∧ s2 = t2(es)




Lemma 6.5. Let Ω be a BS derivation tree of the form

C 5

C 1 C 2C 3

(inf 1)....

C 4C 6

(inf 2)

where C 1 and C 2 are optional (i.e. the inferences inf 1 and inf 2 may have only one

premise).

Assume that the clause C 1 is of the form Γ1, L1, M 1 | T 1 where the literal that

takes part in inf 1 is M 1. Assume also that a given selection function selects L1 from

C 1, and that no inference between inf 1 and inf 2 takes place with L1, which is a literal that sup2 takes place with. The derivation Ω can then be transformed to a derivation

C 5 C 1C 3

(inf 2)C 2

C 4(inf 1)

....C 6

in which the inference inf 2

takes place with the literal L1 (which is selected in C 1) and

the inference inf 1

with the literal M 1. Moreover, the clauses C 6 and C 6 are variants.

Proof. For easier readability, in the derivation trees below, selected literals are under-

lined, while the ones that inferences take part with are boxed. If a clause has neither

boxed nor underlined literals, it is because the information about the selected literals

is not essential for the proof.

Assume, without a loss of generality, that the inference inf 1 is superposition and

inf 2 equality solution. Let C 1 be of the form Γ1, s1 ≈ t1, l1 ≈ r1 | T 1 and C 2 of the

form Γ2, s2[l] ≈ t2 | T 2. Let a given selection function be such that it selects s1 ≈ t1

in C 1 and does not select l1 ≈ r1, which the inference inf 1 takes part with.

The proof is by induction on the number of inferences between inf 1 and inf 2.

If this number is 0, then C 3 is the same as C 4 and C 6 is the conclusion of equality

solution applied on C 3, Thus, the derivation is Ω can be written as

Γ1, s1 ≈ t1, l1 ≈ r1 | T 1 Γ2, s2[l] ≈ t2 | T 2

Γ1, Γ2, s1 ≈ t1, s2[r2] ≈ t2 | T 1 ∧ T 2 ∧ l1 = l (sup)

Γ1, Γ2, s2[r2] ≈ t2 | T 1 ∧ T 2 ∧ l1 = l ∧ s1 = t1(es)




which, by applying the permutation rule sup-es, can be transformed to

Γ1, s1 ≈ t1 , l1 ≈ r1 | T 1

Γ1, l1 ≈ r1 | T 1 ∧ s1 = t1(es)

Γ2, s2[l] ≈ t2 | T 2

Γ1, Γ2, s2[r2] ≈ t2 | T 1 ∧ T 2 ∧ l1 = l ∧ s1 = t1(sup)

and the lemma holds.

Alternatively, let the number of inferences between inf 1 and inf 2 be n. In this

case, assume w.l.o.g. that the inference that precedes inf 2 is equality solution. The

derivation Ω is of the form:

Γ1, s1 ≈ t1, l1 ≈ r1 | T 1 Γ2, s2[l] ≈ t2 | T 2

Γ1, Γ2, s1 ≈ t1, s2[r2] ≈ t2 | T 1 ∧ T 2 ∧ l1 = l (sup)

....Γ4, s4 ≈ t4, s1 ≈ t1 | T 1 ∧ T 2 ∧ l1 = l ∧ T 4

Γ4, s1 ≈ t1 |T 1 ∧ T 2 ∧ l1 = l ∧ T 4 ∧ s4 = t4(es)

Γ4 |T 1 ∧ T 2 ∧ l1 = l ∧ T 4 ∧ s1 = t1 ∧ s4 = t4(es)

By applying the permutation rule es-es on Ω, it can be transformed to a derivation

Ω of the form

Γ1, s1 ≈ t1, l1 ≈ r1 | T 1 Γ2, s2[l] ≈ t2 | T 2

Γ1, Γ2, s1 ≈ t1, s2[r2] ≈ t2 | T 1 ∧ T 2 ∧ l1 = l (sup)

.

...Γ4, s4 ≈ t4, s1 ≈ t1 | T 1 ∧ T 2 ∧ l1 = l ∧ T 4

Γ4, s4 ≈ t4 |T 1 ∧ T 2 ∧ l1 = l ∧ T 4 ∧ s1 = t1(es)

Γ4 |T 1 ∧ T 2 ∧ l1 = l ∧ T 4 ∧ s1 = t1 ∧ s4 = t4(es)

The statement of the lemma follows after applying the induction hypothesis to the

sub-derivation obtained by removing the final inference from Ω .

Theorem 6.6. Let S be a set of constrained first-order clauses that has a refutation by

BS , which not necessarily employs a selection strategy. Then there exists a refutation

compatible with any selection strategy.

Proof. Let Ω be a refutation from S . Consider now a given arbitrary selection, and

mark the literals of the clauses of the derivation Ω that are selected. We call misused

any clause in which the literal that takes part in an inference is not the one selected

by the selection function. A clause C is well-used if it is not misused and there are

no misused clauses in the sub-derivation of Ω rooted by C . We use induction on the

number of well-used clauses.




Assume that Ω contains misused clauses and that it is of the form:

Ω5C 5

Ω1C 1

Ω2C 2

C 3(sup1)

....C 4

C 6(sup2)

....

such that C 1 is misused and there are no misused clauses in Ω1. This is without a loss

of generality, and represents only one of a number of essentially similar scenarios in

which misused clauses can appear. Assume that the clause C 1 is Γ1, s1[l

] ≈ t1, l1 ≈r1 | T 1, such that the selected literal is s1[l] ≈ t1 and the one used in the inference

is l1 ≈ r1. Also assume that the clause C 5 be of the form Γ5, l2 ≈ r2 | T 5. Let the

inference sup2 take place with the literal s1[l] ≈ t1 and assume that there are no

other inferences with the same literal between sup1 and sup2 (therefore there are no

inferences into different positions of the same literal). The last assumption makes it

possible to apply Lemma 6.5 to the sub-derivation of Ω which is rooted by C 6. This

effect is a transformation of Ω into Ω:

Ω5

C 5

Ω1

C 1C 3(sup1) Ω2

C 2

C 4(sup2)

....C 6....

where C 6 and C 6 are variants, and the clause C 1 is well-used. Because no permuta-

tion rule is applied to an inference that has a well-used clause as its conclusion, the

transformation has not changed the property to be well-used of any clause from Ω.

Finally, the transformation has not added to the number of clauses in the refutationand therefore the induction hypothesis applies.

Since in the case of derivations with Horn clauses the factoring inference never

appears, the following statement easily follows from the previous theorem.

Corollary 6.7. Basic superposition with equality and ordering constraints for Horn

clauses is complete with arbitrary selection.



6.2 Arbitrary selection for superposition on closures 100

This result can not be generalized for arbitrary clauses. In the case where all

refutations involve factoring, incompleteness for arbitrary selection strategies alreadyappears in the propositional case (see Example 6.2).

6.2 Arbitrary selection for superposition on closures

As it has been pointed out several times, the transformations presented in the previous

chapter, like the transformations in [BG01b], are based on the implementation of

the basic strategy with ordering constraint inheritance. The question is whether

similar transformations can be applied for basic superposition on closures, given from

[BGLS92]. The example below gives a negative answer, showing that under the

weaker ordering inheritance strategy determined by closures, the same transformation

technique can not be applied, and that Theorem 6.6 can not be reformulated for basic

superposition on closures.

Example 6.8. Let BSC denote a basic superposition inference system over closures,

sup and es denote superposition and equality solution inference rules, respectively.

Consider the following BSC -derivation over closures:

f (x, y) ≈ f (g(z), h(z)), h(x) ≈ g(y) · [x → g(y1), y → h(x1)] h(u) ≈ v · [v → g(v1)]

f (x, y) ≈ f (g(z), h(z)), g(y) ≈ v · [x → g(y1), y → h(x1), v → g(v1)] (s)

f (x, y) ≈ f (g(z), h(z)) · [x → g(y1), y → h(x1)] (es)

· ε (es)

This is a correct BSC -derivation for every reduction ordering .

Let f (x, y) ≈ f (g(z), h(z)) be a selected literal. The result of transforming this

derivation in the style suggested in the previous section is:

f (x, y) ≈ f (g(z), h(z)), h(x) ≈ g(y) · [x → g(y1), y → h(x1)]

h(x) ≈ g(y) · [x → g(z), y → h(z)] (es)

h(u) ≈ v · [v → g(v1)]

g(y) ≈ v · [v → g(v1), y → h(z)] (s)

· ε (es)

This is a BS C -derivation iff h(g(z)) g(h(z)). If is defined as the lexicographic

path ordering where g > h, this derivation is not a BSC -derivation because of the

violation of the ordering conditions.

In [Lyn97], Lynch shows completeness of arbitrary selection strategy for basic

superposition on Horn closures using the model generation technique instead of proof

transformations. Although the authors of [BG01b] point out some severe flaws in this

completeness proof, the answer on whether arbitrary selection is complete for basic



6.3 Arbitrary selection and tautology elimination 101

superposition on Horn closures is not known – Example 6.8 is not a counterexample

for Lynch’s claim.

6.3 Arbitrary selection and tautology elimination

Example 5.4 shows that arbitrary selection is not complete for superposition on Horn

clauses with tautology elimination. Analyzing redundancy notions for the results in

the chapters that follow, we posed the question about redundancy of tautologies in

basic superposition on flat Horn clauses (a clause is flat if its literals contain non-

variable terms only at top positions). Although this question remains unanswered,

our efforts gave rise to a related result about arbitrary selection for resolution.

Theorem 6.9. Resolution of Horn clauses with arbitrary selection is complete with

tautology elimination.

Proof. Let Ω be (a tree-representation of) a refutation by resolution of an unsatisfiable

set S . Moreover, let Ω be compatible with a given selection function. The proof is by

induction on the number of clauses in the derivation tree.

The base case is when Ω is the shortest non-trivial derivation. Such derivation

contains only a resolution inference with premises ¬A and A, neither of which is a

tautology.

Assume that there are tautologies in Ω. Therefore, Ω contains more than one

application of resolution, and is of the form:

Ω4Γ4, ¬A

Ω1Γ1, A, ¬B

Ω2Γ2, ¬A, B

Γ1, Γ2, ¬A, A (res)

....Γ1, Γ2, ¬A, A

Ω3A, Γ3

Γ1, Γ2, Γ3, A (res)

....Γ1 , Γ2, Γ3, A

Γ1 , Γ2, Γ3, Γ4 (res)....

Note that, since the conclusion of Ω is the empty clause, both literals A and ¬A of

the clause Γ1, Γ2, ¬A, A have to be resolved upon. Consider the following derivation,

which can be obtained by firstly resolving the conclusions of Ω3 and Ω4, and then



6.3 Arbitrary selection and tautology elimination 102

applying the inferences from Ω which take place with the literals from Γ 3:

Ω4Γ4, ¬A

Ω3Γ3, A

Γ3, Γ4(res)

....Γ3, Γ4

Because the clause Γ3, Γ4 is a subsumer of Γ1, Γ2, Γ3, Γ4, and there is a refutation

which involves the latter clause, there has to be a refutation which involves the clause

Γ3, Γ4:Ω4

Γ4, ¬AΩ3

Γ3, A

Γ3, Γ4 (res)....

Γ3, Γ4....

This refutation, denoted by Ω, can certainly be chosen so that it contains fewer

clauses than Ω. However, the derivation Ω may not be compatible with the selection

function. By Theorem 6.6, Ω can be transformed into Ω which is compatible with the

selection function and has the same number of nodes as Ω . Therefore, the derivation

Ω contains fewer clauses in its tree-representation, and the induction hypothesis

applies.



Chapter 7

Basic folding revisited

In [DV95], Degtyarev and Voronkov suggest the equality elimination method for the

Horn fragment of first-order logic. This method transforms a set of clauses, which

may contain both equality and predicate literals into a set of purely predicate clauses.

Differently to the Brand’s method, see [Bra75]), where equality is eliminated by ea-

gerly performing inferences where one premise is always an equivalence axiom, [DV95]

suggests that the equality is eliminated using basic superposition to systematically

“solve” all equations before reasoning with non-equality atoms. This solving of the

equations in the original set of clauses is done by applying so-called regular strategy

of derivations, described in Chapter 5. The essential difference between these twomethods is that, while Brand treats equality indirectly, by adding the congruence

axioms to the theory, the latter approach is based on treating the equality directly,

as a part of the language.

The problem with applying basic superposition is that does not guarantee ter-

mination – there may be infinite chains of superposition. In order to eliminate this

problem, Degtytarev and Voronkov in [DV96], and Degtyarev, Koval and Voronkov

in [DKV95], enrich the superposition calculus by introducing a new inference rule,

called basic folding . By adding the folding rule, they ensure termination of the equality

elimination procedure, thus making it possible to obtain elegant finite sets of purely

predicate clauses from sets of clauses that contain equality literals. In [DKV95, DV96],

the authors show that superposition with basic folding is a terminating, sound and

complete calculus of Horn clauses.

The results presented in this chapter strengthen the results presented in [DKV95,

DV96], by extending them in two directions. Firstly, the completeness result of basic

folding of Horn clauses is shown to hold in presence of tautology elimination and

103



104

subsumption. The completeness result given in [DKV95, DV96] does not tell any-

thing about completeness of basic folding in presence of deletion rules. Although thepossibility of tautology elimination and subsumption is only hinted at in the closing,

chapter of [DKV95], it only addresses tautologies of the form Γ, t ≈ t, and gives no

formal proof to this claim.

Secondly, this improved completeness result, with tautology deletion and sub-

sumption, is further generalized to the case of basic folding of full first-order clauses.

Analyzing the definition of the basic folding rule in [DKV95, DV96], it is by no means

restricted to Horn clauses. Furthermore, having that basic superposition of general

clauses is complete, and given (see Chapter 5) that the regular strategy is complete

for general first-order clauses with tautology deletion and subsumption, refutational

completeness of basic folding with subsumption and tautology deletion is a rather

intuitive result.

The method used to prove the results of this chapter is based on transformations

of derivation trees. Having said that, it is particularly exceptional that the results

address elimination of some state-of-art redundancies, like tautologies and subsumed

clauses. The related results which are also proved using transformations of derivation

trees, like [RW69, Bra75, DV95, BG01b], do not address elimination of redundancies

at all.

It is important, before going any further, to point out that the definition of com-

pleteness (of a calculus) used by the authors of [DV96] does not match the standard

definition of completeness given in Section 2.4 of this thesis. Instead, a calculus is

considered complete if it can be used to derive, from a set S of clauses with equality,

a set of purely predicate clauses S , which is somehow equivalent to S .

In the Horn case, S is equivalent to S iff existence of a correct answer to S

implies existence of a correct answer to S . The notion of answers to sets of clauses

is transferred from logic programming, in the following way.

An answer to a set of closures S is any tuple t1, . . . , tn of terms, where

n is the arity of the predicate query in S . It is a correct answer if

S query(x1, . . . , xn) · σ , so that the terms t1, . . . , tn are instances of

x1σ , . . . , xnσ, respectively.

The reason for using this definition of completeness in case of Horn clauses is that

basic folding was initially introduced in the logic programming framework, where the

key is to compute answers to logic programs.



7.1 The basic folding calculus 105

To clarify the previous completeness definition, one may want to know a bit more

about the relation between the computed answers to the sets of closures S and S . Itis possible to strengthen the definition by requiring that these answers are equivalent

with respect to an equational theory defined by S . However, this strengthening does

not mean much in practice: the problem of checking equality with respect to an

equational theory given by S is undecidable.

In case of general first-order clauses, the concept of answers to sets of clauses does

not apply. Consider an example.

Example 7.1. The following set of clauses is clearly unsatisfiable:

→ P (a), P (b)P (x) →

If a distinguished predicate query(x) is added to the latter clause, like it is added to

goal clauses in case of logic programs, the following set of clauses is obtained:

→ P (a), P (b)

P (x) → query(x)

The answer substitution for this set of clauses can clearly not be computed, the reason

being that the former clause is not Horn, and has to be resolved against the latter

(“goal clause”) twice, which requires that the variable x is simultaneously substitutedby a and b.

The conclusion is that, in case of general first-order clauses, S and S are equiva-

lent iff unsatisfiability of S implies unsatisfiability of S .

The proof of the results of this chapter are organized as follows. First comes

a proof of refutational completeness of basic folding of Horn clauses with tautology

elimination and subsumption, followed by a proof of the same completeness result to

the general first-order case.

7.1 The basic folding calculus

The basic folding calculus BF is defined on p-flat closures. It contains the rules

of the calculus EBFP (see Section 3.6) plus the basic folding rule, introduced by

Definition 7.5.

Before proceeding, it should be emphasized that the basic folding rule is defined

on closures whose clause parts are p-flat (for the definition, refer to Section 2.2.



7.2 Completeness of basic folding of Horn clauses 108

The definition of the folding rule can be modified to obtain a more efficient rule.

From [DV96], it follows that this can be done in at least three ways. For example,the following form of the folding rule is also complete:

S S \ C ∪ C

The underlying idea of this form of the rule is that the definition of the folding term t

(the second folding descendant, E t) need not be introduced provided that it already

exists in S .

Secondly, folding can be applied into all occurrences of the folded term t, not only

one. This is emphasized in Example 6.1 in [DV96], where it is conjectured that such

modification of the folding rule preserves completeness, although it would take a fairly

more complicated proof (than the one in [DV96]) to show it.

Recently, a similar idea of ”folding” a closure by introducing the definition for a

special part of it has been used in the decomposition inference rule of [HMS04] to

obtain basic superposition-based decision procedures for some complex description

logics.

7.2 Completeness of basic folding of Horn clauses

There are two important features of basic folding which need to be pointed out before

going any further into proving refutational completeness of the calculus BF . Firstly,

basic folding was originally introduced in the framework of logic programming, where

emphasis is in computing answers to logic programs. Secondly, the aim of basic folding

is to eliminate equality from a logic program (and obtain a finite logic program without

equality). It is therefore not a surprise that the authors of [DV96] use these two facts

in their completeness statement.

Definition 7.7 (completeness). The calculus I of closures is complete if, for any fair

derivation S 0 ⇒ S 1 ⇒ · · · with a limit S ω (where S 0 contains a definition of thepredicate query), existence of a correct answer to S 0 implies existence of a correct

answer to S −ω .

The notion of answers to a set of closures is straightforwardly transferred from

logic programming to theorem proving with Horn closures.

Definition 7.8 (answer to a set of closures). An answer to a set of closures S is any

tuple t1, . . . , tn of terms, where n is the arity of the predicate query in S . It is a




correct answer if S query(x1, . . . , xn) · σ, so that the terms t1, . . . , tn are instances

of x1σ , . . . , xnσ, respectively.

Note that it is always possible to obtain S from P . Each program clause from P

is trivially translated to a Horn closure with the substitution part being the empty

substitution. Thus obtained set of closures is then transformed to a set of p-flat

closures S . The latter transformation is done by modifying the clause part of each

closure to a p-flat clause, as explained in Section 2.2.

A roadmap of the completeness proof for Horn clauses

The rest of the section contains a completeness proof, with respect to Definition 7.7,

of the calculus BF of Horn clauses. Because of its length, the completeness proof is

split into a number of lemmas. In order to grasp it quicker and easier, this section

contains its informal description, which points out the proof’s most important parts

and shows the way in which they fit together. Consider the completeness statement

for basic folding (the claim of Theorem 7.24).

Let ∆ be a fair BF -derivation S 0 ⇒ S 1 ⇒ · · · from a set of initial closures

S 0, and let S ω be a limit of ∆. If S 0 has a correct answer then S −ω has a

correct answer, too.

The most important intermediate results needed to prove the completeness statement

are formulated in Lemma 7.20, Lemma 7.22 and Lemma 7.23. The best way of

showing how they fit together is giving a draft of the completeness proof. In order to

keep this draft simple, it will not address any redundancy elimination aspects of the

proof. They are addressed in a separate paragraph.

Lemma 7.23 shows that if S 0 has a correct answer, then S ω also has a correct an-

swer, which is derivable from S ω by basic superposition, applying the regular strategy

of derivations. This means that there exists a regular derivation Ω, by basic super-

position, of a clause query(x) · σ, for some substitution σ, where x is x1, . . . , xn.

Equivalently, there exists a derivation (by the equality inference rules of the basicsuperposition calculus) of a set S of purely predicate clauses, so that query(x) · σ is

derivable from S by resolution.

To show the completeness, it is necessary to derive that the existence of a correct

answer to S ω implies existence of a correct answer to S −ω . This would already hold if

the set S was a sub-set of S −ω . In general, however, this is not the case. Therefore, it

is required to prove that there exists a set S of purely predicate clauses, which is a

sub-set of S ω, such that S is derived from S ω by applying the equality inferences of




the basic superposition calculus, and such that there is a correct solution to S . To

show that there exists a correct solution to S , it has to be shown that there exists asubstitution σ, so that a closure query(x) · σ is derivable from S by resolution.

At this point the proof branches, depending on whether there are any substitu-

tion inferences in Ω. If not, the job of proving the completeness is done by invoking

Lemma 7.22. However, if there are some superposition inferences, the completeness

proof is done by induction on the number of the inferences by superposition. Assume

that C 1 and C 2 are premises of a superposition inference with a conclusion C , and

that C 1 and C 2 belong to S ω. Either clause C belongs to S ω, in which case the in-

duction takes care of the rest of the proof (by considering the derivation without the

considered superposition inference), or Lemma 7.20 applies before the induction step

takes care of the proof again. In case C 1 and C 2 do not belong to S ω, the proof can

be reduced to the latter case after implementing a simple transformation.

Another very important ingredient of the completeness proof of the basic fold-

ing calculus of Horn clauses is Lemma 7.12, which states that the relation “a fold-

ing descendant or a proper subsumer of” is well-founded. This lemma is used in

Lemma 7.20, Lemma 7.22 and Lemma 7.23, as well as in Theorem 7.24, which puts

these three lemmas in context.

The other lemmas in the chapter are there to serve Lemma 7.20, Lemma 7.22 and

Lemma 7.23. Having said this, all the intermediate results of Section 7.2.3 are there

to show that Lemma 7.20 holds. Similarly, the lemmas in Section 7.2.4 are highlighted

only to make the proof of Lemma 7.22 more readable.

The biggest strength of the proof given in this chapter is that it allows for tautol-

ogy elimination and subsumption in derivations by BF . This very fact is the main

difference between this proof and the proof given in [DKV95]. The key in keeping

completeness while allowing tautology elimination and subsumption is in showing

that the regular strategy of derivations by basic superposition is complete with the

same deletion rules. It can therefore be said that the main strength of the proof givenin this section lies in the result formulated by Theorem 5.14. However, to preserve

the strength of the result of Theorem 5.14, it is necessary to verify that tautologies

do not appear by involving the basic folding rule in Lemma 7.20, Lemma 7.22 and

Lemma 7.23, which are the other main building blocks of the proof of Theorem 7.24.

Performing inferences using the basic folding rule does not affect the completeness

(of basic folding) with subsumption, since the “real” reasoning is still done using the

basic superposition calculus.




After cancelling out the common atoms, the relation (2) reduces to

nvp(t[y1, . . . , ym]σ) > nvp(E t(x0, x1ρ , . . . xnρ)) (3)

One of the conditions for the term t from the folding definition is that its variables

x1, . . . , xn are below the positions of variables of t. More precisely, for each xi where

i ∈ 1, . . . , n, and its position pi in t, exactly one of the following holds:

1. there exists a variable y j , j ∈ 1, . . . , m in t, such that pi is below its position.

Since the variables y j do not appear in vars(C ) and the domain of σ is vars(C ),

it follows that y j σ = y j = xiρ, which means that xiρ is a variable;

2. there exists a variable z ∈ y1, . . . , ym in t, so that pi is below its position pz.Let pi = pz.q . From the definition of basic folding, tσ = t ρ, which implies that

xiρ = z σ |q. Thus, if xiρ is non-variable, then zσ |q, is non-variable, too.

The previous two items show that the non-variable terms x1ρ , . . . , xnρ exist in t at

disjoint positions. Added that the top position of the term t is non-variable and that

t is not appear in E t(x0, x1ρ , . . . xnρ), it can be derived that the left hand side of (3)

is indeed greater than its right hand side.

What follows is the main result of the section. Note that well-foundedness of the

relation “a folding descendant or a proper subsumer of” implies well-foundedness of

the relations “a proper subsumer of“ and “a folding descendant of”.

Lemma 7.12 (well-foundedness). The relation “a proper subsumer or a folding de-

scendant of” is well-founded.

Proof. Every closure C · σ can be assigned a complexity measure p, n, m, denoted

by cm(C ). The elements of cm(C ) are:

• the sum of non-variable positions in literals of Cσ, denoted by p,

• the number of literals in C , denoted by n, and

• a multiset of integers k1, . . . , km, denoted by m, where var(Cσ) = x1, . . . xm

and each xi occurs ki times.

The lexicographic order on such triples, defined over well-founded relations >, >

and >mul on the components of cm(C ) is well founded. Let a closure C · σ be

either a proper subsumer of C · σ or its folding descendant. The following shows that

cm(C ) < cm(C ).




• Assume that C · σ is a proper subsumer of C · σ, with an injection ϕ and a

substitution η (see Definition 3.4.4). Let L be a literal of C . By Definition 3.4.4,ϕ(L)σ = Lση and non-variable positions in ϕ(L) are non-variable positions in

L, too. For every variable x and substitution σ , the relation nvp(x) ≥ nvp(xσ)

holds true. Therefore,

nvp(ϕ(L)σ) = nvp(Lση) ≥ nvp(Lσ) (1)

i.e. the number of non-variable positions in ϕ(L)σ is greater or equal to the

number of non-variable positions in Lσ. From Definition 3.4.4, it follows that

the number of literals in a subsumer is not greater than the number of literals

in the subsumed clause. Bearing this in mind, the relation (1) rewrites to

π1(cm(C )) ≥ π1(cm(C )) (2)

However, since (2) is not a strict relation, it is necessary to look at other com-

ponents of the complexity measures of C and C . Lets closely examine some

properties of the previously introduced substitutions ϕ and η.

– If ϕ is not a bijection it follows that π2(cm(C )) < π2(cm(C )), and the

lemma holds.

– If the mapping ϕ is a bijection, than π2(cm(C

)) = π2(cm(C )). Regardless,if η maps some variables to non-variable terms, the ≥ on the right of (1) can

be replaced by >, which implies that π1(cm(C )) < π1(cm(C )). Therefore,

the lemma holds.

– If ϕ is a bijection and η is a variable renaming substitution, πi(cm(C )) =

πi(cm(C )) for i ∈ 1, 2. Being a variable renaming substitution, η either

maps different variables to different variables, or there are at least two dis-

tinct variables that are mapped into the same variable. If the former holds,

the closures C and C are variants and therefore not proper subsumers.

Otherwise, π3(cm(C )) < π3(cm(C )) and therefore cm(C ) > cm(C ).

• Assume C · σ be the first folding descendant obtained by folding into C · σ. In

this case, the claim of the lemma follows from Lemma 7.11.

• Finally, assume that C · σ (named Def t) is the second folding descendant

obtained by folding into a term t (= L | p, for some literal L in C · σ). By the

definition of E -definitions of terms, the number of non-variable positions in the

ε-closure Def t is equal to the number of non-variable positions in t . According




to the folding definition, the term t is picked so that there is a substitution ρ

such that tσ = tρ. As shown at the beginning of the proof, for every term t andsubstitution ρ, it holds that nvp(tρ) ≥ nvp(t). Therefore, nvp(tσ) ≥ nvp(t).

According to the folding definition, the choice of the folded term t also implies

the choice of a substitution τ , for which nvp(tτ ) ≥ nv p(t) (equality holds only

in case τ is the empty substitution). It follows that

nvp(tτ σ) ≥ nvp(tσ) ≥ nvp(t) (3)

The leftmost value in (3) is less than or equal to the number of non-variable

positions of C σ, while the rightmost equals to the number of non-variable posi-

tions of C . Consequently, if τ and σ were such that at least one ≥ in (3) waspossible to replace by >, it would directly imply π1(cm(C )) < π1(cm(C )) and

therefore cm(C ) > cm(C ). This is possible if one of the following holds:

– the substitution τ is not the empty substitution, or

– the substitution τ is empty but σ restricted to the variables of t is not a

variable renaming substitution.

The relation π1(cm(C )) < π1(cm(C )) can hold true even if neither of the

previous two items is satisfied. Namely, π1(cm(C )) < π1(cm(C )) if:

– the substitution τ is empty and σ restricted to the variables of t is a

variable substitution, but either there exists a literal L1 in C such that

L1σ contains non-variable positions, or there are non-variable positions in

L1σ other than the positions at or below p;

– the position of t is below the position of an argument of the equality predi-

cate in a literal in C . This literal, and therefore the closure C , has at least

one non-variable position more than the term tτ σ (the position of the

term which contains t), which (from (6)) has at least as many non-variable

positions as t.

However, it is possible that none of the previous items hold true, meaning that

π1(C ) = π1(C ). In this case, τ and σ are, respectively, the empty and a variable

substitution, and t is at a position of an argument of equality. Moreover, all

positions of Cσ, other than the ones at or below the position of t, are variable

positions. Under these assumptions, if C is a closure that consists of only one

literal, cm(C ) > cm(C ).




At this point, in order to show that the lemma holds, consider an arbitrary

chain of closures C (= D0), Def t(= D1), D2, D3, D4, . . ., where each closure is inthe relation “a folding descendant or a proper subsumer of” with the one that

precedes it in the chain. The lemma holds (the induction hypothesis can be

applied) if there exists a closure in the chain whose complexity measure is less

than cm(C ). It can be assumed, without a loss of generality, that all closures

in the chain have the empty substitution part2. The following shows that either

there exists a finite number n such that either cm(Dn+1) < cmC , or the chain

ends with Dn. In the latter case lemma trivially holds.

There are two properties of the above chain which are important for the proof.

Firstly, π1(cm(D0)) ≥ π1(cm(D1)) ≥ · · · . Secondly, each closure contains atmost one literal with non-variable positions. Moreover, each such literal contains

only one non-variable term, which is either the term t3 or some other term

t that contains less non-variable positions. Take the smallest n for which

the closure Dn contains one such t. We show that n is a finite number, or

equivalently, that there is no infinitely many subsequent closures in the chain

that contain a variant of the term t.

If there are two subsequent closures in the chain such that they both contain

the term t, it is because of one of the following reasons:

– one closure is a subsumer of the other with an injection ϕ (see Defini-

tion 3.4.4), where ϕ, restricted on the literals that contain t , is a bijection;

– one closure is the second folding descendant of the other closure, where

both the folding and the folded terms are variants of t.

The latter can happen only once in the chain, since the same definition of a

term can not be introduced twice. The number of subsumptions in the former

case is bounded by the number of predicate literals and the number of different

proper subsumers into the clause t ≈ u (u is a variable), where the subsumers

have the same number of non-variable positions as t

. The latter is bounded bythe number of variable positions in t . Therefore, n is finite and the chain either

terminates, or the closure Dn+1 is the first folding descendant of a basic fold-

2It is easy to show that a subsumer of an ε-closure is an ε-closure. The second folding descendant

of an ε-closure is always an ε-closure. The first folding descendant of an ε-closure is of the form C ·σ

where σ is a variable substitution. However, there is a variant of such a closure Cσ · ε, which can be

used instead.3More precisely, a variant of t. Variants are identified, and therefore t can be used instead of t.




ing inference into Dn, such that π1(cm(D0)) > π1(cm(Dn+1)). Consequently,

cm(C ) > cm(Dn+1) and the induction hypothesis applies.

7.2.2 Relations m−→ and

c−→

Definition 7.13 (relations m−→,

c−→ and

mc−→). It is said that C 1

m−→ C 2 iff the

closures C 1 and C 2 are, respectively, of the form Γ, L[t] · θ and Γ, L[x], x ≈ t · θ and

• x ∈ var(C 1),

• t is not a variable, and

• if L is a negative literal, L[t] and x ≈ t are not variants.

Every term t that satisfies the above conditions is called an m−→-eligible term. It is

said that the relation m−→ takes place with the

m−→-eligible term t.

Two closures C 1 and C 2 are in the relation C 1c

−→ C 2 iff C 2 can be obtained from

C 1 by applying equality solution. It is said that the relation c−→ takes place with the

literal which is “solved” by the underlying equality solution inference.

As usual, r∗−→ denotes the transitive and reflexive closure of the relation

r−→,

where r ∈ m, c. The relation C 1m∗

−→ C 2 defines an m∗−→-chain of closures D0(=

C 1), D1, . . . , Dn(= C 2) where C i

m

−→ C i+1 for i ∈ 0, . . . n − 1.The relations

m−→ and

c−→ are introduced purely to make the completeness result

more readable. However, it is important, at this point, to understand their definitions.

While the definition of c−→ is quite trivial, it is not as straightforward with

m−→.

To understand the introduction of m∗−→, consider an application of the basic folding

inference rule into a closure C .

C

C , Def t BF

Let the closures C and C be, respectively, of the form

Γ, L[tτ ] · θ

and

Γ, L[x0], ¬E t(x0, x1, . . . , xn), ¬S τ · (ρ ∪ θ)

where ρ is calculated according to the definition of the folding rule. A resolution with

the folding descendants as premises yields the closure C of the form

Γ, L[x0], x0 ≈ t, ¬S τ · (ρ ∪ θ)




Consider now a closure C 1, which is obtained by “imitating” basic folding into C with

the same literal t. C 1 is of the form

Γ, L[x0], x0 ≈ t, ¬S τ · θ

and C m∗−→ C 1.

The difference between C and C 1 is that where there is a term t in the former

closure, it is replaced by t in the latter. From the definition of basic folding, in

particular the definition of t, it follows that C is a subsumer of C 1. The previous

consideration can be formulated as the following lemma.

Lemma 7.14. Let C m∗−→ C 1, and let closures C , C and Def t be a premise and

folding descendants of a basic folding inference. The resolvent of C and Def t is a

subsumer of C 1.

By the previous lemma and by Lemma 5.13, whenever there is a refutation which

involves C 1, there exists a refutation which involves C . This property makes it pos-

sible derive Lemma 7.20 (which is a key lemma in proving completeness of EBFP )

from the other lemmas in Section 7.2.3.

The following is a proof of a property which is used in the proof of Lemma 7.19.

Lemma 7.15. Let C ,C and C be closures such that C m−→ C and C c−→ C . Then

there is a closure D such that C c∗−→ D and C

m∗−→ D, i.e. the following diagram

commutes:

C

D

C C

········

········

m c

c∗ m∗

Proof. There are two cases to analyze. In one scenario m−→ and

c−→ are applied to

two different literals in C . In the other, they are applied to the same literal. Since

the cases are similar, the rest of the proof focuses on the slightly more complex case,

the latter one.




Let C be of the form Γ, p[t] ≈ q · θ and both m−→ and

c−→ are applied with the

literal p[t] ≈ q . ThenC : Γ, p[t] ≈ q · θ

C : Γ, p[x] ≈ q, x ≈ t · θ

C : Γ · θσ

where σ is most general unifier of p[t]θ and qθ. Let D be, by definition, the same as

C . On one hand, C m∗−→ D. On the other hand, C

c−→ D c

−→ D , where D is of

the form

Γ, p[x] ≈ q · θ[x → tθ].

Lemma 7.16. Let C , C and C be closures such that C m∗−→ C and C

c∗−→ C . Then

there is a closure D such that C c∗−→ D and C

m∗−→ D, i.e. the following diagram

commutes:

C

D

C C

·······

·

·······

·

m∗ c∗

c∗ m∗

Proof. A closure C can be assigned a reducibility measure rm(C ), defined as a pair

m, c, where m and c are defined as numbers of different closures D such that, respec-

tively, C m−→ D and C

c−→ D. Note that m can equivalently be defined as the number

of m−→-eligible terms in C (see Definition 7.13). The lexicographic combination of >

and > on non-negative integers is a well-founded relation on reducibility measures.

The proof is by induction on r m(C ). If the rm(C ) = 0, 0, the closure C is either

the empty closure or a purely predicate closure (the relation mc−→ is defined on equality

literals only). In either case, the statement of the lemma trivially holds.Assume that r m(C ) = m, c = 0, 0. By definition of the relations

m∗−→ and

c∗−→,

there exist closures C 1 and C 2 such that C 1m

←− C c−→ C 2. By Lemma 7.15, there is

a closure D so that C 1c∗

−→ D m∗←− C 2. To summarize, there exists the diagram:




C

D

C 1 C 2

C C

········

········

m c

c∗ m∗

m∗ c∗

The closure C 2 contains one negative literal less than C . From the definitions of the

reducibility measure and the relation c−→, it immediately follows that π2(rm(C )) >

π2(rm(C 2)). Also, having one literal less, the closure C 2 may contain at least onem

−→-eligible term less than C , i.e. π1(rm(C )) ≥ π1(rm(C 2)). It follows that rm(C 2) <rm(C ).

On the other hand, the closure C 1 certainly contains at least one m−→-eligible term

less than C – one such term is replaced by a variable which is not an m−→-eligible term.

This implies that π1(rm(C )) ≥ π1(rm(C 1)) and therefore rm(C 1) < rm(C ).

By the induction hypothesis, the lemma holds for the closures C 1 and C 2. There-

fore, there are closures D1 and D2 such that C c∗−→ D1

m∗←− D and D c∗

−→ D2m∗

←− C .

Using identical argument as to explain that rm(C 2) < rm(C ), it is easy to infer

that rm(D) < rm(C 1). By the induction hypothesis, there is a closure D so that

D1c∗

−→ D m∗←− D2. Equivalently, there exists the diagram

C

D

C 1 C 2

C C

D1 D2

D

········

········

········

········

········

········

········

········

m c

c∗ m∗

m∗ c∗

c∗ m∗

c∗

m∗

c∗ m∗

which is what is in the claim of the lemma.




7.2.3 Proof transformations involving superposition and m−→

This section contains statements that explain that the structure of a EBFP -derivation

does not essentially change if its premises are replaced by closures which they are in

the m−→ relation with. Bearing in mind that the relation

m−→ is motivated by the

basic folding rule (see Section 7.2.2), the main result of this section, Lemma 7.20, is

an essential building block in the completeness proof of BF .

The statement of the following lemma is often used in the remainder of this chap-

ter.

Lemma 7.17. Let C 1 and C 2 be closures such that C 2 is obtained from C 1 by applying

equality solution. If C 2 is not a tautology than C 1 is not a tautology either.

Proof. Let C 1 be of the form Γ, u ≈ v · σ and let equality solution be applied on the

literal u ≈ v. The clause C 2 is then of the form Γ · σmgu(uσ,vσ).

Assume the opposite, that C 1 is a tautology. This means that its every instance

is a tautology. Take its ground instance of the form Γρ,uρ ≈ vρ, which is also a

ground instance of Γ, u ≈ v · σmgu(uσ,vσ). This implies that the terms uρ and vρ are

identical, which further implies that the clause Γρ is a tautology itself. On the other

hand, Γρ is also an arbitrary ground instance of C 2, which is then a tautology by

definition. This contradicts the assumption that C 2 is not a tautology, and therefore

the statement of the lemma follows.

Lemma 7.18. Let C 1 and C 2 be premises of a superposition (resolution) inference

with a conclusion C 3. Assume that C 2 is a closure such that C 2m∗

−→ C 2. Then there

exist closures C 3 and C 3 such that

• C 3 is obtained by superposition (resolution) with premises C 1 and C 2,

• C 3c∗

−→ C 3 , and

• C 3m∗

−→ C 3 .

In other words, there exists a derivation of the form

C 1 C 2 C 2

C 3 C 3

C 3

m∗

m∗ c∗

·········

·································

·········

·········




If the closures C 1, C 2 and C 3 are not tautologies, then C 2, C 3 and C 3 are not tau-

tologies either.

Proof. The proof is only given for the case the inference between C 1 and C 2 is super-

position. In case of resolution the proof is analogous, even simpler. This is because

resolution takes place only on predicate literals, while, by definition, the relation m∗−→

takes place on equality literals only.

There can be two instances of superposition with the premises C 1 and C 2. In

the first scenario C 1 is the “to” and C 2 the “from” premise. In the other, which is

addressed in the remainder of the proof, the premises are taken the other way round.

Assume that the closures C 1, C 2 and C 3 are of the form

C 1 : Γ1, p ≈ q · θ1

C 2 : Γ2, u[s] ≈ v · θ2

C 3 : Γ1, Γ2, u[q ] ≈ v · (θ1 ∪ θ2)ρ

where ρ is a most general unifier of pθ1 and sθ2.

The proof is by induction on the length of the m−→-chain from C 2 to C 2. In the

base case the length of the m−→-chain is 1. Let C 2

m−→ C 2, so that the folded term t in

C 2 is replaced by x, which results in the introduction of the literal x ≈ t to C 2. The

folded term t either does or does not belong to the literal u ≈ v (the “into” literal of

the superposition inference between C 1 and C 2). The latter case is straightforward,

and hence the rest of the proof considers the former.

Denote the position of s in u by p s, and the position of t in the same occurrence of

u as pt. The two positions can be disjoint (this case is trivial as well) or can overlap.

If pt ≤ ps, i.e. ps = pt.p for some p, then C 2 is of the form

Γ2, u[x] ≈ v, x ≈ t[s] · θ2

where u[x] is obtained after replacing the term t at the position pt by x and where

t |π= s. An application of superposition from C 1 into C 2, with the newly generated

literal x ≈ t[s] as the “to” literal, yields C 3, which is of the form

Γ1, Γ2, u[x] ≈ v, x ≈ t[q ] · (θ1 ∪ θ2)ρ

where ρ is a most general unifier of pθ1 and sθ2 (the same as in the conclusion of

superposition on C 1 and C 2).

The term t[q ] can be a variable or non-variable term. In case t[q ] is not a variable,

consider the literal u[q ] ≈ v of C 3. Since the position of q is ps and pt ≤ ps, the term

u[q ] can also be written as u [t[q ]]. By replacing the term t[q ] by a fresh variable, it is

obvious that C 3m

−→ C 3 . If C 3 is the same as C 3, the statement of the lemma follows.




If t[q ] is a variable, then q has to be a variable, and t[q ] = q ( p = ε). Define C 3 as

C 3. Then C 3c

−→ C 3 by applying equality solution on x ≈ t[q ].Assume now that ps < pt, i.e. pt = ps.p for some p = ε. Then C 2 is of the form

Γ2, u[s[x]] ≈ v, x ≈ t · θ2

where s[x] is obtained by replacing the term t by x in s at the position pt. An

application of superposition from C 1 into C 2 yields C 3 of the form

Γ1, Γ2, u[q ] ≈ v, x ≈ t · (θ1 ∪ θ2)ρ

where ρ is the most general unifier of pθ1 and sθ2. The closure C 3 , obtained from

C 3 by equality solution on the literal x ≈ t, is of the form

Γ1, Γ2, u[q ] ≈ v · (θ1 ∪ θ2)ρσ

where σ is the most general unifier of x(θ1 ∪ θ2)ρ and t(θ1 ∪ θ2)ρ. By definition,

C 3c

−→ C 3 , with C 3 being a variant of C 3.

A note on the claim of the lemma concerning tautologies. From the definition of

the relation m−→ and from the fact that neither C 3 nor C 2 is a tautology, it follows

that C 3 and C 2 can not be tautologies either. Simultaneously, by Lemma 7.17, since

C 3 is not a tautology C 3 is not a tautology either.

To prove the induction step, assume that n > 1. The diagram from the statementof the lemma can be rewritten as:

C 1 C 2 C 21 C 2

C 3 C 31 C 3

C 31 C 32

C 3

sup sup sup

m∗ m

c∗ m∗

m∗ c∗

m∗ c∗

·························································

·································

·········

·········

·········

·········

·········

·········

·········

There exists a closure C 21, such that C 2m∗

−→ C 21m

−→ C 2. By the induction hypothesis,

there exist clauses C 31 and C 31 such that

• C 31 is obtained by applying superposition from C 1 into C 21,

• C 31c∗

−→ C 31, and




• C 3m∗

−→ C 31.

From the consideration on the base of induction, there exist C 3 and C 32 such that

• C 3 is obtained by a pseudo application of superposition from C 1 into C 2,

• C 3c∗

−→ C 32, and

• C 31m∗

−→ C 32.

By Lemma 7.16, there exists a closure C 3 such that C 31m∗

−→ C 3 and C 32c∗

−→ C 3 .

From C 3m∗

−→ C 31m∗

−→ C 3 and C 3c∗

−→ C 32c∗

−→ C 3 it follows that C 3m∗

−→ C 3 and

C 3c∗

−→ C 3 . This proves the part of the lemma that addresses the existence of the

diagram. The part of the claim about tautologies follows in the way similar to what

has been said in the base case. By the induction hypothesis, the closures C 21, C 31 and

C 31 are not tautologies. From the definition of the relation m∗−→, it follows that C 2,

C 32 and C 3 are not tautologies either. Since C 32 is not a tautology, by Lemma 7.17,

the clause C 3 can not be a tautology, which completes the proof.

Lemma 7.19. Let Ω be a regular, tautology-free e-refutation of S ∪ C by EBFP .

Let the conclusion of Ω be a closure D and let C 1 be a closure such that C m∗−→ C 1.

Then there exists a regular, tautology-free EBFP -e-refutation Ω of D from S ∪ C 1,

such that Ω contains as many superposition inferences as Ω.

Proof. The proof is by induction on the number of inferences in the derivation tree.

The base case, when Ω consists of only one closure C (and therefore no inferences) is

trivial. In that case C = D = C 1.

Assume now that Ω contains at least one inference. Since Ω is a derivation from

S ∪ C , the closure C appears in Ω as a leaf. There are three possible inference rules

applicable to C : superposition, equality solution and resolution.

Assume that the rule applied to C is a superposition with a closure C 2. Then Ω

is of the formΩ2

C 2 C D1

(sup)....

D

By Lemma 7.18 there exist closures D1, D

1 such that

• D1 is obtained by an application of superposition of C 1, C 2,

• D1

c∗−→ D

1 ,




• D1m∗

−→ D 1 , and

• neither D1 nor D

1 is a tautology.

C 2 C C 1

D1 D1

D1

supsup

m∗

m∗ c∗

·········

·································

·········

·········

Consider the subderivation Ω1 of Ω obtained by replacing the derivation of D1 by D1:

D1....D

The number of inferences in this derivation is smaller than the number of inferences

in Ω. The derivation Ω1 is regular and contains no tautologies, D1m∗

−→ D 1 , and D

1 is

not a tautology. Therefore, Ω1 is eligible for application of the induction hypothesis,

which results in a derivation Ω 1, which is of the form

D1....D

and has the same multiset of closures in leaves as Ω 1 except that the leaf D1 is replaced

by D 1 . This derivation is also regular and contains no tautologies. The derivation Ω

1

can be extended in the following way

Ω2C 2 C 1

D1

(sup).

...D

1....D

where the subderivation of D1 from D

1 consists of equality solutions. The obtained

derivation is a regular and tautology-free derivation Ω . It is apparent that Ω contains

the same number of applications of superposition as Ω.




such that D m∗−→ D1. The closure D is not a tautology, which implies that neither is

D1 nor D. The former because of the property of the relation m

−→, and the latterbecause it is a subsumer of D1 (all non-variable positions in t are non-variable in t

and tθ = tσ).

By Lemma 7.19, since D m∗−→ D1, there exists a regular, tautology-free EBFP -

derivation Ω of B from S n−1 \ D ∪ D1 with the same number of superposition

inferences as Ω. Having that D is a subsumer of D1, and using Lemma 5.11, it

follows that there exists a regular, tautology-free EBFP -derivation of B from S n−1 \

D ∪ D. Because D is derived from D and Def t , it follows that there is a

tautology-free EBFP -derivation Ω of B from S n−1 \ D ∪ D, Def t(= S n).

Note that Ω is not a regular derivation. Nevertheless, by applying Lemma 5.9

to Ω, it follows that there exists a regular EBFP -derivation of B, which contains

no tautologies and has the same number of superposition inferences as Ω .

The base of induction is trivial – it is the case when the length of the derivation

∆ is 0.

7.2.4 Proof transformations involving c−→

Lemma 7.20 from the previous section is the first necessary ingredient necessary to

show completeness of BF . The rest is contained within the statement of Lemma 7.22,

given below.

Lemma 7.21. Let S 0 and S ω be, respectively, a set of closures and the limit of a fair

BF -derivation S 0 ⇒ S 1 ⇒ · · · (denoted by ∆). Assume that C 1 ∈ S ω is of the form

Γ, s ≈ t · θ

where Γ does not contain equality literals. Assume also that C 2 is the conclusion of

equality solution on C 1, that it is not a tautology, and that it is of the form

Γ · θmgu(sθ,tθ).

Then either C 2 or its proper subsumer belongs to S −ω .

Proof. By fairness of ∆, there exists S j such that C 2 ∈ S j. If C 2 is also in S ω the

lemma holds true.

Otherwise, there is k ≥ j such that C 2 ∈ S k, S k ⇒ S k+1, and C 2 ∈ S k+1. It follows

that S k+1 is obtained from S k either by applying a deletion rule to C 2 or basic folding.

However, basic folding can not be applied, since C 2 is a purely predicate clause. Since,

by assumption, C 2 is not a tautology, the only applicable rule on C 2 is subsumption.




If s1 ≈ t1 is an o-literal, then π1(cm(C )) < π1(cm(C )) and lemma follows by the

induction hypothesis. Let s1 ≈ t1 be an f-literal and assume accordingly, without aloss of generality, that t1 is a variable y. It follows immediately that π3(cm(C )) >

π3(cm(C )). Obviously, the number of o-literals in C equals π1(cm(C )). As for

π2(cm(C )), it is not greater than π2(cm(C )), but neither is it necessarily smaller. To

explain this, notice that the substitution part of C , θρ = θ ∪ [y → s1ρ]. Depending on

whether s1θ contains non-variable positions or not, nvp((s2 ≈ t2, . . . , sn ≈ tn)θρ) can

be less then or equal to π2(cm(C )). Therefore, cm(C ) > cm(C ) and the induction

hypothesis applies.

If C ∈ S ω, then (see Lemma 7.12) there exists a finite sequence of sets S 0, S 1, . . . , S n

where S 0

= C and S i is obtained from S

i−1 (for i ∈ 1, . . . , n) either by replac-

ing a closure from S i by its folding descendants or by its subsumer. If none of the

sets in this sequence (named Π) is obtained by replacing a closure with its folding

descendants, then S n contains only one closure C . By the definition of subsumers,

π1(cm(C )) ≥ π1(cm(C )), π2(cm(C )) ≥ π2(cm(C )) and π3(cm(C )) ≥ π3(cm(C )).

It follows that cm(C ) ≥ cm(C ), and since C ∈ S ω, the proof can be finished by

applying the argument from the previous paragraph.

Finally, let Π contain a set which is obtained by replacing a closure by its folding

descendants. Take the first such set, i.e. the one with the lowest index, and assume

it is S l. All sets that precede in the chain Π are obtained by replacing closures by

their subsumers. Consequently, the set S l−1 contains only one closure C , which is a

subsumer of C . As seen in the previous paragraph, cm(C ) ≥ cm(C ). Let C and

Def t be the folding descendants of C . By the definitions of basic folding and the

complexity measure, π1(cm(C )) ≥ π1(cm(C )). On the other hand, as shown in the

proof of Lemma 7.11, π2(cm(C )) > π2(cm(C )). By the induction hypothesis, there

exists a proper subsumer D of C , which can be obtained from S ω by resolution.

Obviously, the closure D may contain the literal E t(x0, . . . , xm). As it has already

been mentioned, the second descendant of C is of the form

E t(x0, . . . , xm), x0 ≈ t · ε

and therefore satisfies the conditions of Lemma 7.21. It follows that there exists a

closure B ∈ S −ω of the form

E t(x0, x1, . . . , xm) · [x0 → t].

If D contains an occurrence of the predicate E t(x0, . . . , xm), by applying resolution

with premises D and B, a closure is obtained which does not contain occurrences

of E t(x0, . . . , xm). Therefore, its conclusion is a subsumer of Γ · θρ and belongs to




S −ω . Otherwise, if D does not contain occurrences of E t(x0, . . . , xm), it is already a

subsumer of Γ · θρ and therefore D ∈ S −ω .

7.2.5 The completeness statement

Theorem 7.24 is the completeness result for BF . The following lemma is an interme-

diate step.

Lemma 7.23. Let ∆ be a BF derivation S 0 ⇒ S 1 ⇒ · · · from a set of initial closures

S 0 with a limit S ω. If there is a correct answer for S 0 then there is a correct answer

for S ω obtained by a regular, tautology-free EBFP -derivation.

Proof. Let t = t1, . . . , tn be a correct answer for S 0. By definition of a correctanswer to a set of clauses, S 0 query(x1, . . . , xn) · σ and query(t) is an instance of

query(x1, . . . , xn) · σ . By compactness, there is a finite subset C 1, . . . , C n ⊆ S 0

such that C 1, . . . , C n query(x1, . . . , xn) · σ. Consider a set S of clauses D ∈ S ω

obtained by repeated application of the basic folding and basic subsumption rules to

the closures from C 1, . . . , C n. From Lemma 7.12 and Konig’s Lemma it follows that

S is finite. Also, as it has been mentioned in the proof of Lemma 7.12, each closure in

S is such that its substitution part equals to the empty substitution. It is apparent

that set S C i, for all i ∈ 1, . . . , n. As a consequence S ω C i, for all i ∈ 1, . . . , n

and thus S ω query(x1, . . . , xn) · σ. By definition, ¯t is a correct answer for S ω, and

the set S ω ∪ ¬query(x1, . . . , xn) · ε is inconsistent.

By Theorem 5.14, there exists a regular, tautology-free EBFP -refutation Ω of

the set S ω ∪ ¬query(x1, . . . , xn) · ε. Equivalently, there exists a set of purely

predicate clauses S obtainable by equality inference rules of Ω, for which S ∪

¬query(x1, . . . , xn) · ε is inconsistent. This also means that there is a derivation

of query(x1, . . . , xn) · σ1 from S by applying predicate inference rules. Any ground

instance of the tuple x1σ1, . . . , xnσ1 is a correct answer to S , and therefore to S ω.

To conclude, a regular and tautology-free derivation of query(x1, . . . , xn) · σ1 from

S ω is obtained by putting together the derivation of S (from S ω) and the derivation

of query(x1, . . . , xn) · σ1 (from S ).

Theorem 7.24. Let ∆ be a fair BF -derivation S 0 ⇒ S 1 ⇒ · · · from a set of initial

closures S 0, and let S ω be a limit of ∆. If S 0 has a correct answer then S −ω has a

correct answer, too.

Proof. From the assumption that S 0 has a correct answer, by the previous lemma, it

follows that S ω has a correct answer. Denote it by t = t1, . . . , tn. By Lemma 7.23,




there exists a regular, tautology-free EBFP -derivation Ω from S ω of a clause

query(x1, . . . , xn) · σ , such that query(t1, . . . , tn) as its instance. The existence of Ω implies that there exists a set of purely predicate clauses S derivable from S ω

by equality inferences only, such that query(x1, . . . , xn) · σ is derivable from S by

resolution.

Generally, S ⊆ S ω. To prove the lemma, it is sufficient to prove that there exists

a set S ⊆ S ω such that

• S is a set of purely predicate closures obtained by applying equality inference

rules of EBFP ,

• query(x1, . . . , xn) · σ is derivable from S

by resolution, and

• query(t1, . . . , tn) is an instance of query(x1, . . . , xn) · σ.

Note that the derivation which consists of the derivation of S from S ω and the

derivation of query(x1, . . . , xn) · σ from S is a regular derivation. Denote it by Ω. It

can be shown, by induction on the number of superposition inferences in Ω, that Ω

can be transformed into Ω.

If Ω contains no superposition inferences, Lemma 7.22 implies that S = S −ω and

that query(x1, . . . , xn) · σ can be derived from S by resolutions only. Equivalently,

S −ω query(x1, . . . , xn) · σ and t is a correct answer for S −ω .

Assume that there are superposition inferences in Ω, and that Ω exists for every

derivation Ω which contains less than n applications of the superposition inference

rule. Consider now a superposition in Ω, such that there are no other superposition

inferences preceding it. Let the premises and the conclusion of this superposition

inference, denoted by sup, be closures C 1, C 2 and C , respectively.

Assume that C 1 and C 2 are leaves of Ω, and therefore elements of S ω. The

conclusion of sup C may not be in S ω. If it is in S ω, the sub-derivation of Ω which

does not contain the inference sup is a derivation with n − 1 superposition inferences

and the induction hypothesis applies.

If C is not in S ω, from fairness of ∆, C ∈ S i, for some i ≥ 0. The reason it isnot in S ω is that it is replaced by either its folding descendants or discarded (in ∆)

because of the presence of its subsumer. More precisely, there is a set F of closures,

all in the (transitive closure of) relation “a subsumer or a folding descendant of”

with C , such that F ⊆ S ω and S ω C . By Lemma 7.12, F is finite. Therefore,

by Lemma 7.20, there exists a regular, tautology-free derivation of (a subsumer of)

query(x0, . . . , xn) · σ from F ∪ S ω with n − 1 superpositions. Since F ∪ S ω = S ω, the

induction hypothesis applies.



7.3 Completeness of basic folding of general clauses 131

Finally, assume that C 1 and C 2 are not leaves of Ω. Then there exist closures C 1

and C 2 which are leaves of Ω (and therefore in S ω, such that Ω is of the form

C 2.... (es)C 2

C 1.... (es)C 1

C (sup)

Consequently, Ω can be transformed to a derivation Ω of the form

C 1 C 2C

(sup).... (es)

C

The closure C is not a tautology since Ω is a tautology-free derivation. By Lemma 7.17

neither clause between C and C , including C , is a tautology. Therefore, the deriva-

tion Ω is regular with the same number of superpositions as Ω, and tautology-free.

Because the premises of sup are C 1 and C 2 (which are in S ω, the consideration reduces

to the previous case.

7.3 Completeness of basic folding of general clauses

A careful analysis of the result presented in the previous chapter suggests that the

result of refutational completeness of the basic folding calculus can be extended to

the general first-order case. Namely, the definition of the basic folding rule does not

require the clauses to be Horn, and, additionally, basic superposition is refutationally

complete in the general case. It is therefore intuitive to conjecture that basic folding

can be extended to the general first-order case. In order to test this conjecture the

following question has to be asked: which basic superposition calculus to chose for

a definition of the basic folding calculus of general clauses? This choice is perhaps

dependant upon the proof method that is going to be used for proving completeness

of the respective calculus. One choice is to reuse, as much as possible, the proof of Theorem 7.24, trying to adapt it so that it works for general first-order clauses. By

analysing the proof, it can be seen that its basic building block is the result about

completeness of the regular strategy of derivations with tautology elimination and

subsumption. As shown in Chapter 5, the regular strategy is complete for the calcu-

lus EBFP , which, amongst others, contains the tautology deletion and subsumption

rules. To conclude, many preconditions are satisfied for attempting a refutational

completeness proof of EBFP , plus the basic folding rule, by using the same route as




in Theorem 7.24. The first obstacle on this way turns out to be proving an equivalent

of Lemma 7.18 for the calculus EBFP . Consider the lemma again and assume thatevery reference to superposition is a reference to the basic factored overlap calculus

of EBFP .

Lemma 7.18 (revisited) Let C 1 and C 2 be premises of a superposition (resolution)

inference with a conclusion C 3. Assume that C 2 is a closure such that C 2m∗

−→ C 2.

Then there exist closures C 3 and C 3 such that

• C 3 is obtained by superposition (resolution) with premises C 1 and C 2,

• C

3

c∗

−→ C

3 , and

• C 3m∗

−→ C 3 .


C 1 C 2 C 2

C 3 C 3

C 3

m∗

m∗

c∗

·········

·································

·········

·········

If the closures C 1, C 2 and C 3 are not tautologies, then C 2, C 3 and C 3 are not tau-

tologies either.

Proof. (failed revision for the first-order case)

Assume that the closures C 1, C 2 and C 3 are of the form

C 1 : Γ1, p1 ≈ q 1, . . . , pn ≈ q 1 · θ1

C 2 : Γ2, u[s] ≈ v · θ2

C 3 : Γ1, Γ2, u[q 1] ≈ v, . . . , u[q n] ≈ v · (θ1 ∪ θ2)ρ

where ρ is a simultaneous most general unifier of pθ1 and sθ2, . . . pθ1 and sθ2.

The proof of the lemma is given by induction on the length of the m−→-chain from

C 2 to C 2. In the base case the length of the m−→-chain is 1. Let C 2

m−→ C 2, so that

the folded term t in C 2 is replaced by x, which results in the introduction of the literal

x ≈ t to C 2. Assume that the folded term t belongs to the literal u ≈ v (the “into”

literal of the superposition inference between C 1 and C 2), and that the position of s




in u is the same as the position of t in u (i.e. the terms s and t coincide), denoted by

ps. In this case C 2 is of the form

Γ2, u[x] ≈ v, x ≈ s · θ2

where u[x] is obtained after replacing the term s at the position ps by x and where,

again, t = s. An application of superposition from C 1 into C 2, with the newly

generated literal x ≈ t as the “to” literal, yields C 3, which is of the form

Γ1, Γ2, u[x] ≈ v, x ≈ q 1, . . . , x ≈ q n · (θ1 ∪ θ2)ρ

where ρ is a simultaneous most general unifier of p1θ1, pnθ1 and tθ2 (the same as in

the conclusion of superposition on C 1 and C 2).

In order for the diagram (in the formulation of the lemma) to commute, the closure

C 3 should be a result of applying some sort of factoring of the positive literals u[q 1] ≈

v , . . . , u[q n] ≈ v in the clause C 3. However, the diagram allows that C 3c

−→ C 3 , which

means that C 3 has to be obtained from C 3 by applying equality solution.

The conclusion is: the proof of Theorem 7.24 does not work if the basic superpo-

sition calculus consists of EBFP plus the basic folding rule. The reason is that the

conclusion of the superposition inference between C 1 and C 2 contains positive liter-

als, which should be factored upon in order for the diagram to commute. A question

that arises is: how to enable factoring and prove Lemma 7.18 for the full first-order

case? The answer perhaps lies in replacing EBFP by another calculus. This calculus

could contain an explicitly defined factoring rule, or alternatively a superposition rule

that handles factoring in a way similar to the way basic factored overlap rule does.

In order to answer this question, it is necessary to take a deeper look into different

approaches to factoring in basic superposition calculi.

Ordered factoring

Consider first the o-factoring rule, introduced in [HR86] as an inference rule of their

ordered paramodulation calculus.

Definition 7.25 (o-factoring, [HR86]). If L1, . . . , Lk are literals of a clause C and

are unifiable with mgu σ, and for every other atom A ∈ C, L1σ Aσ, then D =

Cσ − L2σ , . . . , Lkσ is an o-factor of C .

The restrictions that the authors pose to applying the rule is that “it is only

applied to clauses on which some inference is about to be applied”. The same rule is




given in [PP91], under the name maximal factoring inference rule . A slightly stronger

version of this rule, in which factored literals can only be positive literals, is definedin [BG90] as positive factoring inference rule.

Importantly for the matters discussed in this chapter, [BG98b] contains a com-

pleteness result for a basic superposition calculus that contains the ordered factoring

rule (the original name of the calculus is strict basic superposition ). Unfortunately,

the authors of [BG98b] also show that strict basic superposition is not compatible

with tautology elimination, which implies that the calculus can not be used to extend

the result formulated in Theorem 7.24 to the first-order case. Completeness with

tautology elimination of the basic folding calculus, and therefore the underlying basic

superposition calculus, is one of the main requirements for this extension of basic

folding to the first-order case.

Equality factoring

An alternative approach to factoring in framework of basic superposition is addressed

in [NR92b]. They define another factoring rule, called equality factoring , as “a gen-

eralization, to the equality case, of normal factoring” (normal factoring refers to the

approach to factoring taken in resolution calculi).

Definition 7.26 (equality factoring, [NR92b]). Equality factoring is the following

rule:Γ, s ≈ t, s ≈ t

Γ, t ≈ t, s ≈ t

where s = s, s ≈ t is the maximal literal in the premise, and s t, s t.

The link with “normal” factoring can be explained in the following way: if s and

s are atoms, then both t and t can be the symbol true and the negative equation

true ≈ true can be omitted from the conclusion. The same rule later appeared in

[BGLS95].

The result [NR92b] shows that basic superposition with equality factoring is com-

plete with basic subsumption, basic simplification and elimination of tautologies in

the basic setting, which makes it a good candidate for the calculus of choice for

showing completeness of basic folding in the general first-order case. However, it is

very difficult to prove completeness of the regular strategy of derivations by basic

superposition with the equality factoring rule. Precisely, it is impossible to derive

a completeness proof based on transformations of derivation trees, as it is done in

Chapter 5. This is best explained by the following example.




Example 7.27. Before going any further, recall that by the regular strategy of deriva-

tions, all equality inferences precede all other inferences, including resolution. Con-sider the following derivation:

c ≈ d

a ≈ b, P (a), a ≈ c P (a), a ≈ d

a ≈ b, a ≈ c, a ≈ d (res)

a ≈ b, c ≈ d, a ≈ d (eq fac)

a ≈ b, d ≈ d, a ≈ d (sup)

and assume that a b c d. Here an application of resolution precedes a

superposition inference and an application of factoring. In order to regularize this

derivation, it would be necessary to push the resolution inference down the derivation

tree, so that it appears after superposition. In a derivation transformed this way

the literals a ≈ c and a ≈ d would not appear in the same clause, and therefore it

would not be possible to apply factoring. The only way the transformed derivation

could have the same conclusion as the original one, superposition would have to take

place into the smallest term of the literal a ≈ c, which is forbidden by definition of

superposition: superposition inferences always take place with the biggest terms of

the selected literals.

Without having completeness of regular derivations by basic superposition, it

is not possible generalise Theorem 7.24 to the first-order case. Considering basic

superposition with equality solution is therefore not a suitable approach.

Factoring embedded in paramodulation-based rules

Finally, consider the approach to factoring used in the calculus EBFP , originated by

[MLS95], where factoring is merged with basic superposition in the same rule, called

basic factored overlap. The superposition calculus that contains the rule, which is

specially designed for improving efficiency of methods that combine superposition and

goal-directed methods (like model elimination), is complete with basic simplification,

basic subsumption and tautology removal in the basic setting. It has been shown,

however, that it looses these nice features when combined with the model elimination

method. Although the presence of the basic factored overlap rule eliminates the need

for a factoring inference rule, it effectively does not factor eligible positive literals.

Instead, it uses them simultaneously in superposition inferences, thus postponing

factoring till the end of the proof. At the opening of this section, it has been shown

that this approach to factoring also fails in extending Theorem 7.24 to the first-order

case.




A solution to the problem is in defining a superposition calculus which does not

contain an explicitly defined factoring rule, but simultaneously treats factoring in away different to the basic factored overlap rule of [MLS95]. Consider the following

modification of the basic factored overlap rule.

Definition 7.28 (basic re-factored overlap inference rule). The rule is defined as

follows.l1 ≈ r1, . . . , ln ≈ rn, Γ1 | T 1 s[l] t, Γ2 | T 2

s[x] t, x ≈ r1, . . . , x ≈ rn, Γ1, Γ2 | T 1 ∧ T 2 ∧ δ (sup)

where

• x is a fresh variable,

• δ stands for (l1 r1 ∧ . . . ∧ ln rn ∧ s t ∧ l1 = l ∧ . . . ∧ ln = l), and

• ∈ ≈, ≈.

Opposite to the basic factored overlap rule of [MLS95], which postpones factoring

until the end of the proof, the above rule “converts” factoring into equality solution.

In other words, the rule always transforms unifiable positive literals into unifiable

negative literals, and thus truly avoids factoring. It turns out that this rule is possible

to integrate with the proof of Theorem 7.24, which is shown in the remainder of this

section.

7.3.1 Calculus EBMP : a new approach to factoring

Consider replacing the basic factored overlap rule of EBFP by the basic re-factored

overlap rule given by Definition 7.28. Name thus obtained calculus EBMP , and its

subset that contains basic re-factored overlap plus equality solution BMP .

Lemma 7.29. BMP is refutationally complete.

Proof. The following shows how the model generation completeness proof given in

[BGLS95] needs to be altered to work for BMP .

Let S be a set of clauses which is saturated by the rules of BMP . Take all ground

instances of the clauses in S , and denote that set by S . Let C be a ground instance

Γ, l ≈ r of a clause C . The equation l ≈ r is called productive, and E C = l ≈ r if

• C is not true in R∗C ,

• l r and l ≈ r is (non-strictly) greater then all literals in Γ,

• l is irreducible by RC , and




• the terms in C which are at positions at which there are variables in C are not

reducible by equations in RC (C is reduced with respect to RC ).

The set RC denotes the union of all E D for D ≺ C , while R∗C denotes the congruence

defined by RC .

The goal is to show that either ∈ S or there is a model for S . Assume now the

opposite, that S does not have a model and that ∈ S . Let R be the union of E D

for all D ∈ S , and let R∗ be the corresponding congruence. Since, by assumption,

R∗ is not a model of S , there must exists the smallest counterexample C ∈ S (with

respect to the clause extension of a given term ordering ) for R∗. It follows that C

is not a productive clause. Note that C must be reduced with respect to R. To show

this, one should first prove that if C is reduced with respect to RC then it is also

reduced with respect to R (a proof can be found in [BGLS95]). Then, if C (which

is a ground instance of C ) was not reduced with respect to RC (and therefore R),

there would exist a clause C , which is reduced a ground instance of C and reduced

with respect to RC (and hence R).

Assume that C is not productive because its greatest literal is negative. It can

be assumed that C is of the form Γ, s ≈ t. Then, either s ≡ t, and therefore equality

solution on C yields a smaller counterexample then C (the clause Γ which is reduced

with respect to R), or s t (w.l.o.g.) and s is reducible by some rule in RC . This

means that there exists a clause C

, smaller than C

, which contributes to RC

withan equation l ≈ r. Assume that C is of the form Γ, l ≈ r1, . . . , l ≈ rn, so that

r1 ≈ ri is true in R∗C for all i ∈ 1, . . . , n, and that for all other literals l ≈ w in

Γ, they are not true in R∗C . Then there is an inference from C into C with the

conclusion DΓ, l ≈ r1, . . . , l ≈ rn Γ, s[l] ≈ t

Γ, Γ, s[u] ≈ t, u ≈ r1, . . . , u ≈ rn

(sup)

such that u is a term which r1 can be reduced to by the equations from RC (the

rewrite system RC is convergent, which is also shown in [BGLS95], and therefore

every term can be rewritten to its canonical form - for r1 it is u). The term u is

arbitrarily chosen because its position is a position of a fresh variable in the conclusion

of the superposition inference.

Γ and Γ are false in, respectively, R∗C and R∗

C . Since every equation in R\RC is

greater then equations in RC , Γ is also false in R∗ (which is proved in [BGLS95]). The

same holds for Γ. Simultaneously, l ≈ r1 and r1 ≈ u are true in R∗. Because s[l] ≈ t

and u ≈ ri, for i ∈ 1, . . . , n, are all false in R∗, the clause s[u] ≈ t, u ≈ r1, . . . , u ≈ rn

has to be false in R∗. Therefore, D is false in R∗.




By assumption, C is reduced with respect to RC . It follows that it is also reduced

with respect to R. As for C , it is also reduced with respect to R∗, except for theliterals l ≈ ri, for i ∈ 1, . . . , n. Consequently, the clause D has to be reduced with

respect to R∗.

Having that D ≺ C , and D is a reduced clause, which is also a counterexample

for R∗, the initial assumption that C is the smallest counterexample for R∗ is wrong.

Finally, assume that C is a minimal counterexample for R∗ and that it is of the

form Γ, l ≈ r . Assume also that l ≈ r is (non-strictly) greater than any other literal

in Γ and that l is reducible by some rule in R. Similarly to the previous case, there

exists a superposition inference into C , which yields a clause that is a counterexample

for R∗, smaller than C and irreducible.

Using a similar argument as in Section 3.6 for showing that EBFP is complete,

the following holds.

Lemma 7.30. EBMP is refutationally complete.

Note that EBMP is given as a calculus of constrained clauses with ordering literal

inheritance. Section 3.6 explains that, if the ordering constraints are taken out of the

constraints and treated as conditions for applying the corresponding inference rule, it

is possible to obtain a complete basic calculus of closures. Therefore, it can be assumed

that there exists a complete calculus EBMP of closures, which is used throughoutthis chapter.

By looking at the proof of completeness of BF , it is apparent that it heavily relies

on completeness of the regular strategy of derivations by EBFP . It order to apply a

transformation method similar to the one used in the previous section, it is necessary

to show that EBMP is also compatible with the regular strategy.

Lemma 7.31. Any unsatisfiable set of flat clauses has a regular refutation by EBMP

of closures, in which tautologies and subsumed closures are redundant.

Proof. Regular derivations by EBMP for constrained clauses can be shown complete

by simply following the proof of Theorem 5.14. The completeness of the regular

strategy for EBMP of closures follows from the consideration given in Section 5.3.

7.3.2 Proof of completeness

It has been pointed out, at the very beginning of this chapter, that the definition of

completeness of basic folding in general case differs from the definition of the basic

folding calculus of Horn clauses (see Definition 7.7). Bearing in mind that basic folding




has equality elimination in its base, i.e. that the aim is to obtain a finite set of purely

predicate closures from a set of closures with equality, refutational completeness isdefined in the following way.

Definition 7.32 (completeness). Let S 0 ⇒ S 1 ⇒ · · · be a fair derivation by a calculus

I and let S ω be its limit. The calculus I is complete if whenever S 0 is unsatisfiable

then S −ω is unsatisfiable, too.

Basic folding calculus for general first-order closures, denoted by BMF , is the

calculus that contains the rules of EBMP (see Section 7.3.1) plus the basic folding

rule introduced in Section 7.1. The rest of this chapter focuses on proving refutational

completeness of BMF .

The completeness proof of BMF very similar to the proof given for basic folding of

Horn clauses. The structure of the proof is the same and every lemma from Section 7.2

can be reused. However, lemmas that address the calculi EBFP and BF should be

rephrased to address the calculi EBMP and BMF . The proofs of these lemmas

also have to be changed in case they describe transformations of derivations which

involve applications of superposition. In particular, these are the lemmas given in

Section 7.2.3 and Section 7.2.5. The following gives proofs of these lemmas.

First-order equivalents of lemmas from Section 7.2.3

Lemma 7.33 (the equivalent for first-order closures of Lemma 7.18). Let C 1 and C 2

be premises of a superposition (resolution) inference with a conclusion C 3. Assume

that C 2 is a closure such that C 2m∗

−→ C 2. Then there exist closure C 3 such that

• C 3 is obtained by superposition (resolution) with premises C 1 and C 2, and

• C 3c∗

−→ C 3.


C 1 C 2 C 2

C 3 C 3

m∗

c∗

· · · · · · · · · · · · · · ·

·········

·································

If the closures C 1, C 2 and C 3 are not tautologies, then C 2 and C 3 are not tautologies

either.




Proof. The case that is considered in this proof is when the inference between C 1

and C 2 is superposition. The resolution case follows more easily, since resolution isperformed on predicate literals, while the relation

m−→ take place with equality literals

only.

Assume, without a loss of generality, that superposition takes place “from” C 1

“into” C 2, and that the inference is into a positive literal of C 2. Let the premises and

the conclusion of this inference be

C 1 : Γ1, l1 ≈ l2, . . . , ln ≈ rn · θ1

C 2 : Γ2, u[s] ≈ v · θ2

C 3 : Γ1, Γ2, u[x] ≈ v, x ≈ r1, . . . , x ≈ rn · (θ1 ∪ θ2)ρ

where ρ is the composition of most general unifiers of liθ1 and sθ2, for all i ∈ 1, . . . , n.The proof is by induction of the length of the

m∗−→-chain between C 2 and C 2.

Assume that it is 1. There are two cases to consider. Firstly, the superposition

inference and m−→ may take place with non-overlapping terms. The proof is then

straightforward, while deep analysis is given in the other case. Let superposition take

place into the term s of the literal u[s] ≈ v, and let the term t, which m−→ takes place

with, be a subterm of u such that it overlaps with s. Let the positions of s and t in

u be, respectively, ps and pt.

If pt < ps then ps = pt.p for some p, and C 2 is of the form

Γ2, u[x] ≈ v, x ≈ t[s] · θ2

Superposition from C 1 into the subterm s of the term t[s] of C 2 yields C 3, which is

of the form

Γ1, Γ2, u[x] ≈ v, x ≈ t[y], y ≈ r1, . . . , y ≈ rn · (θ1 ∪ θ2)ρ

where y is a fresh variable. Equality solution with x ≈ t[y] results in a clause which

is a variant of C 3, and the lemma holds.

If pt > ps then pt = ps.p for some p, then C 2 is of the form

Γ2

, u[s[x]] ≈ v, x ≈ t · θ2

A superposition inference into s[x] produces the clause

Γ1, Γ2, u[y] ≈ v, y ≈ r1, . . . , y ≈ rn, x ≈ t · (θ1 ∪ θ2)ν

where ν is the composition of mgu(s[x]θ2, liσ1), for all i ∈ 1, . . . , n. Equality solu-

tion with x ≈ t results in a closure that is a variant of C 3, and the lemma holds.

Let now the length of the m∗−→-chain between C 2 and C 2 be greater than 1. Then

C 1m∗

−→ C 21m∗

−→ C 2. By induction hypothesis there exists a closure C 31 such that




• C 31 is obtained by superposition with premises C 1 and C 21, and

• C 31c∗

−→ C 3.

Then, from the proof of the base of induction, there exists a closure C 3 such that

• C 3 is obtained by superposition with premises C 1 and C 2, and

• C 3c∗

−→ C 31.

In other words, there exists the diagram

C 1 C 2 C 21 C 2

C 3 C 31 C 3

sup sup sup

m∗ m

c∗ c∗

·························································

·································

·········

·········

· · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·

which proves the lemma.

Lemma 7.34 (the equivalent for first-order closures of Lemma 7.19). Let Ω be a

regular, tautology-free e-refutation of S ∪ C by EBMP . Let the conclusion of Ω be

a closure D and let C 1 be a closure such that C m∗−→ C 1. Then there exists a regular,

tautology-free EBMP -e-refutation Ω of D from S ∪ C 1, such that Ω contains as

many superposition inferences as Ω.

Proof. Assume that the closure C is a premise of a superposition inference, where

other premise is a closure C 2. Then Ω is of the form

Ω2C 2 C

D1(sup)

....D

By Lemma 7.33 there exist a closure D 1 such that

• D

1 is obtained by an application of superposition of C 1, C 2,

• D1

c∗−→ D1, and

• neither D1 nor D

1 is a tautology.

This means that D1 can be derived by superposition from C 1 and C 2 which is followed

by a sequence of equality solution inferences. The derivation of D from D1 can be

copied from Ω, and therefore the lemma holds. The proof is similar if C 1 is a premise

of any other inference by a rule from EBMP .




The claim and the proof of the following two lemmas are identical (up to the names

of the calculi) to the ones of Lemma 7.20 and Lemma 7.22. They stated again forreadability, because they are two main lemmas used to prove completeness of BMF .

Lemma 7.35 (the equivalent for first-order closures of Lemma 7.20). Let Ω be a

regular, tautology-free e-refutation of a set of clauses S by EBMP . Assume that

the conclusion of Ω is B. Let ∆ be a sequence S 0, S 1, . . . , S n of sets of closures,

where S 0 = C . Assume that S i is obtained from S i−1 (for all i ∈ 1, . . . , n)

either by replacing a closure from S i by its folding descendants, or by replacing it

by its subsumer. Then there exists a regular, tautology-free EBMP -e-refutation of a

subsumer of B from S \ C ∪ S n, which contains the same number of applications of

superposition as Ω.

Lemma 7.36 (the equivalent for first-order closures of Lemma 7.22). Let S 0 and S ω

be, respectively, a set of initial closures and the limit of a BF -derivation S 0 ⇒ S 1 ⇒

· · · (denoted by ∆). Let a closure C ∈ S j be of the form

Γ, s1 ≈ t1, s2 ≈ t2, . . . , sn ≈ tn · θ.

Let ρ be a substitution such that siθρ = tiθρ for all i ∈ 1, . . . , n. If it is not a

tautology, the closure

Γ · θρ

or its proper subsumer can be derived from S −ω by resolution.

First-order equivalents of lemmas from Section 7.2.5

Lemma 7.37 (the equivalent for first-order closures of Lemma 7.23). Let ∆ be a

BMF derivation S 0 ⇒ S 1 ⇒ · · · from a set of initial closures S 0 with a limit S ω. If

S 0 is unsatisfiable then there is a regular, tautology-free refutation of S ω.

Proof. By the compactness theorem, since S 0 is unsatisfiable, there exists a finite

unsatisfiable set C 1, . . . , C n ⊆ S 0. Consider a set S of clauses D ∈ S ω obtained by

repeated application of the basic folding and basic subsumption rules to the closures

from C 1, . . . , C n. From well-foundedness of the relation “a folding descendant or a

subsumer of” and Konig’s Lemma, it follows that S is finite. It also holds that S C i,

for all i ∈ 1, . . . , n. As a consequence, S ω S , and therefore S ω is unsatisfiable. By

Lemma 7.31, there exists a regular, tautology-free EBMP -refutation of S ω.

Theorem 7.38 (the equivalent for first-order closures of Theorem 7.24). Let ∆ be a

fair BMF -derivation S 0 ⇒ S 1 ⇒ · · · from a set of initial closures S 0, and let S ω be

a limit of ∆. If S 0 is unsatisfiable then S −ω is unsatisfiable, too.




Proof. From unsatisfiability of S 0, by previous lemma, it follows that S ω is unsatis-

fiable, too. Moreover, there exists regular, tautology-free EBMP -refutation of S ω.This implies that there exists an unsatisfiable set of purely predicate clauses S deriv-

able from S ω by the equational inferences of EBMP .

The set S , however, is not such that S ⊆ S −ω . In the remainder of the proof it is

shown that there exists a purely predicate and unsatisfiable set S ⊆ S ω, obtainable

from S ω by the equality inferences of EBMP .

Assume that S is derived by a derivation Ω. The proof is by induction on the

number n of superposition inferences in Ω. If there are no superposition inferences in

Ω, the lemma follows from Lemma 7.36, in the same way Lemma 7.24 follows from

Lemma 7.22.

Otherwise, take two leaves C 1 and C 2 of Ω which are premises of a superposition

inference sup with a conclusion C . Since C 1 and C 2 are leaves of Ω, they are in S ω.

If the closure C is in S ω, the induction hypothesis applies. Otherwise, from fairness

of ∆, C ∈ C i for some i ≥ 1. Since C ∈ S ω, then there exists a set F ⊆ S ω of

closures which are in (the transitive closure of) the “a folding descendant or a proper

subsumer of” relation with C . By Lemma 7.20, there exists an unsatisfiable set of

purely predicate clauses obtainable from S ω ∪ F by the equality inferences of EBMP .

Moreover, this purely predicate set can be derived using only n − 1 applications of

the superposition inference rule. The induction hypothesis applies.

If there are no two leaves of Ω which are premises of a superposition inference,

then there exists closures C 1 and C 2 such that Ω is of the form

C 2.... (es)C 2

C 1.... (es)C 1

C (sup)

and contains no tautologies. Ω can then be transformed to

C 1 C 2C

(sup).

... (es)C

where, by Lemma 7.17, since C is not a tautology, neither clause between C and

C (including C ) is a tautology. The modified derivation is, therefore, regular and

tautology-free, and contains the same number of superposition inferences as Ω. More-

over, the inference sup has leaves of the derivation as premises, which reduces the

consideration to the previous case.



Chapter 8

Conclusions and future work

This chapter summarizes the results presented in the thesis and indicates avenues for

further explorations.

The thesis contains an investigation of methods for handling equality in first-

order clausal logic based on proof transformations. The prospective taken in the

thesis with respect to the term ‘method for handling equality’ was quite broad. In

particular, this term has been used to refer to any refinement of the paramodulation

calculus, and also to any more complex method based on one such refinement. The

methods that have been of particular interest are those that seem not to fit into the

state-of-art framework for proving refutational completeness, known as the modelgeneration technique. Every method addressed in the thesis has first been looked

at from the point of view of the state-or-art results in the field. This approach has

helped in understanding the reasons why the model generation technique could not

be used to show their completeness. Consequently, this has benefited the process of

developing transformation techniques for proving refutational completeness of these

non-standard methods, so that the methods can keep all desired (and given in the

literature) properties of the state-of-art systems for handling equality. What follows

is a more detailed account of these results.

The first part of the work has been intended to be a brief survey of the most

important ideas related to paramodulation-based theorem proving, and results that

came out of these ideas. Its aim has been to point out the state-of-art features of

paramodulation-based calculi, which have been referred to in the rest of the thesis.

The conclusion has been that an efficient method for handling equality, based on the

saturation procedure, should be a refinement of basic superposition which allows for

removal of redundant clauses and possibly employs an efficient strategy for selecting

literals.

144



145

The first non-standard method for handling equality, which is presented in Chap-ter 4, has been obtained by relaxing the constraints for term orderings that are used

in the superposition inference rule. More precisely, this method has been formulated

as a refinement of superposition which drops all requirements for term orderings, and

uses any total and asymmetric relation instead. Completeness has been shown, using

a transformation technique, for unit and Horn ground clauses. Moreover, these results

have been shown compatible with elimination of tautologies.

An obvious avenue for further work is to extend these results to the general case

for ground clauses. The considered calculus would be a refinement of superposition

where an asymmetric relation is used instead of an ordering. Such refinement, as

discussed in Section 3.4.3, does not fit into the model generation method. It would

therefore be necessary to take a proof tree transformation approach, perhaps simi-

lar to the one demonstrated in Lemma 4.3 and Lemma 4.6 of Chapter 4, where a

derivation by a complete superposition calculus transformed so that superposition

inferences take place with the greatest terms in corresponding literals. This could be

taken even further, to the case of clauses with variables, although one should bear in

mind Example 4.7, which shows that completeness in the ground case is not possible

without inferences into variables.

The next method that has been developed in this thesis is a refinement of basic

superposition which implements the regular strategy of derivations and allows for the

removal of tautologies and subsumed clauses. This is a refinement which has turned

out to be central to the thesis, since it forms the core of the basic folding method,

which has also been addressed in the thesis. The proof transformation technique

used to prove completeness of this refinement has turned to be particularly powerful,

since it allows for elimination of tautologies and subsumed clauses. The result has

been proved for both formalisms that implement the basic strategy: closures and

constrained clauses. Although the closures are a less restrictive formalism, as shown

by Example 3.17, constrained clauses are more difficult to use when it comes toredundancy elimination. These difficulties are a result of the presence of ordering

atomic constraints in the constraint part of the constrained clause.

Consider the main redundancy class that exists for constrained clauses and does

not exist in case of closures. Namely, a clause C | T is redundant if the constraint T

is not satisfiable. Although there are many results that address ordering constraint

solving, like [Com90, Com91, Nie93, CT94, KV00, NR00, KV00], there is still a lot

of research to be done before constrained clauses can be efficiently used in practice.



146

One of the most obvious problems is that each constrained clause has its own

constraint, and that the constraint size can grow very rapidly. By analysing deriva-tions by superposition calculus defined on constrained clauses, it is apparent that the

constraint size of a clause C | T is proportional to the size of its derivation in the tree

form. Through personal communication with Andrei Voronkov, he revealed that the

largest derivation ever found by his Vampire prover reaches 40, 000 steps in the linear

form, which can easily be 21000 in the tree form. This example motivates the first

direction for future research, which can be named constraint simplification : given a

constraint C , which contains atomic ordering constraints, find a “simpler” constraint

equivalent to C .

Assume that a constrained clause C | T is derived from a set of unconstrained

clauses. In this case, it is clear that C is a logical consequence of this set. Therefore,

soundness is not lost when constraints are replaced by larger ones, i.e. it is possible

to use the following rule of (over)approximation :

C | T

C | T

where T is more general than T . Bearing this in mind the problem of constraint

simplification can be reformulated into the following problem of constraint approx-

imation : given a constraint C , which contains atomic ordering constraints, find a

“simpler” approximation to C .

Another problem that is related to redundancy elimination from derivations by

basic superposition of constrained clauses is determining whether a clause satisfies

a given redundancy criterion. For example, checking whether a clause satisfies the

abstract redundancy criterion given by Definition 3.29 is an undecidable problem.

Checking whether a clause is a tautology or subsumed by another clause, as defined

Definition 3.31 is not any easier at all. This brings about another direction for re-

search, the problem of redundancy criteria approximation : define redundancy criteria

for constrained clauses which are truly practical, i.e. which are decidable and easy to

check.Returning to the regular strategy of derivations, an important issue is testing the

efficiency of the method by experimenting. To be able to perform these experiments,

one would have to resort to using a theorem prover that implements a basic super-

position calculus of, preferably, constrained clauses. A straightforward choice of one

such theorem prover, apart from creating one from scratch, is the Saturate system,

described in [NN93]. This is the only existing prover that uses constrained clauses,

for which reasons may have been addressed in the previous few paragraphs. Carry-



147

ing out derivations by the regular strategy using Saturate does not seem to be very

straightforward. On the one hand, the Saturate system implements classical selectionstrategies, i.e. ones that fit into the model generation framework. On the other hand,

as explained in Chapter 5, the regular strategy does not fit into the model generation

framework. A way around this problem would be to extend Saturate (the code is

in Prolog and downloadable) so that it implements a selection strategy by which all

equality literals are selected in each clause that contains them. This solution, from

the point of view of design of the Saturate system, seems rather feasible. Another,

less inventive and much harder way, is to use the system as it is, run it in the mode

that allows for the user to manually select (equality) literals from clause, and hence

enforce the regular strategy by hand.

Whether Saturate is used or a new system is implemented, implementing the

regular strategy opens the door for further enhancement the prover efficiency. Recall

that regular derivations are those in which all equality inference rules are applied

at the beginning of the proof, which results in obtaining a purely predicate set of

clauses. If the initial set of clauses is equationally unsatisfiable, then the obtained

purely predicate set of clauses is also unsatisfiable (see Theorem 5.14). Therefore,

theorem proving using the regular strategy can be seen as a process that consists of two

dependant processes. The first process, called equality elimination, applies equality

inference rules to the clauses that contain equality literals. This process is potentially

infinite, because there may exist infinite chains of superposition, and should therefore

be parallelised with a process that uses resolution for reasoning on purely predicate

clauses. One of the most important issues connected to this parallel implementation

is how to connect the two processes, i.e. how often to feed the resolution engine with

purely predicate clauses obtained during the equality elimination phase.

Refinements of basic superposition which implement arbitrary strategies for literal

selection have also been presented as non-standard methods for handling equality. It

has been shown in this thesis that these strategies are complete for full first-order

clauses, provided that there is a refutation without factorization. However, the trans-

formation method method that has been used the result does not give any answerregarding redundancy elimination. Although Example 5.4 shows that arbitrary selec-

tion is not complete with tautology elimination, it could be interesting to follow the

idea of flattening (like it is done in the same example for tautology elimination for reg-

ular derivations) and check whether arbitrary selection is compatible with tautology

elimination for flat clauses.

Notice that the regular strategy of derivations falls into the set of arbitrary, non-

classical selection strategies. It is an interesting question to study which other non-



148

classical selection strategies (for derivations by basic superposition) are complete with

tautology elimination and subsumption. More precisely, is there a criterion that a non-classical selection strategy would have to satisfy in order to be complete in presence

of tautology elimination and subsumption?

Another, seemingly really tough question comes from an informal conversation

with Andrei Voronkov during JELIA’06, who suggested that their prover Vampire

uses selection strategies which are likely to be incomplete. As an instance, Vampire

uses a selection strategy that picks the longest, by the number of characters, literals

in the clause. This motivates the following question: is there an identifiable line be-

tween complete and incomplete selection strategies for a given calculus and a given

term ordering.

The last method that has been presented in the thesis is the basic folding method.

Theorem 7.38 gives a refutational completeness proof for a basic folding calculus with

tautology elimination and subsumption. Note though, that the basic folding calculi of

Horn and general clauses are defined for closures, and not constrained clauses. Basic

folding, as the authors of [DKV95] correctly claim, is possible to define for constrained

clauses. However, as depicted by the following example, basic folding of constrained

clauses in presence of subsumption looses the property of termination.

Example 8.1. Consider the following set of constrained clauses, assuming that f anda are function symbols such that a ≺ f , where ≺ is a reduction ordering on ground

terms.C 1 : → f (x) ≈ a | x < f (a)

C 2 : → f (x) ≈ a | x < f (f (a))...

C k : → f (x) ≈ a | x < f k(a)...

Following the definition of the relation “basic subsumer”, given in Definition 3.31, the

above clauses form an infinite chain of basic subsumers C 1 C 2 · · · C k · · · .

This example reiterates the earlier mentioned question about approximating the

redundancy criterion given by Definition 3.29 for constrained clauses. One way of

discovering these approximations could be by looking at the applications that would

benefit from them, like basic folding. More precisely, it would be interesting to look

at practical redundancy criteria, that would not affect completeness, soundness and

well-foundedness of the basic folding calculus of constrained clauses.



149

Another big question related to basic folding addresses the way the basic fold-

ing rule is applied. As it is defined, there are no restrictions on the way the basicfolding rule should be applied. Therefore, the folded terms are chosen completely

non-deterministicly. On the one hand, the brute force strategy of applying the rule

as long as there terms which are not flat, ensures termination of the basic folding

calculus (see [DKV95]). However, this approach generates many new clauses and

results in very large sets of purely predicate clauses once the equality elimination

part of reasoning by the basic folding calculus has been completed. One suggestion

for future work is investigating optimal strategies for applying the basic folding rule.

For example, it may be beneficial to fold only those terms that repeatedly appear in

equality literals of “many” different clauses. To add even more efficiency, this new

strategy could be made even more efficient if different occurrences of the same term

are folded simultaneously.



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