Combinatory Categorial Grammaras a Subdirectional Combinatory Logic�

BEKKI Daisuke

Ochanomizu UniversityFaculty of Science, Department of Information Science ��

Abstract. This paper aims to provide a logical background for Com-binatory Categorial Grammar (CCG) and its typological discussions.Based on the Curry-Howard correspondence between Gentzen-style proofsystems and Lambek Lamda Calculi, and those between Hilbert-styleproof systems and substructural BCWK-logic, I define a new class oflogic which provides subclasses for each of the substructural combinatorylogics, called Subdirectional Combinatory Logic, and propose that CCGis a subdirectional logic of a relevance logic (Combinatory Hypothesis).This hypothesis allows us to discuss typological parameters in univer-sal grammar in terms of the presence/absence of a certain directionalcombinators.

1 The Aim: CCG as a Logic of Universal Grammar

Classical Categorial Grammar (CG: Ajdukiewicz (1935), Bar-Hillel (1953)) haslong been considered, rightly or wrongly by both theoretical and computationallinguists, to be just a toy grammar which only logicians who are not seriousabout natural language have pursued in relation to proof theories, or just a frag-ment of grammar as Montague modestly called his version of categorial grammar(Montague (1973)).

However, recent study has revealed that this is not the case for CombinatoryCategorial Grammar (CCG: Ades and Steedman (1982), Steedman (1996, 2000),Steedman and Baldridge (2007)), a categorial grammar enriched with combina-tors. It has been proven that CCG is able to provide suitable devices to describea robust grammar to parse real texts such as those from large-scale corpora andthe world-wide web (Hockenmaier (2003); Hockenmaier and Steedman (2005)),and spell out even DRT-style semantic representations (Bos et al. (2004)).� This research was partly supported by the Ministry of Education, Science, Sports

and Culture, Grant-in-Aid for Young Scientists (A), 22680013, 2010-203. I also thankto Hiroakira Ono, Sachio Hirokawa, Hideki Aso, Kenichi Asai, Ken Shiotani, KristinaLiefke, Eric McCready, Alastair Butler, Kei Yoshimoto and other audiences of the7th international workshop of Logic and Engineering of Natural Language Semantics(November 18-19th, 2010, Tokyo) and the symposium on Categorial Grammar andProof Theory (December 23rd, 2010, Tokyo) for many valuable comments.

�� 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan.

On the other hand, CCG imposes proper restriction on non-local dependen-cies such as wh-movements so that they obey the island constraints in the senseof Ross (1967) as well as on local dependencies such as control and anaphorabinding where the so-called c-command relation is involved1. This means thatCCG is not a mere overgenerating grammar, and its power to provide a robustgrammar comes from the fact that the devices it is equipped with represent somedeep aspects of natural language.

This suggests that we had better pay special attention to CCG as a candidatefor the proper formalization of human language faculty, or universal grammar.Although it is true that CCG still faces many empirical challenges, it is a naturalstep for us to move on to the study of the conceptual side of CCG; for instance,addressing such questions as “which class of logic/proof system/type theorycorresponds to CCG?”

Such questions themselves presuppose the well-known result concerning CGand Lambek calculi (Lambek (1958)): since CG is known to be equivalent toa non-associative Lambek calculus (NL) and CCG is an extended categorialgrammar, CCG is a stronger system than NL. Jager (2001) demonstrates theequivalence between associative Lambek calculus (L) and a certain version ofCCG with two combinators, B and T.

In order to assess these results correctly, we need some background knowledgeabout the relation between L and NL, the Gentzen-style and Hilbert-style proofsystems, Curry-Howard isomorphisms between these notions, and substructurallogics2, which are summarized in Fig. 1 and Fig. 2.

2 Background: Substructural Logics, Curry-HowardIsomorphisms and Grammars

2.1 Gentzen-style Proofs, Lambek Lambda Calculi and FL

Full Lambek calculus (FL) is a class of logic obtained by expanding L withconjunctions and disjunctions. In other words, L is an implicational fragmentof FL. After the emergence of substructural logics in 1990s, FL has been givena major role as the most basic (namely, the weakest) class of Ono’s hierarchy(Ono (1990)).

The following rules are structural rules available in LJ, which are called weak-ening on the left and right (w), exchange (e) and contraction (w), respectively.Substructural logics are logics in which one or all of them do not hold.

(w)Γ � CA,Γ � C

(w)Γ �Γ � A

(e)Γ,B,A,Δ � CΓ,A,B,Δ � C

(c)A,A, Γ � CA,Γ � C

1 In regards to English grammar, see Steedman (1996), Chapters 2 and 3. In the caseof Japanese grammar, refer to Bekki (2010) for an exhaustive discussion.

2 Morrill (1994) and Buszkowski (2010) argue the relation between classical categorialgrammar and substructural logics, but they do not mention CCG.

Logic Type system Grammar

Gentzen-style proof systems Typed Lambda Calculi Lambek CalculiHilbert-style proof systems Combinatory Logics CCG

Fig. 1. Curry-Howard Isomorphisms and Grammars

Intuitionistic Logic = FLecw

= BCWK-logic

����������������

�����������������

Affine Logic = FLew

= BCK-logic

����������������

Relevant Logic = FLec

= BCWI-logic

�����������������

Linear Logic = FLe

= BCI-logic

(Associative) Full Lambek = FL= BI-logic

Fig. 2. Ono (1990)’s hierarchy of Substructural Logics

One of the key concepts of substructural logics is that considering variantsof proof systems in which a certain structural rule does not hold may lead tonew logical operators. For example, under the presence of the exchange rule, thefollowing two introduction rules for implication “→” are equivalent.

(→I)A,Γ � BΓ � A→ B

(→I)Γ,A � BΓ � A→ B

However, this is not the case when the exchange rule is absent, which moti-vates the system with two distinct implications, which correspond to the respec-tive introduction rules above.

(\I)A,Γ � BΓ � B\A

(/I)Γ,A � BΓ � B/A

Under the presence of exchange, A\B ⇔ A/B is provable; thus, we can givethem the same name, “B → A”, just as in LJ.

Many non-classical logics find their own places in Ono’s hierarchy. For ex-ample, Relevant logic (Priest (2001)), Affine logic and Linear logic (Girard(1987)) are a weakening-free, contraction-free and weakening-and-contraction-free Gentzen-style proof systems, respectively, and FL is a Gentzen system withno structural rules at all (but is still associative).

Through Curry-Howard isomorphism, every Gentzen-style proof system inOno’s hierarchy has a corresponding lambda calculus. In FL, each lambda termnot only represents a proof for a given judgment, but also a syntactic deriva-tion in linguistic terms. The type theories for FLs are called Lambek lambda

calculi (Buszkowski (1987), Wansing (1993), Morrill (1994), Moortgat (1997),Buszkowski (1997), Polakow and Pfenning (1999), Restall (2000), Jager (2005),Ogata (2008), Morrill (2010)).

2.2 Non-associative Lambek Lambda Calculi

Although FL is the weakest system in Ono’s hierarchy, Non-associative FullLambek NL (Aarts and Trautwein (1995), de Groote (1999), de Groote andLamarche (2002), Wansing (2002)) is an even weaker system where associativelaw does not hold in the premises/contexts of sequents. For example, the follow-ing proof of the combinatory rule (> B) of CCG in L which relies on associativity(namely, Γ, (Δ,C) = (Γ,Δ), C) is not a proof of NL.

(/I)

(/E)Γ � A/B (/E)

Δ � B/C (Ax)C � C

Δ,C � BΓ,Δ,C � A

Γ,Δ � A/CThe type theory which corresponds to NL is called Non-associative Lambek

Lambda Calculus.

2.3 Hillbert-style Proofs, Combinatory Logics and CCG

In the case of Hilbert-style proof systems, all axioms are provided in formsof combinators. For example, an implicational fragment of intuitionistic logicconsists of the axioms S and K. Since S is equivalent to the combination of B,C and W (see Theorem 1), the axioms for intuitionistic logic can be replacedwith B, C, W and K. Thus LJ is equivalent to the Hilbert system known asBCWK-logic. When the deduction theorem is available, the structural rules(e), (c) and (w) in LJ correspond to the combinators C, W and K, respectively.Therefore, as indicated in Figure 2, Relevant logic, Affine logic, Linear logic andFull Lambek correspond to BCWI-logic, BCK-logic, BCI-logic and BI-logic,respectively, which are K-free, W-free, KW-free, and all-structural-rules-freeHilbert systems, respectively.

Theorem 1 (Combinators in CL).

S = B(B(BW)C)(BB)I = SKK

B = S(KS)KW = SS(KI)

= SS(SK)C = S(BBS)(KK)

B′ = CB

C∗ = CI

Φ = B(BS)BΨ = B(BW(BC))(BB(BB))

= Φ(Φ(ΦB))B(KK)

3 Subdirectional Combinatory Logic

In this paper, I introduce a new class of logic, called Subdirectional CombinatoryLogic, which provides subclasses for each of the substructural combinatory logics,and define CCG as a certain subclass of SB-logic. This allows us to discusstypological parameters systematically in terms of combinatory logics as we willsee in Section 5.

3.1 Syntax

Definition 2 (Directed Combinators). A directional combinator is a com-binator in Subdirectional Combinatory Logic which can be obtained by replacingeach occurrence of implications that appear in a certain combinator in CL witheither / or \.

Example 3 (Directed Combinators for K).

K// : (A/B)/AK/\ : (A/B)\A

K\/ : (A\B)/AK\\ : (A\B)\A

A subclass of Subdirectional Combinatory Logic is determined by a set ofground types and a set of directional combinators.

Example 4 (CCG). CCG is a Subdirectional Combinatory Logic determined by:

the set of ground types :{NP , N, S, S

}

the set of directional combinators :{B/,B\,S/,S\

}

Definition 5 (Syntax of Subdirectional Combinatory Logic). Given aset of ground types GT and a set of directional combinators C, the sets of types(τ) and the sets of raw terms (Λ) for a subdirectional combinatory logic arerecursively defined as follows:

τ ::= γ | τ/τ | τ\τ |Λ ::= x | c | Λ�

Λ | Λ�Λ

where γ ∈ GT and c ∈ C.

I will also use the following notations to describe cases where the directionof a given type/term/rule is underspecified.

Definition 6 (Underspecified notations).

τ/\σ def≡ τ/σ or τ\σM

��N

def≡ M�N or M

�N

(<>)def≡ (<) or (>)

As for the category A\B, we obey the convention standard in CCG in whichit takes an object of category B on its left, rather than the convention standardin Lambek calculi or classical categorial grammar in which it takes an object ofcategory A on its left.

The typing rules of Subdirectional Combinatory Logic are defined as follows,which is Curry-Howard correspondent with a substructural Hilbert-style sequentcalculus: (>) and (<) are left and right Modus Ponens rules, and the combinatorsB/, B\, S/ and S\ are axioms.

Definition 7 (Type assignment axioms of Subdirectional CombinatoryLogic). The typing of directional combinators which instantiates one of the fol-lowing schema is an axiom of Subdirectional Combinatory Logic.

B/ : (A/C)/(B/C)/\(A/B)B\ : (A\C)\(B\C)/\(A\B)S/ : (A/C)\(B/C)/\(A\B/C)S\ : (A\C)/(B\C)/\(A/B\C)

Definition 8 (Type assignment rules of Subdirectional CombinatoryLogic). Terms in Subdirectional Combinatory Logic are typed by the followingrules, where Γ,Δ are any contexts, M,N are any raw terms, and A,B,C areany types.

(VAR)x : A � x : A

(AX)an axiom in Definition 7

(>)Γ �M : A/B Δ � N : B

Γ,Δ �M�N : A

(<)Δ � N : B Γ �M : A\B

Δ,Γ �M�N : A

It should be noted that we have to adopt so-called “structural independentforms” for the typing rules, instead of “structural sharing forms”, in order toformalize natural language syntax.

3.2 Equational theory for Subdirectional Combinatory Logic

The equational theory for Subdirectional Combinatory Logic consists of the fol-lowing axioms and rules.

Definition 9 (Axioms of Subdirectional Combinatory Logic).

B/��f

�g

�x = f

�(g

�x)

B\��f�g�x = f

�(g

�x)

S/��f�g

�x = (f

�x)

�(g

�x)

S\��f

�g�x = (f

�x)

�(g

�x)

Definition 10 (Equational rules of Subdirectional Combinatory Logic).For any term L,M,N , the following inferences are valid.

(=R)M = M

(=S)N = M

M = N(=T )

L = M M = N

L = N

(=F�)M = N

F�M = F

�N

(=F�)M = N

F�M = F

�N

(=A�)M = N

M�A = N

�A

(=A�)M = N

M�A = N

�A

3.3 Deriving combinatory rules as admissible rules

It is straightforward to deduce from Definition 7 and Definition 8 that the fol-lowing rules are admissible in Subdirectional Combinatory Logic. The first set ofadmissible rules are functional composition rules and the second set are crossedfunctional substitution rules in the sense of Steedman (1996, 2000).

Theorem 11 (Functional Composition Rules).

(>B)Γ �M : A/B Δ � N : B/C

Γ,Δ � B/��M

�N : A/C

(<B)Δ � N : B\C Γ �M : A\B

Δ,Γ � B\��M

�N : A\C

Proof.

(>)

(<>)

� B/ : (A/C)/(B/C)/\(A/B) Γ �M : A/B

Γ � B/��M : (A/C)/(B/C) Δ � N : B/C

Γ,Δ � B/��M

�N : A/C

(<)

Δ � N : B\C (<>)

Γ �M : A\B � B\ : (A\C)\(B\C)/\(A\B)

à � B\��M : (A\C)\(B\C)

Γ,Δ � B\��M

�N : A\C

Theorem 12 (Crossed Functional Substitution Rules).

(>S)Γ �M : A/B\C Δ � N : B\C

Γ,Δ � S\��M

�N : A\C

(<S)Δ � N : B/C Γ �M : A\B/C

Δ,Γ � S/��M

�N : A/C

Proof.

(>)

(<>)

S\ : (A\C)/(B\C)/\(A/B\C) Γ �M : A/B\C

Γ � S\��M : (A\C)/(B\C) Δ � N : B\C

Γ,Δ � S\��M

�N : A\C

(<)

Δ � N : B/C(<>)

Γ �M : A\B/C S/ : (A/C)\(B/C)/\(A\B/C)

à � S/��M : (A/C)\(B/C)

Δ,Γ � S/��M

�N : A/C

4 Combinatory Hypothesis

Steedman (1996, 2000) employs the combinators B, T, S and Φ for his ver-sion of CCG. Meanwhile, it is established in combinatory logic that the com-binators S and Φ can be derived from the combinations of B, C and W (seeTheorem 1). Thus, if we put T aside (we will come back to the status of Tin Section 5.2), Steedman’s version of CCG can be regarded as a subclass ofBCWI-logic, namely, a Hilbert-style relevance logic, which leads to the follow-ing working hypothesis:

Proposition 13 (Combinatory hypothesis for universal grammar). Uni-versal grammar is a subdirectional logic of BCWI-logic.

One of the aims of this paper is to give some preliminary consideration to thestatus of CCG as a substructural/subdirectional logic under the CombinatoryHypothesis. From the linguistic point of view, the discussion of the status ofCCG as a substructural/subdirectional logic leads us to new research in whichtypological discussion can be carried out in terms of substructural/subdirectionallogics.

For example, the combinatory rule B is the most general syntactic rule amongall languages, but languages which have a reflexivization rule such as Englishmay include the combinator W (Jacobson (1990), Szabolci (1989) and Steedman(1997)), while languages which allow scrambling such as Japanese may includethe combinator C (Bekki (forthcoming)). This view of the parameters (in Chom-sky’s sense) of individual languages leads to our second working hypothesis:

Proposition 14 (Combinatory hypothesis for individual grammars).An individual grammar is determined by the parameters whose values are de-scribed as a presence or absence of a certain directional combinator, or sub-structural conditions under which the combinators B, C, W and I are used inthe derivation.

In other words, the parameters of each individual language can be regardedas giving it a status as a certain intermediate logic in Ono’s hierarchy of sub-structural logics.

5 Presence or Absence of Directed Combinators

In this section, I discuss the status of the presence or absence of some directionalcombinators from the perspective of the two Combinatory Hypotheses.

5.1 Absence of K: Universal Grammar is a Relevant logic

The directional counterparts of K combinator (≡ λx.λy.x : A → (B → A) inCombinatory Logic) are not included as the axioms of Subdirectional Combina-tory Logic in Definition 9, but we can consider their potential forms as we haveseen in Example 3.

For the present purpose, let us consider instead the following forms in whichthe directions of their first arguments are underspecified.

K\def≡ λ

��x.λ

�y.x : (A\B)/\A

K/

def≡ λ��x.λ

�y.x : (A/B)/\A

Reduction rules for these directional combinators are as follows:

K/��x

�y = x

K\��x�y = x

It seems that a grammar of any language must not be equipped with anyof the above variations of the combinator K: otherwise, any phrase of any cat-egory can appear in any position in a grammatical sentence. For example, in alanguage in which K\ and/or K/ are available, the following set of rules becomeadmissible.

(>K)Γ �M : A

à � K/��M : A/B

(<K)Γ �M : A

à � K\��M : A\B

Proof.

(<>)

K/ : (A/B)/\A Γ �M : A

à � K/��M : A/B

(<>)

Γ �M : A K\ : (A\B)/\A

à � K\��M : A\B

Then, any word/phrase (for example, runs in the following derivation) canbe inserted in front of/at the back of any saturated phrase, which is apparentlynot the case:

(<)

runsrun : S\NP

(<K)

(>)

John

λ�p.p

�j : T /(T \NP)

runsrun : S\NP

run�j : S

K\��(run

�j) : S\(S\NP)

K\��(run

�j)

�run : S

(<)

(>K)

(>)

John

λ�p.p

�j : T /(T \NP)

runsrun : S\NP

run�j : S

K/��(run

�j) : S/(S\NP)

runsrun : S\NP

K\��(run

�j)

�run : S

These effects of adding K\ or K/ to our grammar are undesirable not onlyin the sense that they allow the unexpected phrases to appear elsewhere, butalso in the sense that the meaning of the unexpected phrases is not reflected atall in the semantic representation, because K\ and K/ just discard their secondarguments.

K\��(run

�j)

�run = run

�j

K/��(run

�j)

�run = run

�j

There seems to be no language which allows this operation to take place;that is, to adjoin a phrase which has no influence to the meaning of the wholesentence. The reason for the lack of this kind of operation seems to be due toGrice’s Maxim of Relevance: “Be relevant” (Grice (1975)). In other words, wecan regard the absence of K in universal grammar as a syntactic realization ofGrice’s Maxim of Relevance.

As indicated in Figure 2, the logical significance of the absence of K is theabsence of the weakening rule. Thus, universal grammar is a subclass of Relevantlogic, in which each premise in a given context must be used at least once.

(MP)

(ASS)φ � φ

(MP)

Γ � ψ (K )� ψ → (φ→ ψ)

Γ � φ→ ψ

φ, Γ � ψ

The resemblance of denomination between Grice’s Maxim and Relevancelogic is not a coincidence, because that the restriction that each word in a givensentence must be used at least once means that a word/phrase which is notsemantically used and thus does not contribute anything to the resulting mean-ing of the sentence in the above derivation is not allowed to appear.

5.2 Absence of I: Unavailability of type raising

Jager (2001)’s version of CCG is BI-logic extended with the combinator T. WhileT stands for “type raising” in the sense of Partee and Rooth (1983), it is knownas a combinator C∗ in Combinatory Logic. Since C∗ = CI (see Theorem 1),Jager (2001)’s system can be regarded as BCI-logic.

However, CCG with the combinator T, namely, CCG with the type raisingoperation allows extraction (wh-movement) from both complex NPs and subjectislands (Ross (1967)).

(1) Complex NP constraint:

* Whoi does Phineas know [NP a girl who is jealous of ti]?

The derivations of the phrases “does Phineas know” and “who is jealous of”are as follows.

>B

>

doesSinv/(Sbase\NPnom)/NPnom

3s

PhineasNPnom|acc

3s

Sinv/(Sbase\NPnom)know

Sbase\NPnom/NPacc

Sinv/NPacc

>

whoN\N/(Sfin\NP)

>B

isSfin\NPnom

3s/(S\NP)

>B

jealousS\NP/PPof

ofPPof /NPacc

S\NP/NPacc

Sfin\NPnom3s

/NPacc

N\N/NPacc

Then the application of the type raising rule to “girl” gives rise to the chunk-ing of whole series of phrases and also to an undesirable extraction from thecomplex NP.

>B

WhoSQ/(Sinv/NPacc)

>B

does Phineas knowSinv/NPacc

>B

aNP3s/N

>B

>T

girlN

T /(T \N)who is jealous ofN\N/NPacc

N/NPacc

NP3s/NPacc

Sinv/NPSQ

On the other hand, CCG without the type raising operation3 correctly blocksthe extractions from these islands, as the following derivations show.3 Here, we have to assign two lexical items of categories T /(T \NP) and T \(T /NP)

to every proper name.

WhoSQ/(Sinv\NPacc)

does Phineas knowSinv/NPacc

aNP3s/N

∗

girlN

who is jealous ofN\N/NPacc

∗

Thus, we are interested in CCG without the type raising operation as acandidate for universal grammar. Therefore, we may also conclude that I isnot available in CCG, assuming that C is available in some grammars such asJapanese.

In the combinatory logics in which both S and K are available, adding I toKS is redundant since I is derivable from K and S (see Theorem 1). This alsoapplies to Subdirectional Combinatory Logic as follows:

I�x = (S\

�K\)

�K\

�x

= (K\�x)

�(K\

�x)

= x

I�x = (S/

�K/)

�K/

�x

= (K/�x)

�(K/

�x)

= x

6 Conclusion

Due to space limitations, my discussion could not cover all of the directionalcombinators. In the future work, I will discuss the status of the other combinatorsin Subdirectional Combinatory Logic and address the following issues:

1. The presence of the combinator W and reflexivization2. The presence of the combinator C and Japanese scrambling3. The presence of the combinator Φ and coordinated structures4. The presence of the combinator S and parasitic gap constructions

Appendix A: Abstraction

In Section 5.1, a derivation contains the lambda abstraction “λ�p.p

�x”, which is

not defined as a term in Subdirectional Combinatory Logic. It is known in theliterature (see Hindley and Seldin (2008)) that in CL, lambda-abstraction con-structions can be represented solely by the combinators. Following that strategy,we can define lambda abstractions by means of directional combinators as fol-lows.

Definition 15 (Abstraction in Subdirectional Combinatory Logic). Forevery term M and every variable x, terms of the form λ

�M. and λ

�x.M are defined

by induction on M :

(1/) λ�x.M

def≡ K/�M if x /∈ fv(M).

(1\) λ�x.M

def≡ K\�M if x /∈ fv(M).

(2/) λ�x.x

def≡ I/

(2\) λ�x.x

def≡ I\(3/) λ

�x.U

�x

def≡ U if x /∈ fv(U).

(3\) λ�x.U

�x

def≡ U if x /∈ fv(U).

(4/) λ�x.U

�V

def≡ S/�(λ

�x.U)

�(λ

�x.V ) if neither (1/) nor (3/) applies.

(4\) λ�x.U

�V

def≡ S\�(λ

�x.U)

�(λ

�x.V ) if neither (1\) nor (3\) applies.

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