The Natural Law as a Recursive Syntactic Operation

3-rd International Workshop on CS and Education in CS

A FIBONACCI-TREE MODEL OF COGNITIVE PROCESSES UNDERLYING LANGUAGE FACULTY

Some pronouncement about us being a part of nature

Velina SlavovaNBU

Alona SoschenMIT


This paper offers a formal mathematical support of the linguistic model of Argument Based Syntax.

The approach developed in this paper is based on the definitions of recursive structures in language. The result is a Tree of Fibonacci with homogeneous nodes, which are syntactically interpreted as paths that establish connections between the elements of an argument-centered structure.

In the recent linguistic research it is suggested that concepts of computational efficiency in natural language are closely related to the principles of a more general character.

Based on the properties of the tree, it is shown that the number of different ways of combining arguments is strictly predetermined. These combinations appear to be related to the principle of efficient growth.


All humans possess the innate Faculty of Language (FL),the reason why small children learn to speak…

Our hypothesis is that recursion and the principles of efficient growth are in the bases of syntactic constructions.

Introduction

Recently, it was shown that certain parts of FL – conceptual-intentional and motor-sensor systems – are not uniquely human:

animals produce signals and form primitive concepts.

It was also assumed that RECURSION is a unique species-specific mechanism. It allows us to connect words into sentences, and sentences into parts of discourse.


1 2 3 5 8 13 21

RECURSIVE SYSTEMS ARE OBSERVED IN NATURE

Leonardo of Pisa (1170-1250)

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 1284, 3881,

Fibi+2= Fibi+1+ Fibi

The numbers of the series of Fibonacci are found surprisingly often in nature.

The number of petals on a flower is often one of the Fibonacci numbers:

Fibonacci numbers in plant sections :

Apples Bananas


Fibonacci numbers…


is known as Golden Ratio; Golden number;

Phi (φ)

5251

251

11

nn

nfib

251

fibi+2= fibi+1+ fibi

WELL KNOWN FACTS…

Fibonacci numbers providethe best whole number approximations to the Golden Ratio

has the solution: , where

Each new seed appears at a certain angle in relation to the preceding one.

Two families of spirals (one in each direction) are then visible: their numbers correspond to the numerator and denominator of one of the fractions which approximates the Golden Ratio : 2/3, 3/5, 5/8, 8/13, 13/21, etc.

The head of a flower is made up of small seeds which are produced at the centre, and then migrate towards the outside to fill best the space !The space is filled the best when this angle is exactly the Golden number, and only this one.

Why Fibonacci numbers in alive nature?


What Nature seems to use is the same pattern to place seeds on a seed head as it used to arrange petals around the edge of a flower AND to place leaves round a stem. What is more, ALL of these maintain their efficiency as the plant continues to grow…

NUMBERS OF EFFICIENT GROWTH


NUMBERS OF EFFICIENT GROWTH


Our hypothesis is that they comply with the principle of efficient growth.

Chomsky (1957) offered rewriting rules for syntax.SYNTACTIC STRUCTURES

Sentence S > Verbal Phrase VP + Nominal Phrase NPNP > Noun N + determiner Phrase DP

S : The boy likes the girl

The boy/ NPV’/ likes the girl

NP/ The girl

likesThe boy

The girl

V’

Syntactic structures are characterized by discreteness and economy.


XP

XP

X XP

Generalized Bare Phrase Structure (X-bar model)

X’

S : The boy likes the girl

The boy/ NPV’/ likes the girl

NP/ The girl

THE CORE PRORERTIES OF SYNTACTIC TREES

L1 Non-terminal XP and X’ are points of growth:XP is a set of terms (final sum)

L2 X’ is a set of terms (intermediate sum)*L3 X is a terminal node

*X’-level is invisible for computation

Specifier

Head X Complement


XP

XP

X XP

X’


The tree that incorporates syntactic constituents (verbal, nominal phrases, etc) is generated in a bottom-up manner, by merging pairs of elements (lexical items). Each item is merged only once; every specifier and every complement positions are filled. Going up to the root, each next operation adds a new element to the already formed pair.

set

term set

set

set


XP

XP

X XP

X’


set

term set

set

set

The tree that incorporates syntactic constituents (verbal, nominal phrases, etc) is generated in a bottom-up manner, by merging pairs of elements (lexical items). Each item is merged only once; every specifier and every complement positions are filled. Going up to the root, each next operation adds a new element to the already formed pair. This structure can be seen as recursive with three types of nodes: XP, X and X’. each XP has two daughters – one XP and one X’, and each X’ has two daughters - one is a XP and the other a X.


XP

XP

1


XP

X’XP

XP X’

1 1

∑

1 1 2


XP

X’XP

XXPX’

XP

XP X’

X

1

1 1

1

2

∑

2 1 3 1


XP

X’XP

XXPX’

XP

X XP XPX’

XPX’

XP X’

X

1

1

2

3

1

1

2

1

2

3

5

1

1

∑


XP

X’XP

XXPX’

XP

X XP XPX’

XPX’

XPX’

X XPX’

X XPX’

XP XP

XP X’

X

1

1

2

3

5

1

1

2

3

1

2

3

5

8

1

1

2

∑


XP

X’XP

XXPX’

XP

X XP XPX’

XPX’

XPX’

X XPX’

X XPX’

XP XP

X XP XPX’

X XPX’ X XPX’

XPX’

XPXPX’

XP

XP X’

X

1

1

2

3

5

8

1

1

2

3

5

1

2

3

5

8

13

1

1

2

3

∑


1. Problem: In a syntactic tree, every head X must have an XP Complement. As a result, there is no line with only terminals (bottomless tree). (Carnie 2002)

XP

X’XP

XXPX’

XP

X XP XPX’

XPX’

XPX’

X XPX’

X XPX’

XP XP

The proposed recursive structure has some difficulties:The tree is bottomless, it grows till the satisfaction of some preliminary determined rule.

Sentences are finite.


The proposed recursive structure has some problems for its application:

2. Problem: The growth of this structure does not correspond to the whole number sequence of efficient growth.

(The hypothesis being that syntactic structures comply with the principle of optimization. )


X’XP

XXPX’

XP

X XP XPX’

XPX’

XPX’

X XPX’

X XP X’

XP XP

X XP XP X’

X XPX’ XXPX’

XPX’

XPXP X’

XP

X’XP

XXPX’

XP

X XP XPX’

XPX’

XPX’

X XPX’

X XP X’

XP XP

X XP XP X’

X XPX’ XXPX’

XPX’

XPXP X’

XP

X’XP

XXPX’

XP

XXP XPX

’

XPX’

XPX’

X XPX’

X XP X’

XP XP

X XP XP X’

X XPX’ XXPX’

XPX’

XPXP X’

XP

X’XP

XXPX’

XP

X

XP XPX’

XPX’

XPX’

XXPX

’X XP X

’XP XP

X XP XP X’

X XPX’ XXPX’

XPX’

XPXP X’

XP

X

X

X X

X X X

X’XP

XPX’

XP

XP XPX’

XPX’

XPX’

XPX’

XP X’

XP XP

XP XP X’

XPX’ XPX’

XPX’

XPXP X’

XP

XP


X

X

X

X X

X X X

X’XP

XPX’

XP

XP XPX’

XPX’

XPX’

XPX’

XP X’

XP XP

XP XP X’

XPX’ XPX’

XPX’

XPXP X’

XP

XP


X

X

X X

X X X

X’XP

XPX’

XP

XP XPX’

XPX’

XPX’

XPX’

XP X’

XP XP

XP XP X’

XPX’ XPX’

XPX’

XPXP X’

XP

In the Fib series, X’Are Not points of branching, they are only point of growth.(Ref: Fibonacci – rabbit’s problem)

XP


XP

X’XP

XXPX’

XP

X XP XPX’

XPX’

XPX’

X XPX’

X XPX’

XP XP

X XP XPX’

X XPX’ X XPX’

XPX’

XPXPX’

XP

2. Problem’s solution: Discard X (heads) and intermediate projections X’ and study the structure determines XP-s (the argument – based structure)


XP

XP

XP XP

XP XP XP

XP XP XPXP XP

XP XP XP XPXP XPXP XP

The XP structure corresponds to the known structureTree of Fibonacci:


XP

XP

XP

XP

XP

XP

XP

XP

XP

XPXP

XP

XP

XP

XP

XP

XP

XP

XP

XP

According to the hypothesis put forward in Soschen (2005, 2006), a general rule governing efficient

growth applies in syntax in such a way that minimal syntactic constituents incorporate

arguments which are related to each other.

Tree of Fibonacci:


h: 0 1 2 3 4 5 6

A XP-Fib tree of Height h consists of a root node XP, to which two sub-trees are attached –

The tree of Height 0 has 0 nodes,The tree of Height 1 has one node XP. - The other is a Fib-tree of Height h-1.

- The one is Fib-tree of Height h-2,

Ø

2. PROPERTIES OF ARGUMENT-BASED Fib-TREE

1. the Fib tree is finite (limited by h). 2. the tree is homogeneous, all its nodes are XP-s, 3. there is a minimal structure defined – a Fib tree with 0 nodes.


XP

XP XP

From the syntactic point of view, the “problem” of the XP-Fib tree is that it has nodes which are not binary branching.

XPXP XPXP

XPXP XPXP

XP

In the syntactic sense, nodes are result of merge (sum) of two elements, the internal nodes of a syntactic construction.


Ø -merge

ØØ -merge

Ø -merge

Ø

XP

XP XP

X3

ØØ -merge

Ø -mergeØØ -merge

XPXP XPXP

XPXP XPXPX4

Ø

X5

ØØ -merge

XPX2

ØX1

Ø -mergeØ

It can be seen as an XP Fib tree in which all non-binary branching nodes (leafs and internal nodes) have additional zero-‘leafs’.


Solution: From the point of view of linguistics, that requires to redefine binarity to include Ø-Merge (singleton set).

The linguistic model includes an operation Ø–merge which produces a XP (singleton set) by merging a terminal node X with Ø. This operation has serious consequences for the general interpretation of syntactic trees, as it provides a rule for producing sets - the starting point in a syntactic treatment.

The bottom of the tree is defined by merging Ø

XØ Ø -merge

XP

The newly introduced type of merge, Ø– merge, is important for distinguishing between Entities X and sets XP. The ‘leaves’ of the XP Fib-tree represent entities.


XP

XP XP

Syntactic trees are generated by successive merge of elements,

X3

ØØ -merge


Ø -merge

XPXP XPXP

XPXP XPXPX4

ØØ

Ø -merge

X5

ØØ -merge

XPX2

ØØ -merge

X1

ØØ -merge

Ø

merge merge

merge

merge

starting from the bottom level.

sentence


XP

XP XP

X3

ØØ -merge


Ø -merge

XPXP XPXP

XPXP XPXPX4

ØØ

Ø -merge

X5

ØØ -merge

XPX2

ØØ -merge

X1

ØØ -merge

Ø

merge merge

merge

merge

‘leaf’ nodes of the XP Fib-tree are Ø–merged terminal nodes (Entities)

Syntactic trees are generated by successive merge of elements,


XP

We define Ø – merge as an operation that produces a XP node of type “XPØ”

All nodes of the tree are of type XPØ,

(they are singleton sets).

Ø -merge XØ

3. HOMOGENEITY OF THE TREE

XØ

XØ

XP

XP

XØ

XP

XØ

XP


XØ

XØ

XP

XP

XØ

XP

XØ

XP

This construction provides a perfect justification of operation type-shift.


XØ

X

XP

XP

XØ

XP

XØ

XP

type shift

type shift

The account for introducing this operation is based on a formal definition ‘non-self-inclusiveness of sets’.

This kind of change from sets XPs to ‘unbreakable’ entities Xs is required in the syntactic model. The shifting of types necessitates two non-equivalent substances; e.g. two XP-s (or two sets) cannot be merged. One of XPs has to be first transformed into an ‘unbreakable’ entity X; after that it can be merged with another XP.


XØ

X

XP

XP

XØ

XP

XØ

XP

type shift

type shift

XP

X entry

Ø entry

Ø -merge XØ

free input X input


XP

XP XP

X3

Ø

XPXP XPXP

XPXP XPXPX4

Ø

X5

ØXP

X2

ØX1

Ø

ØØ

Ø

3. XP FIB-TREE, LINGUSTIC STATEMENTS, AND RESULTS


XP

XP XP

X3

Ø

XPXP XPXP

XPXP XPXPX4

Ø

X5

ØXP

X2

ØX1

Ø


XP

XP XP

X3

Ø

XPXP XPXP

XPXP XPXPX4

Ø

X5

ØXP

X2

ØX1

Ø

Ø

Ø


XP

XP XP

X3

Ø

XPXP XPXP

XPXP XPXPX4

Ø

X5

ØXP

X2

ØX1

Ø


XP

X3

Ø

XPXP

XPXP XP

XPX2

ØX1

Ø


XP

X3

Ø

XPXP

XPXP XP

XPX2

ØX1

Ø Ø


XP

X3

Ø

XPXP

XPXP XP

XPX2

ØX1

Ø Ø

Ø


XP

X3

Ø

XPXP

XPXP XP

XPX2

ØX1

Ø

Ø


XP

XP XP

X3

Ø

XPXP XPXP

XPXP XPXPX4

Ø

X5

ØXP

X2

ØX1

Ø

ØØ

Ø

THE BOTTOM-UP MERGE OF ANY TREE LEADS TO THE TREE OF HEIGHT 3

START!


XP

XP XP

XPXP XPXP

XPXP XPXP

XP

ØØ

Ø

MERGE !


XP

XP XP

XPXP XPXP

XPXP

MERGE !


XP

XP XP

XPXP

XP

The one of the entries is for X.The other is for 0 or for XP

X entryØ

entry

ROOT:

It can be easily shown that the basic tree is the one with h=3.

It merges in a unique way each of its fill-in variants and provides a full set of merge-patterns.


WE HAVE EITHER THIS:

XP

XP XP

XP Ø -merge

XØ

XP

Ø -mergeXØ

XP Ø

Type-shift

Type-shift


X

XP

XP XP

XP Ø -merge

XØ

XP

Ø -mergeXØ

XP

Type-shift

OR THIS:


X2

Ø

XP

XP

E1

a. Infinite iteration: Mary, Mary…

3. XP FIB-TREE, LINGUSTIC STATEMENTS, AND RESULTS

Ø


X2

Ø

XP

XP

E1

b. Mary in Mary smiles.


X2

Ø

XP

XPXP

XPX1

ØE2

E1

c. Two arguments Mary, John inMary loves John


X2

Ø

XP

XPXP

XPX1

ØE2

E1

d. Three arguments Mary, John, apple inMary gave John an apple.

Ø


X2

Ø

XP

XPXP

XPX1

ØE2

E1

Ø

The above schemes represent all possible configurations and relations between arguments in the human theta-role Semantics Space. They show convincingly that the number of arguments in a thematic domain is necessarily limited to three.


The tree expresses thepaths of connecting smaller units in order to produce a larger meaningful unit.

SOMETHING LIKE CONCLUSION

The question of the height of the XP Fib-tree is deeply related to the limits of the human cognitive resources.

It could be suggested that the limits of this structure are determined in the same way as the number of nodes and relations that can be treated by the human brain within a semantically meaningful argument space. XP

XP

XPXP

The analysis of the tree covering the paths of merged arguments has produced some interesting results and supported the hypothesis that a syntactic argument structure complies with more general principles, namely, the principle of efficient growth.


RECURSIVE SYSTEMS ARE OBSERVED IN NATURE

AND IN OUR HEADS

THANK YOU

The Natural Law as a Recursive Syntactic Operation

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