1. A FIBONACCI-TREE MODEL OF COGNITIVE PROCESSES
UNDERLYING LANGUAGE FACULTY
Alona Soschen
MIT
Velina Slavova
NBU
Some pronouncement about us being a part of nature
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This paper offers a formal mathematical support of the linguistic model of Argument Based Syntax.
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.
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.
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.
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Introduction
All humans possess the innate Faculty of Language (FL),
the reason why small children learn to speak…
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.
Our hypothesis is that recursion and the principles of efficient growth are in the bases of syntactic constructions.
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RECURSIVE SYSTEMS ARE OBSERVED IN NATURE
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 1284, 3881,
Leonardo
of Pisa ( )
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
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Fibonacci numbers…
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6. the best whole number approximations to the Golden Ratio
WELL KNOWN FACTS…
is known as
Golden Ratio;
Golden number;
Phi (φ)
fibi+2= fibi+1+ fibi
has the solution:
, where
Why Fibonacci numbers in alive nature?
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 !
Each new seed appears at a certain
angle in relation to the preceding one.
The space is filled the best when this angle is exactly the Golden number, and only this 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.
Fibonacci numbers provide
the best whole number approximations to the Golden Ratio
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7. NUMBERS OF EFFICIENT GROWTH
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…
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8. NUMBERS OF EFFICIENT GROWTH
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Syntactic structures are characterized by discreteness and economy.
Our hypothesis is that they comply with the principle of efficient growth.
SYNTACTIC STRUCTURES
Chomsky (1957) offered rewriting rules for syntax.
Sentence S > Verbal Phrase VP + Nominal Phrase NP
NP > Noun N + determiner Phrase DP
S : The boy likes the girl
V’/ likes the girl
The boy/ NP V’
NP/ The girl
likes
The
boy
The
girl
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10. Generalized Bare Phrase Structure (X-bar model)
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
S : The boy likes the girl XP
V’/ likes the girl
The boy/ NP X’ XP
Specifier
NP/ The girl X XP
Complement
Head X
Generalized Bare Phrase Structure (X-bar model)
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THE CORE PRORERTIES OF SYNTACTIC TREES
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 XP X’
set XP
set
term X XP
set
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THE CORE PRORERTIES OF SYNTACTIC TREES
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.
set XP X’
set XP
set
term X XP
set
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13. 3-rd International Workshop on CS and Education in CSXP XP 1
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14. 3-rd International Workshop on CS and Education in CS∑ XP X’ XP 1 1 XP X’ 1 1 2
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15. 3-rd International Workshop on CS and Education in CS∑ XP X’ X XP 1 1 XP X’ 1 1 2 X’ X 2 1 3 1 XP XP
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16. 3-rd International Workshop on CS and Education in CS∑ XP X’ X XP 1 1 XP X’ 1 1 2 X’ X 2 1 3 1 XP XP X XP X’ XP X’ XP 3 2 5 1
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17. 3-rd International Workshop on CS and Education in CS∑ XP X’ X XP 1 1 XP X’ 1 1 2 X’ X 2 1 3 1 XP XP X XP X’ XP X’ XP 3 2 5 1 X’ X X XP XP X’ XP XP X’ XP 5 3 8 2
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18. 3-rd International Workshop on CS and Education in CS∑ XP X’ X XP 1 1 XP X’ 1 1 2 X’ X 2 1 3 1 XP XP X XP X’ XP X’ XP 3 2 5 1 X’ X XP XP X’ X XP XP X’ XP 5 3 8 2 X X’ X X XP XP X’ XP XP X’ XP X’ XP XP X’ XP 8 5 13 3
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The proposed recursive structure has some difficulties:
The tree is bottomless, it grows till the satisfaction of some preliminary determined rule.
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’ X
Sentences are finite. X’ X XP XP X’ X
3-rd International Workshop on CS and Education in CS XP XP X’ XP X XP X’ XP X’ XP XP X’ XP X’ X XP X XP X’ XP

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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. )
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21. 3-rd International Workshop on CS and Education in CSXP X X’ XP X’ XP X X’ XP X X’ XP X X’ XP X
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22. 3-rd International Workshop on CS and Education in CSXP X X’ XP X
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In the Fib series, X’
Are Not
points of branching,
they are only point of growth.
(Ref: Fibonacci – rabbit’s problem) XP X’ XP X X X X X X X
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2. Problem’s solution: Discard X (heads) and intermediate projections X’ and study the structure determines XP-s (the argument – based structure) XP XP X’ X’ X XP XP X XP X’ XP X’ XP X’ X X’ X XP XP XP XP X’ XP X X’ X’ X X’ X’ X XP XP XP XP XP XP XP X’ XP
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2. Problem’s solution: Discard X (heads) and intermediate projections X’ and study the structure determines XP-s (the argument – based structure) XP XP X’ X’ X XP XP X XP X’ XP X’ XP X’ X X’ X XP XP XP XP X’ XP X X’ X’ X X’ X’ X XP XP XP XP XP XP XP X’ XP
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The XP structure corresponds to the known structure
Tree of Fibonacci: XP
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27. arguments Tree of Fibonacci: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.
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2. PROPERTIES OF ARGUMENT-BASED Fib-TREE
A XP-Fib tree of Height h consists of a root node XP, to which two sub-trees are attached –
- The one is Fib-tree of Height h-2,
- The other is a Fib-tree of Height h-1.
The tree of Height 0 has 0 nodes,
The tree of Height 1 has one node XP. Ø h:
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.
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From the syntactic point of view, the “problem” of the XP-Fib tree is that it has nodes which are not binary branching. XP XP XP XP XP XP XP XP XP XP XP XP
In the syntactic sense, nodes are result of merge (sum) of two elements, the internal nodes of a syntactic construction.
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It can be seen as an XP Fib tree in which all non-binary branching nodes (leafs and internal nodes) have additional zero-‘leafs’. XP XP XP XP XP XP XP
Ø -merge
Ø -merge Ø X4 Ø
Ø -merge Ø XP XP XP XP
Ø -merge
Ø -merge
Ø -merge
Ø -merge Ø X1 Ø X2 Ø Ø X5 XP
Ø -merge Ø X3
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31. The bottom of the tree is defined by merging Ø
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. XP
Ø -merge Ø X
The bottom of the tree is defined by merging Ø
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.
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Syntactic trees are generated by successive merge of elements, XP
sentence
merge XP XP
merge
merge XP XP XP XP
merge
Ø -merge
Ø -merge Ø X4 Ø
Ø -merge Ø XP XP XP XP
Ø -merge
Ø -merge
Ø -merge
Ø -merge Ø X1 Ø X2 Ø Ø X5 XP
starting from the bottom level.
Ø -merge Ø X3
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Syntactic trees are generated by successive merge of elements, XP
merge XP XP
merge
merge XP XP XP XP
merge
Ø -merge
Ø -merge Ø X4 Ø
Ø -merge Ø XP XP XP XP
Ø -merge
Ø -merge
Ø -merge
Ø -merge Ø X1 Ø X2 Ø Ø X5 XP
‘leaf’ nodes of the XP Fib-tree
are Ø–merged terminal nodes
(Entities)
Ø -merge Ø X3
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34. All nodes of the tree are of type XPØ,
3. HOMOGENEITY OF THE TREE
We define Ø – merge as an operation that produces a XP node of type “XPØ” XP XP Ø X XP XP Ø X Ø
Ø -merge X Ø X XP Ø X
All nodes of the tree are of type XPØ,
(they are singleton sets).
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This construction provides a perfect justification of operation type-shift. XP Ø X XP XP Ø X Ø X XP Ø X
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The account for introducing this operation is based on a formal definition
‘non-self-inclusiveness of sets’. XP X
type shift XP XP Ø X Ø X
type shift XP Ø X
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.
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free input
X input
Ø entry
X entry Ø
Ø -merge X XP X
type shift XP XP Ø X Ø X
type shift XP Ø X
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3. XP FIB-TREE, LINGUSTIC STATEMENTS, AND RESULTS XP XP XP XP XP XP XP Ø X4 Ø Ø XP XP XP XP Ø X1 Ø Ø X2 Ø X5 XP Ø X3
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39. 3-rd International Workshop on CS and Education in CSXP XP XP XP XP XP XP Ø X4 XP XP XP XP Ø X1 Ø X2 Ø X5 XP Ø X3
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40. 3-rd International Workshop on CS and Education in CSXP XP XP XP XP XP XP Ø X4 Ø XP XP XP XP Ø X1 Ø X2 Ø Ø X5 XP Ø X3
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41. 3-rd International Workshop on CS and Education in CSXP XP XP XP XP XP XP Ø X4 Ø XP XP XP XP Ø Ø Ø X1 X2 Ø X5 XP Ø X3
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42. 3-rd International Workshop on CS and Education in CSXP XP XP XP XP XP XP Ø X4 XP XP XP XP Ø X1 Ø X2 Ø X5 XP Ø X3
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43. 3-rd International Workshop on CS and Education in CSXP XP XP XP XP XP Ø X1 Ø X2 XP Ø X3
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44. 3-rd International Workshop on CS and Education in CSXP XP XP XP XP XP Ø X1 Ø X2 XP Ø X3
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45. 3-rd International Workshop on CS and Education in CSXP XP XP XP XP XP Ø X1 Ø X2 Ø XP Ø X3
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46. 3-rd International Workshop on CS and Education in CSXP XP XP Ø XP XP XP Ø X1 Ø X2 Ø XP Ø X3
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47. 3-rd International Workshop on CS and Education in CSXP XP XP Ø XP XP XP Ø X1 Ø X2 XP Ø X3
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THE BOTTOM-UP MERGE OF ANY TREE LEADS TO THE TREE OF HEIGHT 3 XP XP XP XP XP XP XP Ø X4 Ø Ø XP XP XP XP Ø X1 Ø Ø X2 Ø X5 XP Ø X3
START!
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MERGE ! XP XP XP XP XP XP XP Ø Ø XP XP XP XP Ø XP
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MERGE ! XP XP XP XP XP XP XP XP XP
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ROOT: XP XP
X entry
Ø entry XP XP
The one of the entries is for X.
The other is for 0 or for XP XP XP
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.
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WE HAVE EITHER THIS: XP
Type-shift XP XP XP Ø X
Type-shift
Ø -merge XP XP Ø Ø X
Ø -merge
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OR THIS: XP
Type-shift XP XP XP Ø X
Ø -merge XP XP X Ø X
Ø -merge
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3. XP FIB-TREE, LINGUSTIC STATEMENTS, AND RESULTS XP Ø XP E1 Ø X2
a. Infinite iteration: Mary, Mary…
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55. 3-rd International Workshop on CS and Education in CSXP XP E1 Ø X2
b. Mary in Mary smiles.
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56. c. Two arguments Mary, John in Mary loves JohnXP XP XP E2 Ø X1 XP E1 Ø X2
c. Two arguments Mary, John in
Mary loves John
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57. d. Three arguments Mary, John, apple in Mary gave John an apple.XP XP XP E2 Ø X1 Ø XP E1 Ø X2
d. Three arguments Mary, John, apple in
Mary gave John an apple.
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58. 3-rd International Workshop on CS and Education in CSXP XP XP E2 Ø X1 Ø XP E1 Ø X2
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.
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SOMETHING LIKE CONCLUSION
The question of the height of the XP Fib-tree is deeply related to the limits of the human cognitive resources.
The tree expresses the
paths of connecting smaller units in order to produce a larger meaningful unit.
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
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. XP XP XP
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RECURSIVE SYSTEMS ARE OBSERVED IN NATURE
AND IN OUR HEADS
THANK YOU
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