Diego Krivochen

University of Reading, UK

School of Psychology and Clinical Language Sciences

What is a function? A function is a relation between a set of inputs and a

set of permissible outputs with the property that each input is related to exactly one output.

(based on Falcade et. al., 2004; Youschkevitch, 1976/1977: 39; May, 1962, among others)

Properties:

Closed to external influence

Operate in polynomial (i.e., finite) time

Alphabet & rules are fixed a priori

Strictly serial (very local access)

Example 1: quadratic functions Axiom: f(x) = x2

This function relates each value of x to its square x2 by means of a definite rule, ‘multiply x by itself ’

Alphabet: ℤ

Halting: only by stipulation (if the memory tape is infinite)

DevelopmentStep 1: f(1) = 12

Step 2: f(2) = 22

Step 3: f(3) = 32

…

Step n: f(n) = n2

The nth step is defined by the axiom alone, as the system has no access to previous information or to what will come next.

Example 2: Σ, F grammars Axioms: S → NP⏜Aux⏜VP VP → V⏜NPNP → Det⏜NDet → theN →man, ballV → hitAux → Ø

Development:NP⏜Aux⏜VPDet⏜N⏜VPDet⏜N⏜Verb⏜NPthe⏜N⏜Verb⏜NPthe⏜man⏜Verb⏜NPthe⏜man⏜hit⏜NPthe⏜man⏜hit⏜Det ⏜Nthe⏜man⏜hit⏜the⏜Nthe⏜man⏜hit⏜the⏜ball

Each line represents a derivational step, which is subjacent to the previous one.

Functions in the theories of syntax Since any language L in which we are likely to be interested is an infinite

set, we can investigate the structure of L only through the study of the finite devices (grammars) which are capable of enumerating its sentences. A grammar of L can be regarded as a function whose range is exactly L. (Chomsky, 1959: 137)

“We must require of such a linguistic theory that it provide for:

(i) an enumeration of the class S1' S2', … of possible sentences

(ii) an enumeration of the class SD1, SD2, … of possible structural descriptions

(iii) an enumeration of the class G1, G2, … of possible generative grammars

(iv) specification of a function f such that SDf(i, j) is the structural description assigned to sentence Si, by grammar Gj, for arbitrary i,j(v) specification of a function m such that m(z) is an integer associated with the grammar G, as its value (with, let us say, lower value indicated by higher number)” Chomsky (1965: 31)

(…) individual neurons can be modeled by finite automata […], and a finite three-dimensional array of such automata can be substituted by one finite automaton […], NLs must be regular. [Type 3] (Kornai, 1985: 4)

An f-structure is a mathematical function that represents the grammatical functions of a sentence […] all f-structures are functions of one argument (…) (Kaplan & Bresnan, 1982: 182-183)

The HPSG lexicon […] consists of roots that are related to stems or fully inflected words. The derivational or inflectional rules may influence part of speech (e.g. adjectival derivation) and/or valence (-able adjectives and passive) […] The stem is mapped to a word and the phonology of the input […] is mapped to the passive form bya function f. (Müller, forthcoming: 16)

This analysis [Pollard & Sag, 1994; below] employs an App(end)-synsems function that appends its second argument (a list of synsems) to a list of the synsem values of its first argument (which is a list of phrases). (Green, 2011: 24)

…and even in ‘performance-oriented theories’

Complexity is a function of the amount of structure that is associated with the terminal elements, or words, of a sentence.(…) complexity is a function of the number of formal units and conventionally associated properties that need to be processed in domains relevant for their processing. Hawkins (2004: 8 / 25)

Rejects UG, but embraces the DTC, based on Miller & Chomsky (1963)

The DTC can also be found in approaches to SLI like Jakubowicz (2011): complexity is a function of operations / derivational steps.

The Minimalist Program We take L [a particular language] to be a generative

procedure that constructs pairs (π, λ) that are interpreted at the articulatory perceptual (A-P) and conceptual-intentional (C-I) interfaces (…). Chomsky, 1995: 219)

phrase structure (…) always completely determines linear order […] Linear Correspondence Axiom: d(A) is a linear ordering of T. (A a set of non-terminals, T a set of terminals) (Kayne, 1994: 3, 6)

Lexicon → Numeration →(⇄)

Computational System ⇉ A-P / C-I

↮ ↮

Conditions over derivations: Inclusiveness Condition: No new features are

introduced by CHL […] permits rearrangement of LIs and of elements constructed in the course of derivation, and deletion of features of LI, but optimally, nothing more. (Chomsky, 2000: 113)

Full Interpretation: There can be no superfluous symbols in representations (Chomsky, 1995: 27)

(…) Yet another [UG condition] imposes "local determinability" conditions (barring "look-ahead," "backtracking," or comparison of alternatives). (Op. Cit.: 99)

Some problems: ‘Combination problem’:

𝑛!

𝑛−𝑘 !𝑘!⇒ 𝑁𝑈𝑀!

𝑁𝑈𝑀−𝐷𝑖

!𝐷𝑖!

‘Uniformity problem’: [X…X…X] ⇒ [X [X [X]]] (also, ‘Lyons’ problem’ → stipulations over labels)

‘Interpretation problem’: Semantic Interpretation > LI + C(HL)

‘Implementational problem’: derivations are at odds with real-time processing. Unidirectional information flow

No temporal dimension

False sense of ‘derivational topology’ (bottom-up / top-down)

Some more problems: HPSG: if syntactic structure projects from lexical items with highly

specified feature matrices, how to account (in a reasonably elegant way) for:

Alternances Idioms Incorporated complex structures

LFG: Entscheidungsproblem

Decidibility Theorem: for any lexical-functional grammar G and for any string s, it is decidable whether s belongs to the language of G (Kaplan & Bresnan, 1982: 267)

However…

An LFG is formally between Type 1 and Type 2 languages.

A possible solution… change the paradigm Interactive Computation (Wegner 1997, 1998; Goldin &

Wegner, 2005, 2007, a.o.):

(…) computation is viewed as an ongoing process that transforms inputs to outputs – e.g., control systems, or

operating systems. (Goldin & Wegner, 2007: 5)

Properties: Open to external influence

Bidirectional information flow

Input-Output entanglement

Computationally…

Replace uniform a-machines with (kind of) c-machines in automaton theory (Turing, 1936: 232)

Replace the static Chomsky Theorem with a dynamic conception of mental processes (Krivochen, forthcoming; Krivochen & Mathiasen, 2012):

Adapting to the input

Able to ‘switch’ between different levels of complexity

Psycholinguistically…

Revisit the AxS model (Townsend & Bever, 2001) under interactive premises

Take the implementational level of the development of a theory seriously when building a formal grammar

Test the claim that computation equals computation of functions separately from the thesis that mental processes are computational (contra Copeland, 2002; Deutsch, 1985; Fitz, 2006; a.o.)