This document discusses the concept of functions in linguistics and syntactic theory. It provides examples of how functions have been modeled in different theories, such as grammars representing functions that map inputs to outputs. The document also discusses problems that have arisen with modeling language as computational functions, and proposes moving toward an interactive computation paradigm that allows for bidirectional information flow and adaptation to inputs.

1. Diego Krivochen
University of Reading, UK
School of Psychology and Clinical Language Sciences

2. 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)

3. 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)

4. Development
Step 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.

5. Example 2: Σ, F grammars
 Axioms:
S → NP⏜Aux⏜VP
VP → V⏜NP
NP → Det⏜N
Det → the
N → man, ball
V → hit
Aux → Ø
 Development:
NP⏜Aux⏜VP
Det⏜N⏜VP
Det⏜N⏜Verb⏜NP
the⏜N⏜Verb⏜NP
the⏜man⏜Verb⏜NP
the⏜man⏜hit⏜NP
the⏜man⏜hit⏜Det ⏜N
the⏜man⏜hit⏜the⏜N
the⏜man⏜hit⏜the⏜ball
Each line represents a derivational step, which is subjacent to
the previous one.

6. 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)

7.  (…) 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 by
a function f. (Müller, forthcoming: 16)

8.  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)

9. …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.

10. 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
↮ ↮

11. 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)

12. 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)

13. 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.

14. 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

15. 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

16. 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.)