Download - Problems of function based syntax
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.)