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Problems of function based syntax

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Diego Krivochen
Diego Krivochen

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.

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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: ? Halting: only by stipulation (if the memory tape is infinite)
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.
Example 2: ¦², F grammars ? Axioms: S ¡ú NP?Aux?VP VP ¡ú V?NP NP ¡ú Det?N Det ¡ú the N ¡ú man, ball V ¡ú hit Aux ¡ú ? ? 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.
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 by a function f. (M¨¹ller, 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 (¦Ð, ¦Ë) 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 ¡ú Numeration ¡ú (?) 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¡¯: ?! ??? !?! ? ???! ?????? !?? ! ? ¡®Uniformity problem¡¯: [X¡­X¡­X] ? [X [X [X]]] (also, ¡®Lyons¡¯ problem¡¯ ¡ú 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 ¨C 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.)
Problems of function based syntax

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Problems of function based syntax

  • 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: ? 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: ¦², F grammars ? Axioms: S ¡ú NP?Aux?VP VP ¡ú V?NP NP ¡ú Det?N Det ¡ú the N ¡ú man, ball V ¡ú hit Aux ¡ú ? ? 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. (M¨¹ller, 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 (¦Ð, ¦Ë) 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 ¡ú Numeration ¡ú (?) 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¡¯: ?! ??? !?! ? ???! ?????? !?? ! ? ¡®Uniformity problem¡¯: [X¡­X¡­X] ? [X [X [X]]] (also, ¡®Lyons¡¯ problem¡¯ ¡ú 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 ¨C 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.)