On the Mathematical Properties of Linguistic Theories

1. On the Mathematical Properties of Linguistic Theories 인지과학 협동과정 석사 정영임

2. 1. Introduction • The development of new formalisms for metatheories of linguistic theories • Decidability • Generative capacity • Recognition complexity • Linguistic theories • Context-free Grammar(CFG) • Transformational Grammar(TG) • Lexical-functional Grammar(LFG) • Generalized phrase structure Grammar(GPSG) • Tree adjuct Grammar(TAG) • Stratificational Grammar(SG)

3. 2. Preliminary Definition • Elementary Definition from Complexity theory • If cw(n) is O(g), then the worst-case time complexity is O(g). -> almost all inputs to M of size n can be processed in time K*g(n) • A1, A2are available algorithms for f, O(g1)and O(g2) are their worst-case complexity and g1≤g2 ->A2 will be the preferable algorithm (∵ K1>K2)

4. 2. Preliminary Definition • Two machine models • Sequential models(Aho et al. 1974) • Singletape and multitape Turning machine(TM), random-access machines(RAM), random-access stored-program machines(RASP) • Polynomially related • Transforming a sequential algorithm to parallel one improves at most a factor K improvement in speed • Parallel models • Polynomial number of processors and circuit depth O(s2)

5. 3. Context-Free Languages • Recognition techniques for CFL • CKY or Dynamic programming(Hays, J.Cocke, Kasami, Younger) • Requires grammar in Chomsky Normal Form • Squares size of input of n length • Earley’s Algorithm • Recognizes CFG in time O(n3) and space O(n2) and unambiguous CFG in time O(n2) • Ruzzo(1979) • Boolean circuits in depth of O(log(n)2) • Parallel recognition is accomplished in O(log(n)2) time • C.f. Possible number of parses in some grammatical sentences of length n: 2n(Church and Patil 1982)

6. 4. Transformational Grammar • Peters and Ritchie(1973a) • Reflects transformations that move, add and delete constituents which are recoverable • Every r.e. set can be generated by applying a set of transformations to CS. • The base grammar can be independent of the language being generated. • The universal base hypothesis is empirically vacuous. • If S is recursive in the CF base, then L is predictable enumerable and exponentially bounded. • If all recursion in the base grammar passes through S and all derivation satisfy the terminal-length-increasing condition, then the generated language is recursive.

7. 4. Transformational Grammar • Rounds(1975) • Language recognition and generation for every recognizable language in exponential time are done in exponential time under the terminal-length-nondecreasing condition and recoverability deletion • NP-complete problems

8. 4. Transformational Grammar • Berwick • A formalization reduces grammaticality to well-formedness conditions on the surface structure is unusual. • In GB grammar G, surface structure s, yield of s w, a constant K -> the number of node in s : K*length(w) • GB languages have the linear growth or arithmetic growth property • Problems in Berwick’s • The formalization is a radical simplification • Recognition complexity under other constraints • No immediate functional for complexity or for weak generation capacity.

9. 5. Lexical-Functional Grammar • Kaplan and Bresnan(1982) • Without making use of transformation • Two levels of syntactic structure: Constituent structure and Functional structure • Berwick(1982) • A set of strings whose recognition problem is NP-complete is and LFL. • The complexity of LFG comes in finding the assignment of truth-values to the variables.

10. 6. Generalized Phrase Structure Grammar • Gerald Gazdar(1982) • 자연언어 특히 영어에 대한 체계적인 설명과 처리에 적합 • 단일화(unification)에 기반한 통사 자질 이론과 규칙 • 일반 원리(universal principle)과 형식적 제약(formal constraint)

11. 6.1. Node admissibility • Interpretation of Context-Free rules • Rewriting rules • Constraints • Rounds(1970) • Top-down FSTA (q, a, n) => (q1, ……, qn) • Bottom-up FSTA (a, n, (q1, ……, qn) => q)

12. 6.2. Metarules • Gazdar(1982) • Rules that apply to rules to produce other rules • E.g. Passive metarules • W: 다중 집합 변수(Multiple variable) VP -> H[2], NP The beast ate the meat. VP -> H[3], NP, PP[to] Lee gave this to Sandy. VP[PAS] -> H[2], (PP(by) The meat was eaten by the beast. VP[PAS] -> H[3], PP[to], (PP(by)] This was given to Sandy by Lee. VP W, NP VP[PAS] W, (PP[by])

13. 6.2. Metarules • Two devices(or constraints) in metarules • Essential variables • Phantom categories

14. 7. Tree Adjunct Grammar • Joshi(1982, 1984) • A TAG consists of two finite sets of finite trees, the center trees and the adjunct trees. • Adjunction operation • CFLs ⊂ TALs ⊂ indexed languages ⊂ CSLs c a c A t a A A + => n t

15. 8. Stratificational Grammar • The Stratification Grammar(Lamb 1966, Gleason 1964) • Strata Linearly ordered and constrained by a realization relation • Realization relation Application of specific pairs of products in the different grammar (e.g. Pairing of syntactic and semantic rules (Montague)) • Two-level stratificiational grammar Rewriting grammar G1 and G2 Relation R: a finite set of pairs(strings: P1, P2) • D1 in G1 is realized by a derivation D2 in G2 if s1 and s2 can be decomposed into substrings s1=u1….un, s2=v1….vn R(ui, vi)

16. 9. Seeking Significance • How to select the most useful metatheorical results among syntactic theories? => To claim that the computationally most restrictive theory is preferable!

17. 9.1. Coverage • Scope(Linebarger 1980) • An item is in the immediate scope of NOT if (1) it occurs only in the proposition which is the entire scope of NOT (2) within the proposition there are no logical elementsintervening between it and NOT • Polarity reverser(Ladusaw 1979) 1. A negative polarity item will be acceptable only if it is in the scope of a polarity-reversing expression 2. For any two expressions α and β, constituent of a sentence S, α is in the scope ofβ with respect to a composition structure of S, S’, iff the interpretation of α is used in the formulation of the argument β’s interpretation in S’ 3. An expression D is a polarity reverser with respect to an interpretation function Φ if and only if, for all expressions X and Y, Φ(X) ⊆ Φ(Y) => Φ(d(Y)) ⊆ Φ(d(X))

18. 9.1. Coverage • Constraint separation • Syntax-semantics boundary (e.g. polarity-sensitive) • Syntax(e.g. GB, LFG) • Separation sometimes has beneficial computational effect. e.g. Separating constraints imposed by CFGs from constraints by indexed grammar => recognition complexity remains low-order polynomial

19. 9.2. Metatheoretical results as lower bounds • What are the minimal generative capacity and recognition complexity of actual languages?

20. 9.3. Metatheoretical results as upper bounds • The class of possible languages could contain languages that are now recursive. • Putnam(1961) • Languages might just happen to be recursive. • Peters and Ritchie(1973) 1. Every TG has an exponentially bounded cycling function, and thus generates only recursive languages, 2. Every natural language has a descriptive adquate TG 3. The complexity of language investigated so far is typical of the class

21. 9.3. Metatheoretical results as upper bounds • O(g)-result • Asymptotic worst-case measures. • Depends on machine model and RAMs.