On the Mathematical Properties of Linguistic Theories

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On the Mathematical Properties of Linguistic
Theories

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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)

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2. Preliminary Definition

• Elementary Definition from Complexity theory
• If cw(n) is O(g), then the worst-case time
  complexity is O(g).
• -gt almost all inputs to M of size n can be
  processed in time Kg(n)
• A1, A2 are available algorithms for f, O(g1)and
  O(g2) are their worst-case complexity and g1g2
• -gt A2 will be the preferable algorithm (? K1gtK2)

context-sensitive CS recursively enumerable r.e.
f(x) a recognition function of a language L a recognition function of a language L a recognition function of a language L
M an algorithm for f an algorithm for f an algorithm for f
c(x) the cost(time and space) of executing M on a specific input x the cost(time and space) of executing M on a specific input x the cost(time and space) of executing M on a specific input x
cw a function whose argument is n(the size of the input to M) a function whose argument is n(the size of the input to M) a function whose argument is n(the size of the input to M)
cw(n) the maximum of c(x), the worst-case complexity function for M the maximum of c(x), the worst-case complexity function for M the maximum of c(x), the worst-case complexity function for M
ce(n) the average of c(x) over all inputs of length n for M, the expected complexity function the average of c(x) over all inputs of length n for M, the expected complexity function the average of c(x) over all inputs of length n for M, the expected complexity function
f is O(g) ngtn0 and f(n)ltKg(n) (K a constant) ngtn0 and f(n)ltKg(n) (K a constant) ngtn0 and f(n)ltKg(n) (K a constant)
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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)

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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
• Earleys 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)

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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.

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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

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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
• -gt the number of node in s Klength(w)
• GB languages have the linear growth or arithmetic
  growth property
• Problems in Berwicks
• The formalization is a radical simplification
• Recognition complexity under other constraints
• No immediate functional for complexity or for
  weak generation capacity.

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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.

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6. Generalized Phrase Structure Grammar

• Gerald Gazdar(1982)
• ???? ?? ??? ?? ???? ??? ??? ??
• ???(unification)? ??? ?? ?? ??? ??
• ?? ??(universal principle)? ??? ??(formal
  constraint)

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6.1. Node admissibility

• Interpretation of Context-Free rules
• Rewriting rules
• Constraints
• Rounds(1970)
• Top-down FSTA
• (q, a, n) gt (q1, , qn)
• Bottom-up FSTA
• (a, n, (q1, , qn) gt q)

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6.2. Metarules

• Gazdar(1982)
• Rules that apply to rules to produce other rules
• E.g. Passive metarules
• W ?? ?? ??(Multiple variable)
• VP -gt H2, NP The beast ate the meat.
• VP -gt H3, NP, PPto Lee gave this to Sandy.
• VPPAS -gt H2, (PP(by) The meat was eaten by
  the beast.
• VPPAS -gt H3, PPto, (PP(by) This was given
  to Sandy by Lee.

VP W, NP VPPAS W,
(PPby)
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6.2. Metarules

• Two devices(or constraints) in metarules
• Essential variables
• Phantom categories

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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

gt
n
t
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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
  s1u1.un, s2v1.vn R(ui, vi)

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9. Seeking Significance

• How to select the most useful metatheorical
  results among syntactic theories?
• gt To claim that the computationally most
  restrictive theory is preferable!

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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
  elements intervening 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 a and ß, constituent
  of a sentence S, a is in the scope ofß with
  respect to a composition structure of S, S, iff
  the interpretation of a is used in the
  formulation of the argument ßs interpretation in
  S
• 3. An expression D is a polarity reverser with
  respect to an interpretation function F if and
  only if, for all expressions X and Y,
• F(X) ? F(Y) gt F(d(Y)) ? F(d(X))

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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
• gt recognition complexity remains low-order
  polynomial

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9.2. Metatheoretical results as lower bounds

• What are the minimal generative capacity and
  recognition complexity of actual languages?

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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

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9.3. Metatheoretical results as upper bounds

• O(g)-result
• Asymptotic worst-case measures.
• Depends on machine model and RAMs.