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23
Parsing InsideOut
, 1998
"... Probabilistic ContextFree Grammars (PCFGs) and variations on them have recently become some of the most common formalisms for parsing. It is common with PCFGs to compute the inside and outside probabilities. When these probabilities are multiplied together and normalized, they produce the probabili ..."
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Cited by 82 (2 self)
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Probabilistic ContextFree Grammars (PCFGs) and variations on them have recently become some of the most common formalisms for parsing. It is common with PCFGs to compute the inside and outside probabilities. When these probabilities are multiplied together and normalized, they produce the probability that any given nonterminal covers any piece of the input sentence. The traditional use of these probabilities is to improve the probabilities of grammar rules. In this thesis we show that these values are useful for solving many other problems in Statistical Natural Language Processing. We give a framework for describing parsers. The framework generalizes the inside and outside values to semirings. It makes it easy to describe parsers that compute a wide variety of interesting quantities, including the inside and outside probabilities, as well as related quantities such as Viterbi probabilities and nbest lists. We also present three novel uses for the inside and outside probabilities. T...
Semiring Parsing
 Computational Linguistics
, 1999
"... this paper is that all five of these commonly computed quantities can be described as elements of complete semirings (Kuich 1997). The relationship between grammars and semirings was discovered by Chomsky and Schtitzenberger (1963), and for parsing with the CKY algorithm, dates back to Teitelbaum ( ..."
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Cited by 64 (1 self)
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this paper is that all five of these commonly computed quantities can be described as elements of complete semirings (Kuich 1997). The relationship between grammars and semirings was discovered by Chomsky and Schtitzenberger (1963), and for parsing with the CKY algorithm, dates back to Teitelbaum (1973). A complete semiring is a set of values over which a multiplicative operator and a commutative additive operator have been defined, and for which infinite summations are defined. For parsing algorithms satisfying certain conditions, the multiplicative and additive operations of any complete semiring can be used in place of/x and , and correct values will be returned. We will give a simple normal form for describing parsers, then precisely define complete semirings, and the conditions for correctness
Kleene Algebra with Domain
, 2003
"... We propose Kleene algebra with domain (KAD), an extension of Kleene algebra with two equational axioms for a domain and a codomain operation, respectively. KAD considerably augments the expressibility of Kleene algebra, in particular for the specification and analysis of state transition systems. We ..."
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Cited by 41 (29 self)
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We propose Kleene algebra with domain (KAD), an extension of Kleene algebra with two equational axioms for a domain and a codomain operation, respectively. KAD considerably augments the expressibility of Kleene algebra, in particular for the specification and analysis of state transition systems. We develop the basic calculus, discuss some related theories and present the most important models of KAD. We demonstrate applicability by two examples: First, an algebraic reconstruction of Noethericity and wellfoundedness. Second, an algebraic reconstruction of propositional Hoare logic.
A Kleene theorem for weighted tree automata
 Theory of Computing Systems
, 2002
"... In this paper we prove Kleene's result for tree series over a commutative and idempotent semiring A (which is not necessarily complete or continuous), i.e., the class of recognizable tree series over A and the class of rational tree series over A are equal. We show the result by direct automatatheo ..."
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Cited by 17 (8 self)
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In this paper we prove Kleene's result for tree series over a commutative and idempotent semiring A (which is not necessarily complete or continuous), i.e., the class of recognizable tree series over A and the class of rational tree series over A are equal. We show the result by direct automatatheoretic constructions and prove their correctness.
Growth and ergodicity of contextfree languages
 Trans. Amer. Math. Soc
, 2002
"... Abstract. A language L over a finite alphabet Σ is called growthsensitive if forbidding any set of subwords F yields a sublanguage L F whose exponential growth rate is smaller than that of L. It is shown that every ergodic unambiguous, nonlinear contextfree language is growthsensitive. “Ergodic ” ..."
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Cited by 11 (7 self)
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Abstract. A language L over a finite alphabet Σ is called growthsensitive if forbidding any set of subwords F yields a sublanguage L F whose exponential growth rate is smaller than that of L. It is shown that every ergodic unambiguous, nonlinear contextfree language is growthsensitive. “Ergodic ” means for a contextfree grammar and language that its dependency digraph is strongly connected. The same result as above holds for the larger class of essentially ergodic contextfree languages, and if growth is considered with respect to the ambiguity degrees, then the assumption of unambiguity may be dropped. The methods combine a construction of grammars for 2block languages with a generating function technique regarding systems of algebraic equations. 1. Introduction and
Bounded Underapproximations
"... We show a new and constructive proof of the following languagetheoretic result: for every contextfree language L, there is a bounded contextfree language L ′ ⊆ L which has the same Parikh (commutative) image as L. Bounded languages, introduced by Ginsburg and Spanier, are subsets of regular lang ..."
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Cited by 8 (1 self)
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We show a new and constructive proof of the following languagetheoretic result: for every contextfree language L, there is a bounded contextfree language L ′ ⊆ L which has the same Parikh (commutative) image as L. Bounded languages, introduced by Ginsburg and Spanier, are subsets of regular languages of the form w ∗ 1w ∗ 2 · · · w ∗ m for some w1,..., wm ∈ Σ ∗. In particular bounded contextfree languages have nice structural and decidability properties. Our proof proceeds in two parts. First, we give a new construction that shows that each context free language L has a subset LN that has the same Parikh image as L and that can be represented as a sequence of substitutions on a linear language. Second, we inductively construct a Parikhequivalent bounded contextfree subset of LN. We show two applications of this result in model checking: to underapproximate the reachable state space of multithreaded procedural programs and to underapproximate the reachable state space of recursive counter programs. The bounded language constructed above provides a decidable underapproximation for the original
Growthsensitivity of contextfree languages
, 2003
"... A language L over a finite alphabet is called growthsensitive if forbidding any set of subwords F yields a sublanguage L F whose exponential growth rate is smaller than that of L. It is shown that every (essentially) ergodic nonlinear contextfree language of convergent type is growthsensitive. ..."
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Cited by 5 (2 self)
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A language L over a finite alphabet is called growthsensitive if forbidding any set of subwords F yields a sublanguage L F whose exponential growth rate is smaller than that of L. It is shown that every (essentially) ergodic nonlinear contextfree language of convergent type is growthsensitive. “Ergodic” means that the dependency digraph of the generating contextfree grammar is strongly connected, and “essentially ergodic” means that there is only one nonregular strong component in that graph. The methods combine (1) an algorithm for constructing from a given grammar one that generates the associated 2block language and (2) a generating function technique regarding systems of algebraic equations. Furthermore, the algorithm of (1) preserves unambiguity as well as the number of nonregular strong components of the dependency digraph.
COMPUTING THE LEAST FIXED POINT OF POSITIVE POLYNOMIAL SYSTEMS
, 2010
"... We consider equation systems of the form X1 = f1(X1,...,Xn),..., Xn = fn(X1,...,Xn), where f1,...,fn are polynomials with positive real coefficients. In vector form we denote such an equation system by X = f(X) andcallf a system of positive polynomials (SPP). Equation systems of this kind appear na ..."
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Cited by 4 (4 self)
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We consider equation systems of the form X1 = f1(X1,...,Xn),..., Xn = fn(X1,...,Xn), where f1,...,fn are polynomials with positive real coefficients. In vector form we denote such an equation system by X = f(X) andcallf a system of positive polynomials (SPP). Equation systems of this kind appear naturally in the analysis of stochastic models like stochastic contextfree grammars (with numerous applications to natural language processing and computational biology), probabilistic programs with procedures, websurfing models with back buttons, and branching processes. The least nonnegative solution μf of an SPP equation X = f(X) is of central interest for these models. Etessami and Yannakakis [J. ACM, 56 (2009), pp. 1–66] have suggested a particular version of Newton’s method to approximate μf. We extend a result of Etessami and Yannakakis and show that Newton’s method starting at 0 always converges to μf. We obtain lower bounds on the convergence speed of the method. For socalled strongly connected SPPs we prove the existence of a threshold kf ∈ N such that for every i ≥ 0the(kf + i)th iteration of Newton’s method has at least i valid bits of μf. The proof yields an explicit bound for kf depending only on syntactic parameters of f. We further show that for arbitrary SPP equations, Newton’s method still converges linearly: there exists a threshold kf and an αf> 0 such that for every i ≥ 0the(kf + αf · i)th iteration of Newton’s method has at least i valid bits of μf. The proof yields an explicit bound for αf;the bound is exponential in the number of equations in X = f(X), but we also show that it is essentially optimal. The proof does not yield any bound for kf, but only proves its existence. Constructing a bound for kf is still an open problem. Finally, we also provide a geometric interpretation of Newton’s method for SPPs.