Parikhʼs theorem: A simple and direct automaton construction

Esparza, Javier; Ganty, Pierre; Kiefer, Stefan; Luttenberger, Michael

doi:10.1016/j.ipl.2011.03.019

Cited by 48 publications

(56 citation statements)

References 16 publications

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“…We show that given a context-free grammar G in program normal form with pr variables, we can construct a regular grammar AG satisfying Π(L(AG)) = Π(L(G)) in O(|Gi| f (pr ) ) time and space (for some function f ). For this we strengthen a recent result of [8], which shows that such a grammar can be constructed in O(|G| f (v) ) time and space, where v is the total number of variables of G. We start by defining the grammar AG. DEFINITION 5.…”

Section: Grammars Of Arbitrary Size: a Polynomial Casesupporting

confidence: 54%

Complexity of pattern-based verification for multithreaded programs

Esparza

Ganty

2011

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Pattern-based verification checks the correctness of the program executions that follow a given pattern, a regular expression over the alphabet of program transitions of the form w * 1 . . . w * n . For multithreaded programs, the alphabet of the pattern is given by the synchronization operations between threads. We study the complexity of pattern-based verification for abstracted multithreaded programs in which, as usual in program analysis, conditions have been replaced by nondeterminism (the technique works also for boolean programs). While unrestricted verification is undecidable for abstracted multithreaded programs with recursive procedures and PSPACE-complete for abstracted multithreaded while-programs, we show that pattern-based verification is NP-complete for both classes. We then conduct a multiparameter analysis in which we study the complexity in the number of threads, the number of procedures per thread, the size of the procedures, and the size of the pattern. We first show that no algorithm for pattern-based verification can be polynomial in the number of threads, procedures per thread, or the size of the pattern (unless P=NP). Then, using recent results about Parikh images of regular languages and semilinear sets, we present an algorithm exponential in the number of threads, procedures per thread, and size of the pattern, but polynomial in the size of the procedures.

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Section: Grammars Of Arbitrary Size: a Polynomial Casesupporting

confidence: 54%

Complexity of pattern-based verification for multithreaded programs

Esparza

Ganty

2011

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“…By suitably combining that result (in particular the polynomial conversion in the case of NFAs accepting nonunary strings) with the above mentioned result from [2] and with a result by Pighizzini, Shallit, and Wang [9] concerning the unary case, we prove that each context-free grammar in Chomsky normal form with h variables can be converted into a Parikh equivalent DFA with 2 O(h 2 ) states. From the results concerning the unary case, it follows that this bound is tight.…”

Section: Introductionmentioning

confidence: 61%

“…We study this equivalence from a descriptional complexity point of view. Recently, Esparza, Ganty, Kiefer, and Luttenberger proved that each context-free grammar in Chomsky normal form (CNFG) with h variables can be converted into a Parikh equivalent NFA with O(4 h ) states [2]. In [4] it was proven that if G generates a bounded language then we can obtain a DFA with 2 h O(1) states, i.e., a number exponential in a polynomial of the number of variables.…”

Section: Introductionmentioning

confidence: 99%

Converting Nondeterministic Automata and Context-Free Grammars into Parikh Equivalent Deterministic Automata

Lavado

Pighizzini

Seki

2012

Developments in Language Theory

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Abstract. We investigate the conversion of nondeterministic finite automata and context-free grammars into Parikh equivalent deterministic finite automata, from a descriptional complexity point of view. We prove that for each nondeterministic automaton with n states there exists a Parikh equivalent deterministic automaton with e O( √ n·ln n) states. Furthermore, this cost is tight. In contrast, if all the strings accepted by the given automaton contain at least two different letters, then a Parikh equivalent deterministic automaton with a polynomial number of states can be found. Concerning context-free grammars, we prove that for each grammar in Chomsky normal form with n variables there exists a Parikh equivalent deterministic automaton with 2 O(n 2 ) states. Even this bound is tight.

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“…Theorem 5 has lead to a simple algorithm for constructing an automaton whose language is Parikh-equivalent to the language of a given context-free grammar [EGKL11]. Theorem 7 was used in [EKL08a] to improve the complexity bound of [CCFR07] for computing the throughput of context-free grammars from O(n 4 ) to O(n 3 ).…”

Section: Discussionmentioning

confidence: 99%