The complexity of equivalence problems for commutative grammars

Huynh, Dung T.

doi:10.1016/s0019-9958(85)80015-2

Cited by 24 publications

(21 citation statements)

References 6 publications

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“…We include the very recent result regarding unfixed alphabets [HH14] that inclusion for regular grammars over an unfixed alphabet is coNEXP hard; it is likely that the proof can be also adapted for universality. On the other hand, it is known from [Huy85] that inclusion and universality for context-free grammars over an unfixed alphabet is in coNEXP. For completeness, we also include N (the non-commutative case -note that the membership problem is actually a different problem in the non-commutative case, since we cannot encode long words succinctly with the length of the word in binary); these results are known from other sources ( [MS72], also see [HU79] for a reference).…”

Section: Complexity Resultsmentioning

confidence: 99%

Complexity of Problems of Commutative Grammars

Kopczyński

2015

Logical Methods in Computer Science

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Abstract. We consider commutative regular and context-free grammars, or, in other words, Parikh images of regular and context-free languages. By using linear algebra and a branching analog of the classic Euler theorem, we show that, under an assumption that the terminal alphabet is fixed, the membership problem for regular grammars (given v in binary and a regular commutative grammar G, does G generate v?) is P, and that the equivalence problem for context free grammars (do G1 and G2 generate the same language?) is in Π P 2 . IntroductionLet Σ be a finite alphabet. By Σ * we denote the set of words over Σ, or finite sequences of elements of Σ. For a word w ∈ Σ * , by Ψ(w) (the Parikh image of w) we denote the function from Σ to non-negative integers N, such that eachContext free and regular languages are one of the most important classes of languages in computer science [HU79]. By a famous result of Parikh [Par66], a subset of N Σ is a Parikh image of a context free language if and only if it is a semilinear set, or a union of finitely many linear sets.In this paper, we explore the complexity of various problems related to Parikh images of context free languages, such as the following:• Membership: Given a context-free grammar G and v ∈ N Σ (given in binary). Is v a member of the Ψ(G), the Parikh image of the language generated by G? • Universality: Given two context-free grammars G, is Ψ(G) equal to N Σ ?• Inclusion: Given two context-free grammars G 1 and G 2 , does Ψ(G 1 ) ⊆ Ψ(G 2 )?• Equality: Given two context-free grammars G 1 and G 2 , does Ψ(G 1 ) = Ψ(G 2 )?• Disjointness: Given two context-free grammars G 1 and G 2 , is Ψ(G 1 ) ∩ Ψ(G 2 ) nonempty?Since in this paper we are never interested in the order of terminals or non-terminals, we treat everything in a commutative way. This allows us to identify the commutative languages (subsets of Σ * ) with their Parikh images (subsets of N Σ ).

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Section: Complexity Resultsmentioning

confidence: 99%

Complexity of Problems of Commutative Grammars

Kopczyński

2015

Logical Methods in Computer Science

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“…This involves checking whether the Parikh Image of one regular language is (not) contained in the Parikh Image of another regular language. It is known that this problem can be solved in nondeterministic exponential time (NEXPTIME) [25]. Proposition 11.…”

Section: Upper Bound For Unrestricted Quiescent Consistencymentioning

confidence: 99%

“…A nondeterministic Turing Machine can solve this problem in PSPACE as follows. First, it guesses a run σ whose length is at 2 Cited in [25]. most the upper bound and checks that σ is end-to-end quiescent.…”

Section: Proposition 13mentioning

confidence: 99%

Decidability and Complexity for Quiescent Consistency

Dongol

Hierons

2016

Proceedings of the 31st Annual ACM/IEEE Symposium on Logic in Computer Science

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Quiescent consistency is a notion of correctness for a concurrent object that gives meaning to the object's behaviours in quiescent states, i.e., states in which none of the object's operations are being executed. The condition enables greater flexibility in object design by allowing more behaviours to be admitted, which in turn allows the algorithms implementing quiescent consistent objects to be more efficient (when executed in a multithreaded environment).Quiescent consistency of an implementation object is defined in terms of a corresponding abstract specification. This gives rise to two important verification questions: membership (checking whether a behaviour of the implementation is allowed by the specification) and correctness (checking whether all behaviours of the implementation are allowed by the specification). In this paper, we consider the membership and correctness conditions for quiescent consistency, as well as a restricted form that assumes an upper limit on the number of events between two quiescent states. We show that the membership problem for unrestricted quiescent consistency is NP-complete and that the correctness problem is decidable, coNEXPTIME-hard, and in EXPSPACE. For the restricted form, we show that membership is in PTIME, while correctness is PSPACE-complete.

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“…It follows that the equivalence problem of commutative context-free grammars is decidable, see also [10]. In fact, as shown in [16], the equivalence problem for both commutative context-free grammars and commutative rational expressions are solvable in nondeterministic exponential time. Thus, the equational theory of continuous idempotent commutative semirings is also decidable in nondeterministic exponential time.…”

Section: For Every µ-Term T There Exists a Rational Term R Such That mentioning

confidence: 99%