Complexity of Problems of Commutative Grammars

Abstract

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 ).

Characterization and complexity results on jumping finite automata

“…The non-disjointness problem is the task to decide, for given automata M 1 and M 2 , whether there exists a word w that is accepted by both M 1 and M 2 , i. e., whether it holds that L(M 1 ) ∩ L(M 2 ) = ∅. In the case of JFA, we encounter a similar situation as for the universal word problem, i. e., it can be decided in polynomial time for fixed alphabets, while it becomes NP-complete in general [50].…”

“…Results of [67] and the fact that, on unary alphabets, classical nondeterministic machines coincide with JFAs, imply that the non-universality problem for JFA is NP-hard even if restricted to JFA with unary alphabets. On the other hand, [50] shows (in terms of a more general model) that non-universality lies in NP for any fixed alphabet size. For the unrestricted variant of non-universality, which is trivially NP-hard as well, no close upper bound of the complexity is known [50].…”

Deterministic Automata for Unordered Trees

“…A proof for membership is just a vertical run, with assorted horizontal runs, checkable in PTIME, hence the upper bound. For L (A) ∩ L (B) = ∅, the polynomial check of [8,7] can be used in our case by replacing the infinite alphabet A by the pair of rules a labeled datavalue would use in the horizontal automata of A and B. To make sure we do not combine two mutually disjunctive rules, we need to make sure in polynomial time that their conjunction is singleton-satisfiable.…”

Context-free commutative grammars with integer counters and resets

“…Apart from the related work mentioned above, closely connected to the problems considered in this paper is the work by Kopczyński and To [23] and Kopczyński [24]. In their work, the complexity of various decision problems for context-free commutative grammars and subclasses thereof has been studied when the number of alphabet symbols (which roughly corresponds to the number of places in the Petri net representation) is fixed.…”