On the Parikh Membership Problem for FAs, PDAs, and CMs

Ibarra, Óscar H.; Ravikumar, Bala

doi:10.1007/978-3-319-04921-2_2

Cited by 2 publications

(2 citation statements)

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“…Therefore, the Parikh membership problem for linear sets is NP-complete. It remains so even under the restriction m = 1 or ||P || ≤ 3, as long as the cardinality of P is allowed to be arbitrarily large [27]. If the cardinality of P is bounded, or if both m and ||P || are bounded, then the problem can be solved in polynomial time [27].…”

Section: Parikh Membership Problemmentioning

confidence: 99%

“…The NP-completeness of this problem was proved for (commutative) CFGs by Esparza [10] and for DFAs by To [42]. As for NFAs, Ibarra and Ravikumar have recently proved an interesting contrast that the problem is NP-complete for NFAs that accept a bounded language, while it can be solved in polynomial time for NFAs that accept a letter-bounded language [27]. They also proposed a pseudopolynomial-time 6 algorithm to solve the Parikh membership problem for reversal-bounded counter machines.…”

Section: Parikh Membership Problemmentioning

confidence: 99%

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Semilinear Sets and Counter Machines: a Brief Survey

Ibarra

Seki

2015

Fundamenta Informaticae

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Semilinear sets are one of the most important concepts in theoretical computer science, as illustrated by the fact that the set of nonnegative integer solutions to any system of Diophantine equations is semilinear. Parikh's theorem enables us to represent any semilinear set as a pushdown automaton (PDA). We summarize recent results on the descriptional complexity of conversions among different representations of a semilinear set: as a vector set (conventional), a finite automaton (FA), a PDA, etc.. We also discuss semilinearity-preserving operations like union, intersection, and complement. We use Parikh's theorem to enlarge the class of finite-state machines that can represent semilinear sets. In particular, we give a simpler proof of a known result that characterizes semilinear sets in terms of machines with reversal-bounded counters. We then investigate the power of such a machine with only one counter in the context of a long-standing conjecture about repetition on words.

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Section: Parikh Membership Problemmentioning

confidence: 99%

Section: Parikh Membership Problemmentioning

confidence: 99%