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

Abstract

We consider the problem of determining if a string w belongs to a language L specified by an automaton (NFA, or PDA augmented by reversal-bounded counters, etc.) where the string w is specified by its Parikh vector. If the automaton (PDA augmented with reversal-bounded counters) is fixed and the Parikh vector is encoded in unary (binary), the problem is in DLOGSPACE (PTIME). When the automaton is part of the input and the Parikh vector is encoded in binary, we show the following results: if the input is an NFA accepting a letter-bounded language (i.e., $\subseteq a_1^* \cdots a_k^*$ for some distinct symbols a ₁, ..., a _k ), the problem is in PTIME, but if the input is an NFA accepting a word-bounded language (i.e., $\subseteq w_1^* \cdots w_m^*$ for some nonnull strings w ₁, ..., w _m ), it is NP-complete. The proofs involve solving systems of linear Diophantine equations with non-negative integer coefficients. As an application of the results, we present efficient algorithms for a generalization of a tiling problem posed recently by Dana Scott. Finally, we give a classification of the complexity of the membership problem for restricted classes of semilinear sets.