Deciding semantic finiteness of pushdown processes and first-order grammars w.r.t. bisimulation equivalence

Petr Jančar

doi:10.1016/j.jcss.2019.10.002

Petr Jančar

Open Access

https://doi.org/10.1016/j.jcss.2019.10.002

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Journal: Journal of Computer and System Sciences	Publication Date: Dec 10, 2019
Citations: 5	License type: publisher-specific-oa

Affiliation: Palacký University, Olomouc

Abstract

The problem if a given configuration of a pushdown automaton (PDA) is bisimilar with some (unspecified) finite-state process is shown to be decidable. The decidability is proven in the framework of first-order grammars, which are given by finite sets of labelled rules that rewrite roots of first-order terms. The framework is equivalent to PDA where also deterministic (i.e. alternative-free) epsilon-steps are allowed, hence to the model for which Sénizergues showed an involved procedure deciding bisimilarity (1998, 2005). Such a procedure is here used as a black-box part of the algorithm.The result extends the decidability of the regularity problem for deterministic PDA that was shown by Stearns (1967), and later improved by Valiant (1975) regarding the complexity. The decidability question for nondeterministic PDA, answered positively here, had been open (as indicated, e.g., by Broadbent and Göller, 2012).

Highlights

The question of deciding semantic equivalences of systems, like language equivalence, has been a frequent topic in computer science
Language equivalence and regularity are undecidable for Pushdown automata (PDA)
Logic, verification, and other areas, a finer equivalence, called bisimulation equivalence or bisimilarity, has emerged as another fundamental behavioural equivalence; on deterministic systems it essentially coincides with language equivalence

Summary

Introduction

The question of deciding semantic equivalences of systems, like language equivalence, has been a frequent topic in computer science. Extrapolating the deterministic case, we might expect that for PDA the “regularity” problem w.r.t. bisimilarity (asking if a given PDA-configuration is bisimilar with a state in a finite-state system) is decidable as well, and that this problem might be easier than the equivalence problem solved in [10]; “only” EXPTIME-hardness is known here (see [7], and [11] for detailed references). This decidability question has been open so far, as indicated in [2] (besides [11]). A full version of this paper is planned to appear as the second version of the paper at http://arxiv.org/abs/1305.0516; it will contain detailed proofs

Basic Notions and Result

Proof of Theorem 1

Simple witnesses of bisim-infiniteness

Each bisim-infinite term has a simple witness

Additional Remarks