Undecidability Results for Bisimilarity on Prefix Rewrite Systems

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Abstract

We answer an open question related to bisimilarity checking on labelled transition systems generated by prefix rewrite rules on words. Stirling (1996, 1998) proved the decidability of bisimilarity for normed pushdown processes. This result was substantially extended by Sénizergues (1998, 2005) who showed the decidability for regular (or equational) graphs of finite out-degree (which include unnormed pushdown processes). The question of decidability of bisimilarity for a more general class of so called Type -1 systems (generated by prefix rewrite rules of the form \(R \ensuremath{\stackrel{a}{\longrightarrow}} w\) where R is a regular language) was left open; this was repeatedly indicated by both Stirling and Sénizergues. Here we answer the question negatively, i.e., we show undecidability of bisimilarity on Type -1 systems, even in the normed case. We complete the picture by considering classes of systems that use rewrite rules of the form \(w \ensuremath{\stackrel{a}{\longrightarrow}} R\) and \(R_1 \ensuremath{\stackrel{a}{\longrightarrow}} R_2\) and show when they yield low undecidability (Π \(^{\rm 0}_{\rm 1}\) -completeness) and when high undecidability (Σ \(^{\rm 1}_{\rm 1}\) -completeness), all with and without the assumption of normedness.