Undecidability Results for Bisimilarity on Prefix Rewrite Systems

Abstract

Abstract.We answer an open question related to bisimilarity checking on labelled transition systems generated by prefix rewrite rules on words. Stirling (1996Stirling ( , 1998 proved the decidability of bisimilarity for normed pushdown processes. This result was substantially extended by Sénizergues (1998Sénizergues ( , 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 a −→ 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 a −→ R and R1 a −→ R2 and show when they yield low undecidability (Π 0 1 -completeness) and when high undecidability (Σ 1 1 -completeness), all with and without the assumption of normedness.

Undecidability of bisimilarity by defender's forcing

“…The classes Type -1a and Type -1b are incomparable with respect to bisimilarity and strictly above Type 0 and below Type -2 systems. (We show this in the report [Jančar and Srba 2006]. )…”

“…We can add that Sénizergues' decidability result applies to so-called regular graphs of finite out-degree. These can be characterized by Type -1a rules R a −→w with the restriction that R is a prefix-free regular language (as discussed in the report [Jančar and Srba 2006]). Hence, informally speaking, the decidability boundary lies between the prefix-free and the unrestricted rules R a −→w.…”