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.

Ping-Pong Protocols as Prefix Grammars and Turchin Relation

“…There exists a hierarchy of prefix rewriting grammars (called Caucal hierarchy) that classifies the grammars by the types of their rewriting rules [5]. The Caucal hierarchy of the prefix rewriting grammars is presented in the following table 2 .…”

Turchin's Relation and Subsequence Relation in Loop Approximation

“…R [1] : f → Λ R [3] : g → Λ R [5] : h → Λ R [2] : f → gf R [4] : g → h R [6] : h → g Γ 0 = f . A first segment of a computation yielded by the grammar G LOG can look as:…”

“…R [3] : x → h, 8 R [4] : x → Λ The computation by G LOG that corresponds to the computation from Example 2 now begins as follows:…”

“…There are no intersections in the right-hand sides of rewrite rules and thus ≈ discerns prefixes that are yielded by distinct rule applications. 4 Now we can find the upper bound of a bad sequence length in a computation yielded by a prefix grammar. The proof uses the following lemma.…”

“…We determine upper bounds of bad sequence length with respect to both relations for sequences generated by a single prefix grammar and for direct products of two sequences generated by prefix grammars. 4. We show that a minimal natural well binary generalization of Turchin's relation on direct products of sequences generated by prefix grammar is the subsequence relation.…”