Undecidability of bisimilarity by defender's forcing

Abstract

Stirling [1996Stirling [ , 1998 proved the decidability of bisimilarity on so-called normed pushdown processes. This result was substantially extended by Sénizergues [1998, 2005] who showed the decidability of bisimilarity for regular (or equational) graphs of finite out-degree; this essentially coincides with weak bisimilarity of processes generated by (unnormed) pushdown automata where the ε-transitions can only deterministically pop the stack. The question of decidability of bisimilarity for the more general class of so called Type -1 systems, which is equivalent to weak bisimilarity on unrestricted ε-popping pushdown processes, was left open. This was repeatedly indicated by both Stirling and Sénizergues. Here we answer the question negatively, that is, we show the undecidability of bisimilarity on Type -1 systems, even in the normed case.We achieve the result by applying a technique we call Defender's Forcing, referring to the bisimulation games. The idea is simple, yet powerful. We demonstrate its versatility by deriving further results in a uniform way. First, we classify several versions of the undecidable problems for prefix rewrite systems (or pushdown automata) as 0 1 -complete or 1 1 -complete. Second, we solve the decidability question for weak bisimilarity on PA (Process Algebra) processes, showing that the problem is undecidable and even 1 1 -complete. Third, we show 1 1 -completeness of weak bisimilarity for so-called parallel pushdown (or multiset) automata, a subclass of (labeled, place/transition) Petri nets.

Equivalences of Pushdown Systems Are Hard

“…. , (n d , n d ))) is implemented as the pair (14). The points 1 and 2 directly correspond to the previous points 1 and 2.…”

Bisimilarity in Fresh-Register Automata

“…In this framework, bisimilarity corresponds to the existence of a winning strategy for Defender. The process of playing a bisimulation game naturally favours Attacker as the decision maker but, thanks to the forcing technique of [12], it is possible to construct transition systems in which Defender effectively ends up making choices.…”

“…From an ALBA M we construct an RA(S#) A that simulates it, with the the binary tape content of M encoded by the register assignment of A. At every step of the bisimulation game, we arrange for Defender to choose transitions from existential states (using Defender forcing [12]) and Attacker to choose from universal states.…”

The algorithmics of bisimilarity

“…For the purpose of this presentation we shall use the so-called Defender's Forcing Technique, which simplifies many of the constructions for hardness results. For a more detailed introduction to this technique consult (Jančar and Srba, 2008).…”

“…However, in Sections 0.3.2-0.3.3, we shall present some well-known results in a new light, using the view of bisimilarity and other co-inductively defined process semantics as games. We shall also highlight some general proof techniques based on tableaux techniques and on the so-called 'Defender's forcing technique' (Jančar and Srba, 2008). We feel that the use of these techniques makes the proofs of some results more transparent than the ones originally presented in the literature.…”