Bisimilarity of One-Counter Processes Is PSPACE-Complete

Abstract

Abstract.A one-counter automaton is a pushdown automaton over a singleton stack alphabet. We prove that the bisimilarity of processes generated by nondeterministic one-counter automata (with no ε-steps) is in PSPACE. This improves the previously known decidability result (Jančar 2000), and matches the known PSPACE lower bound (Srba 2009). We add the PTIME-completeness result for deciding regularity (i.e. finiteness up to bisimilarity) of one-counter processes.

Bisimilarity of Pushdown Automata is Nonelementary

“…It will be convenient to rely on it when representing a k-EXPSPACE computation through PDA configurations. 1 Note that this is easily achieved by including no outgoing rules for …”

“…Observe that whenever • stop and stop • are reached by the players during the bisimulation game, the stacks are out of synch by one -counter. Thus, before allowing for synchronous popping as in popAll, we will rebalance the stacks by using rules that are dual to those for testDec 1 and testTran. Accordingly, we add the following rules:…”

Bisimilarity on Basic Process Algebra is in 2-ExpTime (an explicit proof)

“…Regarding the lower bounds for the (full) BPA problem, Srba [19] showed PSpacehardness, and Kiefer [15] recently shifted this to ExpTime-hardness (using the ExpTimecompleteness of countdown games [14]); he thus also strengthened the lower bound results known for (visibly) pushdown processes [16], [20] and for weak bisimilarity [17]. This was a bit surprising since the bisimulation equivalence problem for related classes of basic parallel processes (generated by commutative context-free grammars) and of one-counter processes were shown PSpace-complete [11], [3]. The mentioned shift of the lower bound is a natural impulse for looking at the complexity again; confirming the upper bound which has been a bit vaguely stated in the literature becomes more important.…”

Simulation Problems Over One-Counter Nets

“…For OCAs/OCNs, strong bisimulation is PSPACE-complete [3,4], while weak bisimulation is undecidable [12]. Strong trace inclusion is undecidable for OCAs [16], and even for OCNs [7], and this trivially carries over to weak trace inclusion.…”