Beyond Language Equivalence on Visibly Pushdown Automata

Abstract

Abstract. We study (bi)simulation-like preorder/equivalence checking on visibly pushdown automata, visibly BPA (Basic Process Algebra) and visibly one-counter automata. We describe generic methods for proving complexity upper and lower bounds for a number of studied preorders and equivalences like simulation, completed simulation, ready simulation, 2-nested simulation preorders/equivalences and bisimulation equivalence. Our main results are that all the mentioned equivalences and preorders are EXPTIME-complete on visibly pushdown automata, PSPACE-complete on visibly one-counter automata and P-complete on visibly BPA. Our PSPACE lower bound for visibly one-counter automata improves also the previously known DP-hardness results for ordinary one-counter automata and one-counter nets. Finally, we study regularity checking problems for visibly pushdown automata and show that they can be decided in polynomial time.

Bisimilarity of One-Counter Processes Is PSPACE-Complete

“…bisimilarity) which asks if, for a given one-counter process, there is a bisimilar state in some finite system. Decidability of this problem was proven in [9] and according to [24] it follows from [1] and [21] that the problem is also hard for P. We give a simpler P-hardness proof, but we also show that the regularity problem is in P, thus establishing its Pcompleteness. It is appropriate to add that Kučera [12] showed a polynomial algorithm deciding bisimilarity between a one-counter process and a (given) finite system state.…”

“…But the PSPACE-hardness result for (visibly) one-counter processes [24] discourages us from doing so; we should be satisfied with solving our problem in polynomial space. Thus a nondeterministic algorithm working in polynomial space is sufficient (since PSPACE=NPSPACE by Savitch's Theorem).…”

“…In an unpublished article [29] Yen analyses the approach of [9], deriving a triply exponential space upper bound. A PSPACE lower bound for bisimilarity is proven by Srba [24]. This lower bound already holds over one-counter automata that cannot test for zero and whose actions can moreover be restricted to be visible (so called visibly one-counter nets), i.e.…”

“…that the label of the action determines if the counter is incremented, decremented, or not modified respectively. For visibly one-counter automata it is proven in [24] that strong bisimilarity is in PSPACE via reduction to the model checking problem of the modal µ-calculus over one-counter processes [20]. For bisimilarity on general one-counter processes, in particular when dropping the visibility restriction, the situation is surely more involved.…”

Bisimulation equivalence and regularity for real-time one-counter automata

“…PSPACE-hardness in Theorem 3 follows from [25], and NL-hardness in Theorem 5 follows from Proposition 1; hence our contribution consists in showing the upper bounds.…”