Beyond Language Equivalence on Visibly Pushdown Automata

We study (bi)simulation-like preorder/equivalence checking on the class of visibly pushdown automata and its natural subclasses 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.Comment: Final version of paper, accepted by LMC

Branching Bisimilarity of Normed BPA Processes is in NEXPTIME

Branching bisimilarity on normed BPA processes was recently shown to be decidable by Yuxi Fu (ICALP 2013) but his proof has not provided any upper complexity bound. We present a simpler approach based on relative prime decompositions that leads to a nondeterministic exponential-time algorithm; this is close to the known exponential-time lower bound.Comment: This is the same text as in July 2014, but only with some acknowledgment added due to administrative need

Equivalence of Deterministic One-Counter Automata is NL-complete

We prove that language equivalence of deterministic one-counter automata is NL-complete. This improves the superpolynomial time complexity upper bound shown by Valiant and Paterson in 1975. Our main contribution is to prove that two deterministic one-counter automata are inequivalent if and only if they can be distinguished by a word of length polynomial in the size of the two input automata

Decidability of a temporal logic problem for Petri nets

AbstractThe paper solves an open problem from [4] by showing a decision algorithm for a temporal logic language L(Q′, GF). It implies the decidability of the problem of the existence of an infinite weakly fair occurence sequence for a given Petri net; thereby an open problem from [2] is solved

Game Characterization of Probabilistic Bisimilarity, and Applications to Pushdown Automata

We study the bisimilarity problem for probabilistic pushdown automata (pPDA) and subclasses thereof. Our definition of pPDA allows both probabilistic and non-deterministic branching, generalising the classical notion of pushdown automata (without epsilon-transitions). We first show a general characterization of probabilistic bisimilarity in terms of two-player games, which naturally reduces checking bisimilarity of probabilistic labelled transition systems to checking bisimilarity of standard (non-deterministic) labelled transition systems. This reduction can be easily implemented in the framework of pPDA, allowing to use known results for standard (non-probabilistic) PDA and their subclasses. A direct use of the reduction incurs an exponential increase of complexity, which does not matter in deriving decidability of bisimilarity for pPDA due to the non-elementary complexity of the problem. In the cases of probabilistic one-counter automata (pOCA), of probabilistic visibly pushdown automata (pvPDA), and of probabilistic basic process algebras (i.e., single-state pPDA) we show that an implicit use of the reduction can avoid the complexity increase; we thus get PSPACE, EXPTIME, and 2-EXPTIME upper bounds, respectively, like for the respective non-probabilistic versions. The bisimilarity problems for OCA and vPDA are known to have matching lower bounds (thus being PSPACE-complete and EXPTIME-complete, respectively); we show that these lower bounds also hold for fully probabilistic versions that do not use non-determinism

Bisimulation Equivalence of First-Order Grammars is ACKERMANN-Complete

Checking whether two pushdown automata with restricted silent actions are weakly bisimilar was shown decidable by S\'enizergues (1998, 2005). We provide the first known complexity upper bound for this famous problem, in the equivalent setting of first-order grammars. This ACKERMANN upper bound is optimal, and we also show that strong bisimilarity is primitive-recursive when the number of states of the automata is fixed

On the Home-Space Problem for Petri Nets and its Ackermannian Complexity

A set of configurations H is a home-space for a set of configurations X of a Petri net if every configuration reachable from (any configuration in) X can reach (some configuration in) H. The semilinear home-space problem for Petri nets asks, given a Petri net and semilinear sets of configurations X, H, if H is a home-space for X. In 1989, David de Frutos Escrig and Colette Johnen proved that the problem is decidable when X is a singleton and H is a finite union of linear sets with the same periods. In this paper, we show that the general (semilinear) problem is decidable. This result is obtained by proving a duality between the reachability problem and the non-home-space problem. In particular, we prove that for any Petri net and any semilinear set of configurations H we can effectively compute a semilinear set C of configurations, called a non-reachability core for H, such that for every set X the set H is not a home-space for X if, and only if, C is reachable from X. We show that the established relation to the reachability problem yields the Ackermann-completeness of the (semilinear) home-space problem. For this we also show that, given a Petri net with an initial marking, the set of minimal reachable markings can be constructed in Ackermannian time