On the Computational Complexity of Verifying One-Counter Processes

Abstract—One-counter processes are pushdown systems over a singleton stack alphabet (plus a stack-bottom symbol). We study the complexity of two closely related verification problems over one-counter processes: model checking with the temporal logic EF, where formulas are given as directed acyclic graphs, and weak bisimilarity checking against finite systems. We show that both problems are P NP-complete. This is achieved by establishing a close correspondence with the membership problem for a natural fragment of Presburger Arithmetic, which we show to be P NP-complete. This fragment is also a suitable representation for the global versions of the problems. We also show that there already exists a fixed EF formula (resp. a fixed finite system) such that model checking (resp. weak bisimulation) over one-counter processes is hard for P NP[log]. However, the complexity drops to P if the onecounter process is fixed. Keywords-Complexity theory, Logic I

Equivalence-Checking on Infinite-State Systems: Techniques and Results

The paper presents a selection of recently developed and/or used techniques for equivalence-checking on infinite-state systems, and an up-to-date overview of existing results (as of September 2004)

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

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

Bisimilarity of Pushdown Systems is Nonelementary

Given two pushdown systems, the bisimilarity problem asks whether they are bisimilar. While this problem is known to be decidable our main result states that it is nonelementary, improving EXPTIME-hardness, which was the previously best known lower bound for this problem. Our lower bound result holds for normed pushdown systems as well

Checking for Open Bisimilarity in the pi-Calculus

This paper deals with algorithmic checking of open bisimilarity in the pi-calculus. Most bisimulation checking algorithms are based on the partition refinement approach. Unfortunately the definition of open bisimulation does not permit us to use a partition refinement approach for open bisimulation checking directly, but in the paper 'A Partition Refinement Algorithm for the pi-Calculus' Marco Pistore and Davide Sangiorgi present an iterative method that makes it possible to check for open bisimilarity using partition refinement. We have implemented the algorithm presented by Marco Pistore and Davide Sangiorgi. Furthermore,we have optimized this algorithm and implemented this optimized algorithm. The time-complexity of this algorithm is the same as the time-complexity for the first algorithm, but performance tests have shown that in many cases the running time of the optimized algorithm is shorter than the running time of the first algorithm. Our implementation of the optimized open bisimulation checker algorithm and a user interface have been integrated in a system called the OBC Workbench.The source code and a manual for it is available from http://www.cs.auc.dk/research/FS/ny/PR-pi/