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)

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

Branching Bisimilarity on Normed BPA Is EXPTIME-complete

We put forward an exponential-time algorithm for deciding branching bisimilarity on normed BPA (Bacis Process Algebra) systems. The decidability of branching (or weak) bisimilarity on normed BPA was once a long standing open problem which was closed by Yuxi Fu. The EXPTIME-hardness is an inference of a slight modification of the reduction presented by Richard Mayr. Our result claims that this problem is EXPTIME-complete.Comment: We correct many typing errors, add several remarks and an interesting toy exampl

A Polynomial Time Algorithm for Deciding Branching Bisimilarity on Totally Normed BPA

Strong bisimilarity on normed BPA is polynomial-time decidable, while weak bisimilarity on totally normed BPA is NP-hard. It is natural to ask where the computational complexity of branching bisimilarity on totally normed BPA lies. This paper confirms that this problem is polynomial-time decidable. To our knowledge, in the presence of silent transitions, this is the first bisimilarity checking algorithm on infinite state systems which runs in polynomial time. This result spots an instance in which branching bisimilarity and weak bisimilarity are both decidable but lie in different complexity classes (unless NP=P), which is not known before. The algorithm takes the partition refinement approach and the final implementation can be thought of as a generalization of the previous algorithm of Czerwi\'{n}ski and Lasota. However, unexpectedly, the correctness of the algorithm cannot be directly generalized from previous works, and the correctness proof turns out to be subtle. The proof depends on the existence of a carefully defined refinement operation fitted for our algorithm and the proposal of elaborately developed techniques, which are quite different from previous works.Comment: 32 page

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

Two Lower Bounds for BPA

Branching bisimilarity of normed Basic Process Algebra (nBPA) was claimed to be EXPTIME-hard in previous papers without any explicit proof. Recently it has been pointed out by Petr Jancar that the claim lacked proper justification. In this paper, we develop a new complete proof for the EXPTIME-hardness of branching bisimilarity of nBPA. We also prove that the associated regularity problem of nBPA is PSPACE-hard. This improves previous P-hard result

Decidability and complexity of equivalences for simple process algebras

In this thesis I study decidability, complexity and structural properties of strong and weak bisimilarity with respect to two process algebras, Basic Process Algebras and Basic Parallel Process Algebras. The decidability of strong bisimilarity for both algebras is an established result. For the subclasses of normed BPA-processes and BPP there even exist polynomial decision procedures. The complexity of deciding strong bisimilarity for the whole class of BPP is unsatisfactory since it is not bounded by any primitive recursive function. Here we present a new approach that encodes BPP as special polynomials and expresses strong bisimulation in terms of polynomial ideals and then uses a theorem about polynomial ideals (Hilbert's Basis Theorem) and an algorithm from computer algebra (Gröbner bases) to construct a new decision procedure. For weak bisimilarity, Hirshfeld found a decision procedure for the subclasses of totally normed BPA-processes and BPP, and Esparza demonstrated a semidecision procedure for general BPP. The remaining questions are still unsolved. Here we provide some lower bounds on the computational complexity of a decision procedure that might exist. For BPP we show that the decidability problem is NP-hard (even for the class of totally normed BPP), for BPA-processes we show that the decidability problem is PSPACE-hard. Finally we study the notion of weak bisimilarity in terms of its inductive definition. We start from the relation containing all pairs of processes and then form a non-increasing chain of relations by eliminating pairs that do not satisfy a certain expansion condition. These relations are labelled by ordinal numbers and are called approximants. We know that this chain eventually converges for some a' such that =a' = =b' = = for all a' w^w, and for BPPA, a' => w.2. For some restricted classes of BPA and BPPA we show that = = =w.2

Process Algebra, CCS, and Bisimulation Decidability

Over the past fifteen years, there has been intensive study of formal systems that can model concurrency and communication. Two such systems are the Calculus of Communicating Systems, and the Algebra of Communicating Processes. The objective of this paper has two aspects; (1) to study the characteristics and features of these two systems, and (2) to investigate two interesting formal proofs concerning issues of decidability of bisimulation equivalence in these systems. An examination of the processes that generate context-free languages as a trace set shows that their bisimulation equivalence is decidable, in contrast to the undecidability of their trace set equivalence. Recent results have also shown that the bisimulation equivalence problem for processes with a limited amount of concurrency is decidable

Resource Bisimilarity in Petri Nets is Decidable

Petri nets are a popular formalism for modeling and analyzing distributed systems. Tokens in Petri net models can represent the control flow state or resources produced/consumed by transition firings. We define a resource as a part (submultiset) of the Petri net marking and call two resources equivalent iff replacing one of them with another in any marking does not change the observable Petri net behavior. We investigate the resource similarity and the resource bisimilarity -- congruent restrictions of the bisimulation equivalence on Petri net markings and prove that the resource bisimilarity is decidable in contrast to the resource similarity.Comment: New version for submission to the journa