On the Complexity of Deciding Behavioural Equivalences and Preorders. A Survey

This paper gives an overview of the computational complexity of all the equivalences in the linear/branching time hierarchy [vG90a] and the preordersin the corresponding hierarchy of preorders. We consider finite state or regular processes as well as infinite-state BPA [BK84b] processes. A distinction, which turns out to be important in the finite-state processes, is that of simulation-like equivalences/preorders vs. trace-like equivalencesand preorders. Here we survey various known complexity results for these relations. For regular processes, all simulation-like equivalences and preorders are decidable in polynomial time whereas all trace-like equivalences and preorders are PSPACE-Complete. We also consider interesting specialclasses of regular processes such as deterministic, determinate, unary, locally unary, and tree-like processes and survey the known complexity results inthese special cases. For infinite-state processes the results are quite different. For the class of context-free processes or BPA processes any preorder or equivalence beyond bisimulation is undecidable but bisimulation equivalence is polynomial timedecidable for normed BPA processes and is known to be elementarily decidable in the general case. For the class of BPP processes, all preorders and equivalences apart from bisimilarity are undecidable. However, bisimilarityis decidable in this case and is known to be decidable in polynomial time for normed BPP processes

Decidability Issues for Petri Nets

This is a survey of some decidability results for Petri nets, covering the last three decades. The presentation is structured around decidability of specific properties, various behavioural equivalences and finally the model checking problem for temporal logics

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)

Verification of parallel systems via decomposition

Recently, Milner and Moller have presented several decomposition results for processes. Inspired by these, we investigate decomposition techniques for the verification of parallel systems. In particular, we consider those of the form q j (I) where p i and q j are (finite) state systems. We provide a decomposition procedure for all p i and q j and give criteria that must be checked on the decomposed processes to see whether (I) does or does not hold. We analyse the complexity of our procedure and show that it is polynomial in n, m and the sizes of p i and q j if there is no communication. We also show that with communication the verification of (I) is co-NP hard, which makes it very unlikely that a polynomial complexity bound exists. But by applying our decomposition technique to Milner's cyclic scheduler we show that verification can become polynomial in space and time for practical examples, where standard techniques are exponential. Note: The authors are supported by the European Communities under ESPRIT Basic Research Action 3006 (CONCUR)

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

Qualitative Logics and Equivalences for Probabilistic Systems

We investigate logics and equivalence relations that capture the qualitative behavior of Markov Decision Processes (MDPs). We present Qualitative Randomized CTL (QRCTL): formulas of this logic can express the fact that certain temporal properties hold over all paths, or with probability 0 or 1, but they do not distinguish among intermediate probability values. We present a symbolic, polynomial time model-checking algorithm for QRCTL on MDPs. The logic QRCTL induces an equivalence relation over states of an MDP that we call qualitative equivalence: informally, two states are qualitatively equivalent if the sets of formulas that hold with probability 0 or 1 at the two states are the same. We show that for finite alternating MDPs, where nondeterministic and probabilistic choices occur in different states, qualitative equivalence coincides with alternating bisimulation, and can thus be computed via efficient partition-refinement algorithms. On the other hand, in non-alternating MDPs the equivalence relations cannot be computed via partition-refinement algorithms, but rather, they require non-local computation. Finally, we consider QRCTL*, that extends QRCTL with nested temporal operators in the same manner in which CTL* extends CTL. We show that QRCTL and QRCTL* induce the same qualitative equivalence on alternating MDPs, while on non-alternating MDPs, the equivalence arising from QRCTL* can be strictly finer. We also provide a full characterization of the relation between qualitative equivalence, bisimulation, and alternating bisimulation, according to whether the MDPs are finite, and to whether their transition relations are finitely-branching.Comment: The paper is accepted for LMC

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