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)

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

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

Towards weak bisimilarity on a class of parallel processes.

A directed labelled graph may be used, at a certain abstraction, to represent a system's behaviour. Its nodes, the possible states the system can be in; its arrows labelled by the actions required to move from one state to another. Processes are, for our purposes, synonymous with these labelled transition systems. With this view a well-studied notion of behavioural equivalence is bisimilarity, where processes are bisimilar when whatever one can do, the other can match, while maintaining bisimilarity. Weak bisimilarity accommodates a notion of silent or internal action. A natural class of labelled transition systems is given by considering the derivations of commutative context-free grammars in Greibach Normal Form: the Basic Parallel Processes (BPP), introduced by Christensen in his PhD thesis. They represent a simple model of communication-free parallel computation, and for them bisimilarity is PSPACE-complete. Weak bisimilarity is believed to be decidable, but only partial results exist. Non-bisimilarity is trivially semidecidable on BPP (each process has finitely many next states, so the state space can be explored until a mis-match is found); the research effort in proving it fully decidable centred on semideciding the positive case. Conversely, weak bisimilarity has been known to be semidecidable for a decade, but no method for semideciding inequivalence has yet been found - the presence of silent actions allows a process to have infinitely many possible successor states, so simple exploration is no longer possible. Weak bisimilarity is defined coinductively, but may be approached, and even reached, by its inductively defined approximants. Game theoretically, these change the Defender's winning condition from survival for infinitely many turns to survival for K turns, for an ordinal k, creating a hierarchy of relations successively closer to full weak bisimilarity. It can be seen that on any set of processes this approximant hierarchy collapses: there will always exist some K such that the kth approximant coincides with weak bisimilarity. One avenue towards the semidecidability of non- weak bisimilarity is the decidability of its approximants. It is a long-standing conjecture that on BPP the weak approximant hierarchy collapses at o x 2. If true, in order to semidecide inequivalence it would suffice to be able to decide the o + n approximants. Again, there exist only limited results: the finite approximants are known to be decidable, but no progress has been made on the wth approximant, and thus far the best proven lower-bound of collapse is w1CK (the least non-recursive ordinal number). We significantly improve this bound to okx2(for a k-variable BPP); a key part of the proof being a novel constructive version of Dickson's Lemma. The distances-to-disablings or DD functions were invented by Jancar in order to prove the PSPACE-completeness of bisimilarity on BPP. At the end of his paper is a conjecture that weak bisimilarity might be amenable to the theory; a suggestion we have taken up. We generalise and extend the DD functions, widening the subset of BPP on which weak bisimilarity is known to be computable, and creating a new means for testing inequivalence. The thesis ends with two conjectures. The first, that our extended DD functions in fact capture weak bisimilarity on full BPP (a corollary of which would be to take the lower bound of approximant collapse to and second, that they are computable, which would enable us to semidecide inequivalence, and hence give us the decidability of weak bisimilarity

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

Weak Bisimulation Approximants

Bisimilarity ∼ and weak bisimilarity ≈ are canonical notions of equivalence between processes, which are defined co-inductively, but may be approached – and even reached – by their (transfinite) inductively-defined approximants ∼α and ≈α. For arbitrary processes this approximation may need to climb arbitrarily high through the infinite ordinals before stabilising. In this paper we consider a simple yet well-studied process algebra, the Basic Parallel Processes (BPP), and investigate for this class of processes the minimal ordinal α such that ≈ = ≈α. The main tool in our investigation is a novel proof of Dickson’s Lemma. Unlike classical proofs, the proof we provide gives rise to a tight ordinal bound, of ω n, on the order type of non-increasing sequences of n-tuples of natural numbers. With this we are able to reduce a long-standing bound on the approximation hierarchy for weak bisimilarity ≈ over BPP, and show that ≈ = ≈ω ω