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

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

Equivalence of infinite-state systems with silent steps

This dissertation contributes to analysis methods for infinite-state systems. The dissertation focuses on equivalence testing for two relevant classes of infinite-state systems: commutative context-free processes, and one-counter automata. As for equivalence notions, we investigate the classical bisimulation and simulation equivalences. The important point is that we allow for silent steps in the model, abstracting away from internal, unobservable actions. Very few decidability results have been known so far for bisimulation or simulation equivalence for infinite-state systems with silent steps, as presence of silent steps makes the equivalence problem arguably harder to solve. A standard technique for bisimulation or simulation equivalence testing is to use the hierarchy of approximants. For an effective decision procedure the hierarchy must stabilize (converge) at level omega, the first limit ordinal, which is not the case for the models investigated in this thesis. However, according to a long-standing conjecture, the community believed that the convergence actually takes place at level omega+ omega in the class of commutative context free processes. We disprove the conjecture and provide a lower bound of omega * omega for the convergence level. We also show that all previously known positive decidability results for BPPs can be re-proven uniformly using the improved approximants techniques. Moreover dissertation contains an unsuccesfull attack on one of the main open problems in the area: decidability of weak bisimulation equivalence for commutative context-free processes. Our technical development of this section is not sufficient to solve the problem, but we believe it is a serious step towards a solution. Furtermore, we are able to show decidability of branching (stuttering) bisimulation equivalence, a slightly more discriminating variant of bisimulation equivalence. It is worth emphesizing that, until today, our result is the only known decidability result for bisimulation equivalence in a class of inifinite-state systems with silent steps that is not known to admit convergence of (some variant of) standard approximants at level omega. Finally we consider weak simulation equivalence over one-counter automata without zero tests (allowing zero tests implies undecidability). While weak bisimulation equivalence is known to be undecidable in this class, we prove a surprising result that weak simulation equivalence is actually decidable. Thus we provide a first example going against a trend, widely-believed by the community, that simulation equivalence tends to be computationally harder than bisimulation equivalence. In short words, the dissertation contains three new results, each of them solving a non-trivial open problem about equivalence testing of infinite-state systems with silent steps

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 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