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

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

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

Logic and Automata

Mathematical logic and automata theory are two scientific disciplines with a fundamentally close relationship. The authors of Logic and Automata take the occasion of the sixtieth birthday of Wolfgang Thomas to present a tour d'horizon of automata theory and logic. The twenty papers in this volume cover many different facets of logic and automata theory, emphasizing the connections to other disciplines such as games, algorithms, and semigroup theory, as well as discussing current challenges in the field

Branching bisimilarity of normed BPA processes as a rational monoid

The paper presents an elaborated and simplified version of the structural result for branching bisimilarity on normed BPA (Basic Process Algebra) processes that was the crux of a conference paper by Czerwinski and Jancar (arxiv 7/2014 and LiCS 2015). That paper focused on the computational complexity, and a NEXPTIME-upper bound has been derived; the authors built on the ideas by Fu (ICALP 2013), and strengthened his decidability result. Later He and Huang announced the EXPTIME-completeness of this problem (arxiv 1/2015, and LiCS 2015), giving a technical proof for the EXPTIME membership. He and Huang indirectly acknowledge the decomposition ideas by Czerwinski and Jancar on which they also built, but it is difficult to separate their starting point from their new ideas. One aim here is to present the previous decomposition result of Czerwinski and Jancar in a technically new framework, noting that branching bisimulation equivalence on normed BPA processes corresponds to a rational monoid (in the sense of [Sakarovitch, 1987]); in particular it is shown that the mentioned equivalence can be decided by normal-form computing deterministic finite transducers. Another aim is to provide a complete description, including an informal overview, that should also make clear how Fu's ideas were used, and to give all proofs in a form that should be readable and easily verifiable.Web of Science134art. no. 1