Decidability and coincidence of equivalences for concurrency

There are two fundamental problems concerning equivalence relations in con-currency. One is: for which system classes is a given equivalence decidable? The second is: when do two equivalences coincide? Two well-known equivalences are history preserving bisimilarity (hpb) and hereditary history preserving bisimi-larity (hhpb). These are both ‘independence ’ equivalences: they reflect causal dependencies between events. Hhpb is obtained from hpb by adding a ‘back-tracking ’ requirement. This seemingly small change makes hhpb computationally far harder: hpb is well-known to be decidable for finite-state systems, whereas the decidability of hhpb has been a renowned open problem for several years; only recently it has been shown undecidable. The main aim of this thesis is to gain insights into the decidability problem for hhpb, and to analyse when it coincides with hpb; less technically, we might say, to analyse the power of the interplay between concurrency, causality, and conflict. We first examine the backtracking condition, and see that it has two dimen

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

An algebra of behavioural types

Special thanks to Gérard Boudol, Ilaria Castellani, Silvano Dal Zilio, and Massimo Merro, for fruitful discussions and careful reading of parts of this document. Several anonymous referees made useful comments.We propose a process algebra, the Algebra of Behavioural Types, as a language for typing concurrent objects. A type is a higher-order labelled transition system that characterises all possible life cycles of a concurrent object. States represent interfaces of objects; state transitions model the dynamic change of object interfaces. Moreover, a type provides an internal view of the objects that inhabits it: a synchronous one, since transitions correspond to message reception. To capture this internal view of objects we define a notion of bisimulation, strong on labels and weak on silent actions. We study several algebraic laws that characterise this equivalence, and obtain completeness results for image-finite types.publishersversionpublishe

Behavioural Preorders on Stochastic Systems - Logical, Topological, and Computational Aspects

Computer systems can be found everywhere: in space, in our homes, in our cars, in our pockets, and sometimes even in our own bodies. For concerns of safety, economy, and convenience, it is important that such systems work correctly. However, it is a notoriously difficult task to ensure that the software running on computers behaves correctly. One approach to ease this task is that of model checking, where a model of the system is made using some mathematical formalism. Requirements expressed in a formal language can then be verified against the model in order to give guarantees that the model satisfies the requirements. For many computer systems, time is an important factor. As such, we need our formalisms and requirement languages to be able to incorporate real time. We therefore develop formalisms and algorithms that allow us to compare and express properties about real-time systems. We first introduce a logical formalism for reasoning about upper and lower bounds on time, and study the properties of this formalism, including axiomatisation and algorithms for checking when a formula is satisfied. We then consider the question of when a system is faster than another system. We show that this is a difficult question which can not be answered in general, but we identify special cases where this question can be answered. We also show that under this notion of faster-than, a local increase in speed may lead to a global decrease in speed, and we take step towards avoiding this. Finally, we consider how to compare the real-time behaviour of systems not just qualitatively, but also quantitatively. Thus, we are interested in knowing how much one system is faster or slower than another system. This is done by introducing a distance between systems. We show how to compute this distance and that it behaves well with respect to certain properties.Comment: PhD dissertation from Aalborg Universit

Syntactic approaches to negative results in process algebras and modal logics

Concurrency as a phenomenon is observed in most of the current computer science trends. However the inherent complexity of analyzing the behavior of such a system is incremented due to the many different models of concurrency, the variety of applications and architectures, as well as the wide spectrum of specification languages and demanded correctness criteria. For the scope of this thesis we focus on state based models of concurrent computation, and on modal logics as specification languages. First we study syntactically the process algebras that describe several different concurrent behaviors, by analyzing their equational theories. Here, we use well-established techniques from the equational logic of processes to older and newer setups, and then transition to the use of more general and novel methods for the syntactical analysis of models of concurrent programs and specification languages. Our main contributions are several positive and negative axiomatizability results over various process algebraic languages and equivalences, along with some complexity results over the satisfiability of multi-agent modal logic with recursion, as a specification language.Samhliða sem fyrirbæri sést í flestum núverandi tölvunarfræði stefnur. Hins vegar er eðlislægt flókið að greina hegðun slíks kerfis- tem er aukið vegna margra mismunandi gerða samhliða, fjölbreytileikans af forritum og arkitektúr, svo og breitt svið forskrifta mælikvarða og kröfðust réttmætisviðmiða. Fyrir umfang þessarar ritgerðar leggjum við áherslu á ástandsbundin líkön af samhliða útreikningum og á formlegum rökfræði sem forskrift tungumálum. Fyrst skoðum við setningafræðilega ferlialgebrurnar sem lýsa nokkrum mismunandi samhliða hegðun, með því að greina jöfnukenningar þeirra. Hér notum við rótgróin tækni mynda jöfnunarrökfræði ferla til eldri og nýrri uppsetningar, og síðan umskipti yfir í notkun almennari og nýrra aðferða fyrir setningafræðileg greining á líkönum samhliða forrita og forskriftartungumála. Helstu framlög okkar eru nokkrar jákvæðar og neikvæðar niðurstöður um axiomatizability yfir ýmis ferli algebrumál og jafngildi, ásamt nokkrum samSveigjanleiki leiðir af því að fullnægjanleiki fjölþátta formrökfræði með endurkomu, sem a forskrift tungumál.RANNIS: `Open Problems in the Equational Logic of Processes’ (OPEL) (grant No 196050-051) Reykjavik University research fund: `Runtime and Equational Verification of Concurrent Programs' (ReVoCoP) (grant No 222021