1. INTRODUCTION ST semantics, originally defined in [van Glabbeek and Vaandrager 1987] over Petri Nets, is one of the most studied noninterleaving semantics. The main reason is that ST semantics is the less informative semantics that is a congruence for

Abstract History preserving bisimilarity (hp-bisimilarity) and hereditary history preserving bisimilarity (hhp-bisimilarity) are behavioural equivalences taking into account causal relationships between events of concurrent systems. Their prominent feature is being preserved under action refinement, an operation important for the top-down design of concurrent systems. We show that--unlike hp-bisimilarity--checking hhpbisimilarity for finite labelled asynchronous transition systems is not decidable, by a reduction from the halting problem of 2-counter machines. To make the proof more transparent we introduce an intermediate problem of checking domino bisimilarity for origin constrained tiling systems, whose undecidability is interesting in its own right. We also argue that the undecidability of hhp-bisimilarity holds for finite labelled 1-safe Petri nets. 1 Introduction The notion of behavioural equivalence that has attracted most attention in con-currency theory is bisimilarity, originally introduced by Park [20] and Milner [15]; concurrent programs are considered to have the same meaning if they are bisim-ilar. The prominent role of bisimilarity is due to many pleasant properties it enjoys; we mention a few of them here. A process of checking whether two transition systems are bisimilar can beseen as a two player game which is in fact an Ehrenfeucht-Fra"iss'e type of game

This thesis is concerned with formal semantics and models for concurrent computational systems, that is, systems consisting of a number of parallel computing sequential systems, interacting with each other and the environment. A formal semantics gives meaning to computational systems by describing their behaviour in a mathematical model. For concurrent systems the interesting aspect of their computation is often how they interact with the environment during a computation and not in which state they terminate, indeed they may not be intended to terminate at all. For this reason they are often referred to as reactive systems, to distinguish them from traditional calculational systems, as e.g. a program calculating your income tax, for which the interesting behaviour is the answer it gives when (or if) it terminates, in other words the (possibly partial) function it computes between input and output. Church's thesis tells us that regardless of whether we choose the lambda calculus, Turing machines, or almost any modern programming language such as C or Java to describe calculational systems, we are able to describe exactly the same class of functions. However, there is no agreement on observable behaviour for concurrent reactive systems, and consequently there is no correspondent to Church's thesis. A result of this fact is that an overwhelming number of di#erent and often competing notions of observable behaviours, primitive operations, languages and mathematical models for describing their semantics, have been proposed in the litterature on concurrency.

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Abstract. Hereditary history preserving bisimulation is a natural extension of bisimulation to the setting of so-called “true ” concurrency. Somewhat surprisingly, this extension turns out to be undecidable, in general, for finite-state concurrent systems. In this paper, we show that for a substantial and useful class of finite-state concurrent systems— those whose semantics can be described in terms of Mazurkiewicz traces— hereditary history preserving is decidable. 1

Abstract. The idea of composition and decomposition to obtain computability results is particularly relevant for true-concurrency. In contrast to the interleaving world, where composition and decomposition must be considered with respect to a process algebra operator, e.g. parallel composition, we can directly recognize whether a truly-concurrent model such as a labelled asynchronous transition system or a 1-safe Petri net can be dissected into independent ‘chunks of behaviour’. In this paper we introduce the corresponding concept ‘decomposition into independent components’, and investigate how it translates into truly-concurrent bisimulation equivalences. We prove that, under a natural restriction, history preserving (hp), hereditary hp (hhp), and coherent hhp (chhp) bisimilarity are decomposable with respect to prime decompositions. Apart from giving a general proof technique our decomposition theory leads to several coincidence results. In particular, we resolve that hp, hhp, and chhp bisimilarity coincide for ‘normal form ’ basic parallel processes. 1

Abstract. We propose a polynomial-time decision procedure for hereditary history preserving bisimilarity (hhp-b) on Basic Parallel Processes (BPP). Furthermore, we give a sound and complete equational axiomatization for the equivalence. Both results are derived from a decomposition property of hhp-b, which is the main technical contribution of the paper. Altogether, our results complement previous work on complexity and decomposition of classical and historypreserving bisimilarity on BPP. 1

bisimilarity

Abstract. History preserving (hp) and hereditary hp (hhp) bisimilarity are two equivalences for concurrent systems that reflect causal dependencies between events. hp bisimilarity is well-known to be decidable for finite-state systems, whereas the decidability of hhp bisimilarity had been a renowned open problem for several years until it was finally proved undecidable. Recently, the following positive result has been obtained: hhp bisimilarity coincides with hp bisimilarity for a class of live free choice systems. To our knowledge, this is the only positive result for a class with a reasonable amount of interplay between concurrency and of the proof that stands behind this result. The aim is to provide an overview of the modules involved and how they interact in the key results. We will not directly be concerned with hp and hhp bisimilarity in this paper, but with an auxiliary bisimilarity and its hereditary version. 1

Abstract. We propose a logic for true concurrency whose formulae predicate about events in computations and their causal dependencies. The induced logical equivalence is hereditary history preserving bisimilarity, and fragments of the logic can be identified which correspond to other true concurrent behavioural equivalences in the literature: step, pomset and history preserving bisimilarity. Standard Hennessy-Milner logic, thus (interleaving) bisimilarity, is also recovered as a fragment. We believe that this contributes to a rational presentation of the true concurrent spectrum and to a deeper understanding of the relations between the involved behavioural equivalences. 1