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, and thus (interleaving) bisimilarity, is also recovered as a fragment. We also propose an extension of the logic with fixpoint operators, thus allowing to describe causal and concurrency properties of infinite computations. We believe that this work contributes to a rational presentation of the true concurrent spectrum and to a deeper understanding of the relations between the involved behavioural equivalences.
In the semantics of concurrent and distributed systems, a major dichotomy opposes the interleaving approaches, where concurrency of actions is reduced to the non-deterministic choice among their possible sequentialisations, to true concurrent approaches, where concurrency is taken as a primitive notion. In both cases, on top of the operational models a number of behavioural equivalences have been defined by abstracting from aspects which are considered unobservable [vG01,vGG01].
For the interleaving world, a systematic and impressive picture is taken in the linear-time branching-time spectrum [vG01]. Quite interestingly, the equivalences in the spectrum can be uniformly characterised in logical terms. Bisimilarity, the finest equivalence, corresponds to Hennessy-Milner (HM) logic: two processes are bisimilar if and only if they satisfy the same HM logic formulae [HM85]. Coarser equivalences correspond to suitable fragments of HM logic, as discussed in [vG01].
In the true concurrent world, relying on models like event structures or transition systems with independence [WN95], several behavioural equivalences have been defined. Hereditary history preserving (hhp-)bisimilarity [Bed91], the finest equivalence in the spectrum in [vGG01], has been shown to arise as a canonical behavioural equivalence when considering partially ordered computations [JNW96] (Their abstract notion of bisimilarity instantiates to hhp-bisimilarity when taking the category of pomsets as the path category.) Coarser equivalences like history preserving (hp-)bisimilarity [RT88,DDNM88,BDKP91], pomset and step bisimilarity have also been widely studied. Correspondingly, a number of logics have been studied, but, to the best of our knowledge, a unifying logical framework for the main true concurrent equivalences is still missing. The huge amount of work on the topic makes it impossible to give a complete account of related approaches. Just to give a few references (see Section 7 for a wider discussion), [DNF90] proposes a general framework encompassing a number of temporal and modal logics that characterise interleaving bisimilarity as well as pomset bisimilarity and weak hhp-bisimilarity, a weakening of hhp-bisimilarity studied, e.g., in [DNF90,PLS94,Che92]. However, finer equivalences are not considered and a single unitary logic is missing. Hp-bisimilarity has been studied in the setting of Petri nets and shown to be decidable for finite 1-safe Petri nets in [Vog91]. A decidability result for finite-state Petri nets is obtained also in [MP97] by means of an encoding of into history dependent (HD-)automata. Concerning hhp-bisimilarity, several logics with modalities corresponding to the “retraction” or “backward” execution of computations have been proposed [HS85,Bed91,NC95,PU11]. When a system does not exhibit autoconcurrency, i.e., when two instances of the same action are never enabled in parallel, such logics are shown to capture hhp-bisimilarity. Relaxing this restriction requires to move to an event based logic, where specific events executed in the past can be retracted [Bed91,NC95,PU11].
In this paper we propose a behavioural logic for concurrency and we show that it allows us to characterise a relevant part of the true concurrent spectrum. More specifically, the full logic L is shown to capture hhp-bisimilarity, the finest behavioural equivalence in the spectrum in [vGG01]. Then suitable fragments of the logic are shown to scale down to the characterisation of other coarser equivalences: history preserving, pomset and step bisimilarity. Standard HM logic, and thus (interleaving) bisimilarity, is also recovered as a fragment.
Our logic allows us to predicate about events in computations together with their causal and independence relations. It is interpreted over prime event structures [NPW81,Win87], one of the most widely known event-based models of computation, where the dependencies between events are expressed in terms of causality and (binary) conflict. It could naturally be interpreted over any formalism with explicit notions of event, causality and consistency. A formula is evaluated in a configuration representing the current state of the computation, and it predicates on the possible future evolutions starting from that state. The logic is event-based in the sense that it contains an operator acting as a binder: it asserts the existence of an event satisfying suitable requirements and it binds the event to a variable so that the event can be referred to later in the formula. In this respect, it is reminiscent of the modal analogue of independence-friendly modal logic as considered in [BF02].
The logic contains two main operators. The formula (x, y < a z)ϕ declares that an a-labelled future event exists, which causally depends on the event bound to x, and is independent from the event bound to y. Such an event is bound to variable z so that it can be later referred to in ϕ. In general, x and y can be replaced by tuples
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