Behavioural Equivalence for Infinite Systems—Partially Decidable!

For finite-state systems non-interleaving equivalences are computationallyat least as hard as interleaving equivalences. In this paper we showthat when moving to infinite-state systems, this situation may changedramatically.We compare standard language equivalence for process description languages with two generalizations based on traditional approaches capturing non-interleaving behaviour, pomsets representing global causal dependency, and locality representing spatial distribution of events.We first study equivalences on Basic Parallel Processes, BPP, a processcalculus equivalent to communication free Petri nets. For this simpleprocess language our two notions of non-interleaving equivalences agree.More interestingly, we show that they are decidable, contrasting a result ofHirshfeld that standard interleaving language equivalence is undecidable.Our result is inspired by a recent result of Esparza and Kiehn, showingthe same phenomenon in the setting of model checking.We follow up investigating to which extent the result extends to largersubsets of CCS and TCSP. We discover a significant difference betweenour non-interleaving equivalences. We show that for a certain non-trivialsubclass of processes between BPP and TCSP, not only are the two equivalences different, but one (locality) is decidable whereas the other (pomsets) is not. The decidability result for locality is proved by a reduction to the reachability problem for Petri nets

Behavioural Equivalence for Infinite Systems—Partially Decidable!

For finite-state systems non-interleaving equivalences are computationallyat least as hard as interleaving equivalences. In this paper we showthat when moving to infinite-state systems, this situation may changedramatically.We compare standard language equivalence for process description languages with two generalizations based on traditional approaches capturing non-interleaving behaviour, pomsets representing global causal dependency, and locality representing spatial distribution of events.We first study equivalences on Basic Parallel Processes, BPP, a processcalculus equivalent to communication free Petri nets. For this simpleprocess language our two notions of non-interleaving equivalences agree.More interestingly, we show that they are decidable, contrasting a result ofHirshfeld that standard interleaving language equivalence is undecidable.Our result is inspired by a recent result of Esparza and Kiehn, showingthe same phenomenon in the setting of model checking.We follow up investigating to which extent the result extends to largersubsets of CCS and TCSP. We discover a significant difference betweenour non-interleaving equivalences. We show that for a certain non-trivialsubclass of processes between BPP and TCSP, not only are the two equivalences different, but one (locality) is decidable whereas the other (pomsets) is not. The decidability result for locality is proved by a reduction to the reachability problem for Petri nets

Further Results on Partial Order Equivalences on Infinite Systems

In [26], we investigated decidability issues for standard language equivalence for process description languages with two generalisations based on traditional approachesfor capturing non-interleaving behaviour: pomset equivalence reflecting global causal dependency, and location equivalence reflecting spatial distribution of events. In this paper, we continue by investigating the role played by TCSP-style renaming and hiding combinators with respect to decidability. One result of [26] was that in contrast to pomset equivalence, location equivalence remained decidable for a class of processes consisting of finite sets of BPP processes communicating in a TCSP manner. Here, we show that location equivalence becomes undecidable when either renaming or hiding is added to this class of processes. Furthermore, we investigate the weak versions of location and pomset equivalences.We show that for BPP with prefixing, both weak pomset and weak location equivalence are decidable. Moreover, we show that weak location equivalence is undecidable for BPP semantically extended with CCS communication

Reverse Bisimulations on Stable Configuration Structures

The relationships between various equivalences on configuration structures, including interleaving bisimulation (IB), step bisimulation (SB) and hereditary history-preserving (HH) bisimulation, have been investigated by van Glabbeek and Goltz (and later Fecher). Since HH bisimulation may be characterised by the use of reverse as well as forward transitions, it is of interest to investigate forms of IB and SB where both forward and reverse transitions are allowed. We give various characterisations of reverse SB, showing that forward steps do not add extra power. We strengthen Bednarczyk's result that, in the absence of auto-concurrency, reverse IB is as strong as HH bisimulation, by showing that we need only exclude auto-concurrent events at the same depth in the configuration

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

A Logic for True Concurrency

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.Comment: 31 pages, a preliminary version appeared in CONCUR 201

True Concurrency Can Be Easy

Net bisimilarity is a behavioral equivalence for finite Petri nets, which is equivalent to structure-preserving bisimilarity and causal-net bisimilarity, but with a much simpler definition, which is a smooth generalization of the definition of standard bisimilarity on Labeled Transition Systems. We show that it can be characterized logically by means of a suitable modal logic, called NML (acronym of net modal logic): two markings are net bisimilar if and only if they satisfy the same NML formulae

A Process Calculus for Expressing Finite Place/Transition Petri Nets

We introduce the process calculus Multi-CCS, which extends conservatively CCS with an operator of strong prefixing able to model atomic sequences of actions as well as multiparty synchronization. Multi-CCS is equipped with a labeled transition system semantics, which makes use of a minimal structural congruence. Multi-CCS is also equipped with an unsafe P/T Petri net semantics by means of a novel technique. This is the first rich process calculus, including CCS as a subcalculus, which receives a semantics in terms of unsafe, labeled P/T nets. The main result of the paper is that a class of Multi-CCS processes, called finite-net processes, is able to represent all finite (reduced) P/T nets.Comment: In Proceedings EXPRESS'10, arXiv:1011.601