Prime Event Structures Research Articles

Branching cells for asymmetric event structures

This paper introduces branching cells as elementary units of independent choices in the model of Asymmetric Event Structures (AES), extending a previous work on branching cells for Prime Event Structures. Branching cells consist of subAES of the surrounding AES. Their maximal configurations are shown to tile any maximal configuration of the surrounding AES in a dynamic way.Branching cells for AES are developed in order to allow the analysis of an optimization procedure in the context of QoS management of web services, presented in a companion paper. Other applications of branching cells include the ability to add a probabilistic layer to AES in a natural fashion where concurrency meets probabilistic independence of choices in distinct and parallel branching cells.

Unified Mathematical Framework for Slicing and Symmetry Reduction over Event Structures

Nonclassical slicing and symmetry reduction can act as efficient structural abstract methods for pruning state space when dealing with verification problems. In this paper, we mainly address theoretical and algorithmic aspects for nonclassical slicing and symmetry reduction over prime event structures. We propose sliced and symmetric quotient reduction models of event structures and present their corresponding algorithms. To construct the underlying foundation of the proposed methodologies, we introduce strong and weak conflict concepts and a pair of mutually inverse operators and extend permutation group based symmetry notion of event structures. We have established a unified mathematical framework for slicing and symmetry reduction, and further investigated the translation, isomorphism, and equivalence relationship and other related basic facts from a theoretical point of view. The framework may provide useful guidance and theoretical exploration for overcoming verification challenges. This paper also demonstrates their practical applications by two cases.

Adding Priority to Event Structures

Event Structures (ESs) are mainly concerned with the representation of causal relationships between events, usually accompanied by other event relations capturing conflicts and disabling. Among the most prominent variants of ESs are Prime ESs, Bundle ESs, Stable ESs, and Dual ESs, which differ in their causality models and event relations. Yet, some application domains require further kinds of relations between events. Here, we add the possibility to express priority relationships among events. We exemplify our approach on Prime, Bundle, Extended Bundle, and Dual ESs. Technically, we enhance these variants in the same way. For each variant, we then study the interference between priority and the other event relations. From this, we extract the redundant priority pairs-notably differing for the types of ESs-that enable us to provide a comparison between the extensions. We also exhibit that priority considerably complicates the definition of partial orders in ESs.

Declarative Event-Based Workflow as Distributed Dynamic Condition Response Graphs

We present Dynamic Condition Response Graphs (DCR Graphs) as a declarative, event-based process model inspired by the workflow language employed by our industrial partner and conservatively generalizing prime event structures. A dynamic condition response graph is a directed graph with nodes representing the events that can happen and arrows representing four relations between events: condition, response, include, and exclude. Distributed DCR Graphs is then obtained by assigning roles to events and principals. We give a graphical notation inspired by related work by van der Aalst et al. We exemplify the use of distributed DCR Graphs on a simple workflow taken from a field study at a Danish hospital, pointing out their flexibility compared to imperative workflow models. Finally we provide a mapping from DCR Graphs to Buchi-automata.

Unfolding Semantics of Petri Nets Based on Token Flows

In this paper we propose two new unfolding semantics for general Petri nets combining the concept of prime event structures with the idea of token flows developed in [13]. In contrast to the standard unfolding based on branching processes, one of the presented unfolding models avoids to represent isomorphic processes while the other additionally reduces the number of (possibly nonisomorphic) processes with isomorphic underlying runs. We show that both the proposed unfolding models still represent the complete partial order behavior. Moreover, in both cases it is possible to construct complete finite prefixes for bounded nets through applying the known theory of cut-off events. In particular, canonical prefixes w.r.t. a given cutting context can be defined and computed for bounded nets. We present implementations of construction algorithms for complete finite prefixes of both the unfolding models. Experimental results show that the computed prefixes are much smaller and can be constructed significantly faster than in the case of the standard unfolding.

Reversibility and Models for Concurrency

There is a growing interest in models of reversible computation driven by exciting application areas such as bio-systems and quantum computing. Reversible process algebras RCCS [Danos, V. and J. Krivine, Reversible communicating systems, in: P. Gardner and N. Yoshida, editors, Proceedings of the 15th International Conference on Concurrency Theory CONCUR 2004, LNCS 3170 (2004), pp. 292–307] and CCSK [Phillips, I.C.C. and I. Ulidowski, Reversing algebraic process calculi, in: Proceedings of 9th International Conference on Foundations of Software Science and Computation Structures, FOSSACS 2006, LNCS 3921 (2006), pp. 246–260. Extended version accepted by Journal of Logic and Algebraic Programming] were developed and general techniques for reversing other process operators were proposed. The paper shows that the notion of reversibility can bridge the gap between some interleaving models and non-interleaving models of concurrency, and makes them interchangeable. We prove that transition systems associated with reversible process algebras are equivalent as models to labelled prime event structures. Furthermore, we show that forward-reverse bisimulation corresponds to hereditary history-preserving bisimulation in the setting with no auto-concurrency and no auto-causation.

A projective formalism applied to topological and probabilistic event structures

This paper introduces projective systems for topological and probabilistic event structures. The projective formalism is used for studying the domain of configurations of a prime event structure and its space of maximal elements. This is done from both a topological and a probabilistic viewpoint. We give probability measure extension theorems in this framework.

Domain and event structure semantics for Petri nets with read and inhibitor arcs

We propose a functorial concurrent semantics for Petri nets extended with read and inhibitor arcs, that we call inhibitor nets. Along the lines of the seminal work by Winskel on safe (ordinary) nets, the truly concurrent semantics is given at a categorical level via a chain of coreflections leading from the category SW- IN of semi-weighted inhibitor nets to the category Dom of finitary prime algebraic domains (equivalent to the category PES of prime event structures). As an intermediate semantic model, we introduce inhibitor event structures, an event-based model able to faithfully capture the dependencies among events which arise in the presence of read and inhibitor arcs. Inhibitor event structures generalise several event structure models in the literature, like prime, asymmetric and bundle event structures.

Domain and event structure semantics for Petri nets with read and inhibitor arcs*1

We propose a functorial concurrent semantics for Petri nets extended with read and inhibitor arcs, that we call inhibitor nets . Along the lines of the seminal work by Winskel on safe (ordinary) nets, the truly concurrent semantics is given at a categorical level via a chain of coreflections leading from the category SW - IN of semi-weighted inhibitor nets to the category Dom of finitary prime algebraic domains (equivalent to the category PES of prime event structures). As an intermediate semantic model, we introduce inhibitor event structures , an event-based model able to faithfully capture the dependencies among events which arise in the presence of read and inhibitor arcs. Inhibitor event structures generalise several event structure models in the literature, like prime, asymmetric and bundle event structures.

Relating conflict-free stable transition and event models via redex families

We describe an event-style (or poset) semantics for conflict-free rewrite systems, including term and graph rewriting (possibly with bound variables), the λ-calculus, and other stable transition systems with a residual relation. Our interpretation is based on considering redex-families as events. It treats permutation-equivalent reductions as representing the same concurrent computation. Due to erasure of redexes, event structures are inadequate for such an interpretation. We therefore extend the prime event structure model in two different but equivalent ways: by axiomatizing permutation-equivalence on finite configurations, and by axiomatizing the erasure of events, for the conflict-free case, and show that these extended models are equivalent to stable transition models with axiomatized residual and family relations. We then construct finitary prime algebraic domains from the set of configurations in these extended models by defining orderings relative to stable sets of ‘results’. All useful sets of results for which the normalization (by neededness) theorem can be proved are stable.

Contextual Petri Nets, Asymmetric Event Structures, and Processes

We present an event structure semantics for contextual nets, an extension of P/T Petri nets where transitions can check for the presence of tokens without consuming them (read-only operations). A basic role is played by asymmetric event structures, a generalization of Winskel's prime event structures where symmetric conflict is replaced by a relation modelling asymmetric conflict or weak causality, used to represent a new kind of dependency between events arising in contextual nets. Extending Winskel's seminal work on safe nets, the truly concurrent event-based semantics of contextual nets is given at categorical level via a chain of coreflections leading from the category SW-CN of semi-weighted contextual nets to the category Dom of finitary prime algebraic domains. First an unfolding construction generates from a contextual net a corresponding occurrence contextual net, from where an asymmetric event structure is extracted. Then the configurations of the asymmetric event structure, endowed with a suitable order, are shown to form a finitary prime algebraic domain. We also investigate the relation between the proposed unfolding semantics and several deterministic process semantics for contextual nets in the literature. In particular, the domain obtained via the unfolding is characterized as the collection of the deterministic processes of the net endowed with a kind of prefix ordering.

Model checking of causal knowledge formulas

A model checking algorithm for a variant of the temporal logic of causal knowledge [25] is given. The temporal language is interpreted over labelled prime event structures. Knowledge operators express agents' history knowledge and the knowledge acquired about other agents. The temporal modalities correspond to the immediate causality and causality. For systems represented by deterministic Asynchronous Automata we prove that the complexity of the model checking algorithm for formula ϕ over automaton A of N-agents is Φϕ∣ × ∣A∣ × ∣GA∣ × 2O(N3logN), where ∣GA∣ is the size of the global state space of A and A is the alphabet of actions. Partial order reductions are used in order to make the model checking algorithm more efficient.

Event Structures for Local Traces

Local traces and local event structures have been introduced in order to lift the semantical theory of 1-safe Petri nets to the level of more general Petri nets. In this paper, we cut out Petri nets and establish a direct link between these two classes. Whereas it is relatively easy to associate a local independence relation to a local event structure, the opposite connection is much less clear. The problem here is the identification of the events in a local trace language. We consider various extensions of the classical technique which relates Mazurkiewicz' traces to prime event structures, one of which leads to a precise connection in terms of a coreflection between main subclasses of each model.

An event structure semantics for general Petri nets

In this paper we address the following question: What type of event structures are suitable for representing the behaviour of general Petri nets? As a partial answer to this question we define a new class of event structures called local event structures and identify a subclass called UL-event structures. We propose that UL-event structures are appropriate for capturing the behaviour of general Petri nets. Our answer is a partial one in that in the proposed event structure semantics, auto-concurrency is filtered out from the behaviour of Petri nets. It turns out that this limited event structure semantics for Petri nets is nevertheless a non-trivial and conservative extension of the (prime) event structure semantics of 1-safe Petri nets provided in Nielsen et al. (1981). We also show that the strong relationship between prime event structures and 1-safe Petri nets established in a categorical framework in Winskel (1987) can be extended to the present setting, provided we restrict our attention to the subclass of Petri nets whose behaviours do not exhibit any auto-concurrency. Finally, we show that Winskel's general and stable event structures can be smoothly related to local event structures and that similarly prime event structures can be related to UL-event structures.

Transition-Systems, Event Structures, and Unfoldings

A subclass of transition systems called elementary transition systems can be identified with the help of axioms based on a structural notion called regions. Elementary transition systems have been shown to be the transition system model of a basic system model of net theory called elementary net systems. Here we show that by smoothly strengthening the regional axioms for elementary transition systems, one obtains a subclass called occurrence transition system. We then prove that occurrence transition systems are the transition system model of yet another basic model of concurrency, namely, prime event structures. We then propose an operation of unfolding elementary transition systems into occurrence transition systems, We prove that it is "correct" in a strong categorical sense.

A Logical Study of Distributed Transition Systems

We extend labelled transition systems to distributed transition systems by labelling the transition relation with a finite set of actions, representing the fact that the actions occur as a concurrent step . We design an addition-based temporal logic in which one can explicity talk about steps. The logic is studied to establish a variety of positive and negative results in terms of axiomatizability and decidability. Our positive results show that the step notion is amenable to logical treatment via standard techniques. They also help us to obtain a logical characterization of two well known models for distributed systems: labelled elementary net systems and labelled prime event structures. Our negative results show that demanding deterministic structures when dealing with a "non-interleaved" notion of transitions is, from a logical standpoint, very expressive. They also show that another well known model of distributed systems called asynchronous transition systems exhibits a surprising amount of expressive power in a natural logical setting.

On the nature of events: another perspective in concurrency

We introduce yet another event-based formalism, that of event automata, which unifies various concepts to be found in the literature. A characteristic property of these automata is that every state bears information about the events which have happened, and every state change coincides with the happening of an event. With configurations being reachable states, we show that other formalisms, such as (prime or flow) event structures and geometric automata, can be embedded in our model. Various constructions on event automata, such as the partially synchronous products, specialize to those of other formalisms. We discuss various new concepts such as possible events, asymmetric conflict and quiescent states, all of which are of an intensional nature, and which more faithfully model certain phenomena of event-based protocols. Finally, we distinguish between specifications, and automata which satisfy a specification. This distinction seems to open new perspectives for the analysis of refinement and abstraction. We prove compositionality of satisfaction with regard to some synchronization operators.

A hybrid model for intelligent control systems

Hybrid continuous-time and discrete-event system models are developed for applications to intelligent robot and autonomous control systems. Certain general properties, including the interface structure between the discrete event part and the continuous part of hybrid systems, are studied. Petri nets are proposed as the model for the discrete event sub-system (DES), and state equations with a set of binary parameters as the model for the continuous time sub-system (CTS) in a hybrid system. The quantized CTS is formulated in prime event structures, which can also be modeled by a condition/event Petri net. Therefore, a complete hybrid system can be fit into an extended Petri net formalism, which provides a unified framework for hybrid system description. The validity of the proposed model is demonstrated by modeling a two-tank system.

DISTRIBUTIVE EVALUATIONS OF λ-CALCULUS

In this paper we address the problem of encoding evaluation strategies for the λ-calculus into prime event structures. In order for this to be possible the derivation spaces yielded by the evaluation mechanism must be prime algebraic cpo's. This requirement is not met by permutation equivalence (the standard concurrent semantics with which λ-calculus is equipped) since the derivation spaces it yields are upper semi-lattices. We solve this problem by taking the coarsest congruence contained in permutation equivalence such that permutations of disjoint reductions are equated and the downward closure of every derivation is a distributive lattice. This equivalence, called distributive permutation equivalence, is characterized directly by restricting permutations of redexes to those sets U which are distributive, i.e. for every u ∈ U, the development of every V ⊆ (U\{u}) does not duplicate or delete u. A simple consequence of our results is that the derivation spaces of the call-by-value λ-calculus are distributive lattices. Finally, we show that a sequential evaluation mechanism can not, in general, be effectively transformed into a maximally distributive one.

The poset of infinitary traces

Partially commutative monoids, also called trace monoids, are among the most-studied formalisms to describe the behaviour of distributed systems. In order to model systems which never stop, we have to consider an extension of traces, namely infinite traces. Finite-trace monoids are strongly related to partial-order sets ( PoSets), domains and event structures, which are other models to describe the behaviour of distributed systems. The aim of this paper is to establish similar connexions between infinite-trace monoids, PoSets and event structures. We prove that the set of finite and infinite traces with the prefix order is a Scott domain and a coherently complete prime algebraic PoSet. Moreover, we establish a representation theorem between the class of finite- and infinite-trace PoSets and a subclass of labelled prime event structures.