Asymmetric Event Structures Research Articles

Minimisation of event structures

Event structures are fundamental models in concurrency theory, providing a representation of events in computation and of their relations, notably concurrency, conflict and causality. In this paper we present a theory of minimisation for event structures. Working in a class of event structures that generalises many stable event structure models in the literature (e.g., prime, asymmetric, flow and bundle event structures), we study a notion of behaviour-preserving quotient, referred to as a folding, taking (hereditary) history-preserving bisimilarity as a reference behavioural equivalence. We show that for any event structure a folding producing a uniquely determined minimal quotient always exists. We observe that each event structure can be seen as the folding of a prime event structure, and that all foldings between general event structures arise from foldings of (suitably defined) corresponding prime event structures. This gives a special relevance to foldings in the class of prime event structures, which are studied in detail. We identify folding conditions for prime and asymmetric event structures, and show that also prime event structures always admit a unique minimal quotient (while this is not the case for various other event structure models).

Towards a categorical representation of reversible event structures

We study categories for reversible computing, focussing on reversible forms of event structures. Event structures are a well-established model of true concurrency. There exist a number of forms of event structures, including prime event structures, asymmetric event structures, and general event structures. More recently, reversible forms of these types of event structure have been defined. We formulate corresponding categories and functors between them. We show that products and coproducts exist in many cases.We define stable reversible general event structures and stable configuration systems, and we obtain an isomorphism between the subcategory of the former in normal form and the finitely enabled subcategory of the latter.In most work on reversible computing, including reversible process calculi, a causality condition is posited, meaning that the cause of an event may not be reversed before the event itself. Since reversible event structures are not assumed to be causal in general, we also define causal subcategories of these event structures.

Towards a Categorical Representation of Reversible Event Structures

We study categories for reversible computing, focussing on reversible forms of event structures. Event structures are a well-established model of true concurrency. There exist a number of forms of event structures, including prime event structures, asymmetric event structures, and general event structures. More recently, reversible forms of these types of event structures have been defined. We formulate corresponding categories and functors between them. We show that products and co-products exist in many cases. In most work on reversible computing, including reversible process calculi, a cause-respecting condition is posited, meaning that the cause of an event may not be reversed before the event itself. Since reversible event structures are not assumed to be cause-respecting in general, we also define cause-respecting subcategories of these event structures. Our longer-term aim is to formulate event structure semantics for reversible process calculi.

Reduction of event structures under history preserving bisimulation

Event structures represent concurrent processes in terms of events and dependency relations between events like causality and conflict. Since the introduction of prime event structures, many variants of event structures have been proposed with different dependency relations and, hence, with differences in their expressive power. One of the possible benefits of using a more expressive event structure model is that of obtaining a more compact representation for the same behaviour using a smaller number of events. This article addresses the problem of reducing the size of an event structure while preserving its behaviour under a classical notion of behavioural equivalence in the true concurrency spectrum, namely history preserving bisimulation. In particular, we investigate this problem on two generalisations of prime event structures: asymmetric event structures, which rely on an asymmetric form of conflict, and flow event structures, which support a form of disjunctive causality. We single out conditions under which distinct events in an event structure can be seen as occurrences of the same activity in different contexts and thus can be folded into a single event without altering the original behaviour. By iterating the folding operation, any finite event structure can be reduced to a minimal form, behaviourally equivalent to the original one. This is not unique in general, as it depends on the order on which the folding operations are applied.

Reversibility and asymmetric conflict in event structures

Reversible computation has attracted increasing interest in recent years, with applications in hardware, software and biochemistry. We introduce reversible forms of prime event structures and asymmetric event structures. In order to control the manner in which events are reversed, we use asymmetric conflict on events. We prove a number of results about reachable configurations; for instance, we show under what conditions reachable configurations which are finite are reachable by purely finite means. We discuss, with examples, reversing in causal order, where an event is only reversed once all events it caused have been reversed, as well as forms of non-causal reversing.

Application of branching cells to QoS aware service orchestrations

By allowing service calls to be guarded by contexts, Asymmetric Event Structures (AES for short) and contextual nets are a convenient framework to model composite Web services or service orchestrations. We equip AES with QoS domains as a framework to capture a number of QoS metrics and their combination. We use the resulting model to formalize QoS-based late service binding in composite services. When subject to QoS-based late service binding, composite services may be non-monotonic with respect to QoS, meaning that strictly improving the QoS of a service may strictly decrease the end-to-end QoS of the composite service, an embarrassing feature for QoS-aware management; we study this issue. Branching cells of AES play a central role in this study.

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.

Unfolding semantics of graph transformation

Several attempts have been made of extending to graph grammars the unfolding semantics originally developed by Winskel for (safe) Petri nets, but only partial results were obtained. In this paper, we fully extend Winskel’s approach to single-pushout grammars providing them with a categorical concurrent semantics expressed as a coreflection between the category of (semi-weighted) graph grammars and the category of prime algebraic domains, which factorises through the category of occurrence grammars and the category of asymmetric event structures. For general, possibly nonsemi-weighted single-pushout grammars, we define an analogous functorial concurrent semantics, which, however, is not characterised as an adjunction. Similar results can be obtained for double-pushout graph grammars, under the assumptions that nodes are never deleted.

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.