Asynchronous Transition Systems Research Articles

Bisimulations Respecting Duration and Causality for the Non-interleaving Applied π-Calculus

This paper shows how we can make use of an asynchronous transition system, whose transitions are labelled with events and which is equipped with a notion of independence of events, to define non-interleaving semantics for the applied $\pi$-calculus. The most important notions we define are: Start-Termination or ST-bisimilarity, preserving duration of events; and History-Preserving or HP- bisimilarity, preserving causality. We point out that corresponding similarity preorders expose clearly distinctions between these semantics. We draw particular attention to the distinguishing power of HP failure similarity, and discuss how it affects the attacker threat model against which we verify security and privacy properties. We also compare existing notions of located bisimilarity to the definitions we introduce.

Event structures for the reversible early internal π-calculus

The π-calculus is a widely used process calculus, which models communications between processes and allows the passing of communication links. Various operational semantics of the π-calculus have been proposed, which can be classified according to whether transitions are unlabelled (so-called reductions) or labelled. With labelled transitions, we can distinguish early and late semantics. The early version allows a process to receive names it already knows from the environment, while the late semantics and reduction semantics do not. All existing reversible versions of the π-calculus use reduction or late semantics, despite the early semantics of the (forward-only) π-calculus being more widely used than the late. We introduce two reversible forms of the internal π-calculus; these are the first to use early semantics. The internal π-calculus is a subset of the π-calculus where every link sent by an output is private, yielding greater symmetry between inputs and outputs. One of the new reversible calculi uses static reversibility, where performing an action does not change the structure of the process, and the other uses dynamic reversibility, where performing an action moves it to a separate history. We show an operational correspondence between the two calculi. For the static calculus we define denotational event structure semantics, which generate an event structure inductively on the structure on the process. For the dynamic calculus we define operational event structure semantics, which generate an event structure based on a labelled asynchronous transition system. We describe a correspondence between the resulting event structures.

A calculus of concurrent graph-rewriting processes

State-of-the-art approaches to controlled graph rewriting focus on the specification of an external control layer over graph-rewriting rule applications and on the input-output semantics of the resulting systems, leading to a decreased relevance of many interesting operational aspects of graph transformation, as studied in the classical theory of algebraic graph rewriting.We propose a novel approach to controlled graph rewriting where we aim at bridging the gap between these two complementary approaches, defining an operational semantics for which classical concepts and results related to independence and parallelism of graph derivations can be recast.The calculus we propose is based on a control layer specified using (a fragment of) Milner's Calculus of Communicating Systems (CCS), where the actions specify the application of graph transformation rules to the current state, according to the Double-Pushout approach (DPO).In particular, we address the following aspects for our controlled graph-rewriting processes: (i) expressiveness of the control language, compared to a minimally complete reference language, Graph Programs, proposed by Habel and Plump; (ii) process equivalence based on labeled transition systems and a structural operational semantics; (iii) a unifying treatment of action independence, considering the corresponding notions from both process algebra and graph-rewriting theory; and (iv) parallelism and concurrency based on the notion of asynchronous (labeled) transition systems, obtained from an asynchronous version of the proposed calculus.

A stable non-interleaving early operational semantics for the pi-calculus

We give the first non-interleaving early operational semantics for the pi-calculus which generalises the standard interleaving semantics and unfolds to the stable model of prime event structures. Our starting point is the non-interleaving semantics given for CCS by Mukund and Nielsen, where the so-called structural (prefixing or subject) causality and events are defined from a notion of locations derived from the syntactic structure of the process terms. We conservatively extend this semantics with a notion of extruder histories, from which we infer the so-called link (name or object) causality and events introduced by the dynamic communication topology of the pi-calculus. We prove that the semantics generalises both the standard interleaving early semantics for the pi-calculus and the non-interleaving semantics for CCS. In particular, it gives rise to a labelled asynchronous transition system unfolding to prime event structures.

Formal Relationships Between Geometrical and Classical Models for Concurrency

A wide variety of models for concurrent programs has been proposed during the past decades, each one focusing on various aspects of computations: trace equivalence, causality between events, conflicts and schedules due to resource accesses, etc. More recently, models with a geometrical flavor have been introduced, based on the notion of cubical set. These models are very rich and expressive since they can represent commutation between any number of events, thus generalizing the principle of true concurrency. While they are emerging as a central tool in concurrency, which is very promising because they make possible the use of techniques from algebraic topology in order to study concurrent computations, they have not yet been precisely related to the previous models, and the purpose of this paper is to fill this gap. In particular, we describe an adjunction between Petri nets and cubical sets which extends the previously known adjunction between Petri nets and asynchronous transition systems by Nielsen and Winskel.

Studying a mathematical model of parallel computation by algebraic topology methods

Under study is one of the mathematical models of parallel computational processes, i.e., an asynchronous transition system. Some methods are proposed for calculating the homology groups and the Poincare polynomial of a finite asynchronous transition system. Conditions are obtained for the decomposability of an asynchronous transition system as a parallel product.

A Framework for Component Categories

This paper provides further developments in the study of the component categories which have been introduced in [Fajstrup, L., E. Goubault, E. Haucourt and M. Rauszen, Component categories and the fundamental category, APCS 12 (2004), pp. 81-108]. In particular, the component category functor is seen as a left adjoint hence preserves the pushouts. This property is applied to prove a Van Kampen like theorem for component categories. This last point is very important to make effective calculations. The original purpose of component categories is to suitably reduce the size of the fundamental categories which are the directed counterpart of classical fundamental groupoids (see [Higgins, P.J., ''Categories and Groupoids,'' Mathematical Studies 32, Van Nostrand Reinhold, 1971, libre d'accs sur internet l'adresse http://www.tac.mta.ca/tac/reprints/]). In concrete examples, the fundamental category is as ''big'' as R while the component category is ''finitely generated''. We take advantage of this fact to define the cohomology of a directed geometrical shape as the cohomology of its component category. The cohomology of small categories is defined in [Baues, H.J. and G. Wirsching, Cohomology of small categories, Journal of Pure and Applied Algebra 38 (1985)] and [Baues, H.J., ''Combinatorial Homotopy and 4-Dimensional Complexes,'' De Gruyter expositions in Mathematics 2, Walter de Gruyter, 1991]. Still, in the recent paper [Husainov, A.A., On the homolgy of small categories and asynchronous transition systems, Homology, Homotopy and Applications 6 (2004), pp. 439-471], the homology of small categories is defined in a very similar way and applied to the study of asynchronous transition systems.

On the Leech dimension of a free partially commutative monoid

We prove that the Leech dimension of any free partially commutative monoid is equal to the supremum of numbers of its mutually commuting generators. As a consequence, we conrm a conjecture that if a free partially commutative monoid does not contain more than n mutually commuting generators, then it is of homological dimension 6 n. We apply this result to the homological dimension of asynchronous transition systems. We positively answer the question whether the homological dimension of an asynchronous transition system is not greater than the maximal number of its mutually independent events.

Concurrent Logic and Automata Combined: A Semantics for Components

In this paper, we describe a true-concurrent hierarchical logic interpreted over concurrent automata. Concurrent automata constitute a special kind of asynchronous transition system (ATS) used for modelling the behaviour of components as understood in component-based software development. Here, a component-based system consists of several interacting components whereby each component manages calls to and from the component using ports to ensure encapsulation. Further, a component can be complex and made of several simpler interacting components. When a complex component receives a request through one of its ports, the port delegates the request to an internal component. Our logic allows us to describe the different views we can have on the system. For example, the overall component interactions, whether they occur sequentially, simultaneously or in parallel, and how each component internally manages the received requests (possibly expressed at different levels of detail). Using concurrent automata as an underlying formalism we guarantee that the expressiveness of the logic is preserved in the model. In future work, we plan to integrate our truly-concurrent approach into the Edinburgh Concurrency Workbench.

Modeling and verification of an air traffic concept of operations

A high level model of the concept of operations of NASA's Small Aircraft Transportation System for Higher Volume Operations (SATS-HVO) is presented. The model is a non-deterministic, asynchronous transition system. It provides a robust notion of safety that relies on the logic of the concept rather than on physical constraints such as aircraft performances. Several safety properties were established on this model. The modeling and verification effort resulted in the identification of 9 issues, including one major flaw, in the original concept. Ten recommendations were made to the SATS-HVO concept development working group. All the recommendations were accepted and incorporated into the current concept of operations. The model was written in PVS. The verification is performed using an explicit state exploration algorithm written and proven correct in PVS.

Models for name-passing processes: interleaving and causal

We study syntax-free models for name-passing processes. For interleaving semantics, we identify the indexing structure required of an early labelled transition system to support the usual π -calculus operations, defining Indexed Labelled Transition Systems . For non-interleaving causal semantics we define Indexed Labelled Asynchronous Transition Systems , smoothly generalizing both our interleaving model and the standard Asynchronous Transition Systems model for CCS-like calculi. In each case we relate a denotational semantics to an operational view, for bisimulation and causal bisimulation respectively. We establish completeness properties of, and adjunctions between, categories of the two models. Alternative indexing structures and possible applications are also discussed. These are first steps towards a uniform understanding of the semantics and operations of name-passing calculi.

On the homology of small categories and asynchronous transition systems

This work is devoted to an interpretation and computation of the first homology groups of the small category given by a rewriting system. It is shown that the elements of the first homology group may be regarded as the equivalence classes of the flows in a graph of the rewriting system. This is applied to calculating the homology groups of asynchronous transition systems and Petri nets. Examples of calculations are given.

Undecidability of domino games and hhp-bisimilarity

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 that they are preserved under action refinement, an operation important for the top-down design of concurrent systems. It is shown that, in contrast to hp-bisimilarity, checking hhp-bisimilarity for finite labelled asynchronous transition systems is undecidable, by a reduction from the halting problem of 2-counter machines. To make the proof more transparent a novel intermediate problem of checking domino bisimilarity for origin constrained tiling systems is introduced and shown undecidable. It is also shown that the unlabelled domino bisimilarity problem is undecidable, which implies undecidability of hhp-bisimilarity for unlabelled finite asynchronous systems. Moreover, it is argued that the undecidability of hhp-bisimilarity holds for finite elementary net systems and 1-safe Petri nets.

Hereditary History Preserving Bisimilarity is Undecidable

We show undecidability of hereditary history preserving bisimilarity<br />for finite asynchronous transition systems by a reduction from the halting<br />problem of deterministic 2-counter machines. To make the proof more<br />transparent we introduce an intermediate problem of checking domino<br />bisimilarity for origin constrained tiling systems. First we reduce the<br />halting problem of deterministic 2-counter machines to origin constrained<br />domino bisimilarity. Then we show how to model domino bisimulations as<br />hereditary history preserving bisimulations for finite asynchronous transitions<br />systems. We also argue that the undecidability result holds for<br />finite 1-safe Petri nets, which can be seen as a proper subclass of finite<br />asynchronous transition systems.

Hereditary History Preserving Simulation is Undecidable

We show undecidability of hereditary history preserving simulation<br />for finite asynchronous transition systems by a reduction from the halting<br />problem of deterministic Turing machines. To make the proof more<br />transparent we introduce an intermediate problem of deciding the winner<br />in domino snake games. First we reduce the halting problem of deterministic<br />Turing machines to domino snake games. Then we show how to<br />model a domino snake game by a hereditary history simulation game on<br />a pair of finite asynchronous transition systems.

Transition Systems with Independence and Multi-Arcs

We extend the model of transition systems with independence in order to provide it with a feature relevant in the noninterleaving analysis of concurrent systems, namely multi-arcs. Moreover, we study the relationships between the category of transition systems with independence and multi-arcs and the category of labeled asynchronous transition systems, extending the results recently obtained by the authors for (simple) transition systems with independence (cf. Proc. CONCUR’96), and yielding a precise characterisation of transition systems with independence and multi-arcs in terms of (event-maximal, diamond-extensional) labelled asynchronous transition systems.

Timing and causality in process algebra

There has been considerable controversy in concurrency theory between theinterleaving' andtrue concurrency' schools. The former school advocates associating a transition system with a process which captures concurrent execution via the interleaving of occurrences; the latter adopts more complex semantic structures to avoid reducing concurrency to interleaving. In this paper we show that the two approaches are not irreconcilable. We define a timed process algebra where occurrences are associated with intervals of time, and give it a transition system semantics. This semantics has many of the advant- ages of the interleaving approach; the algebra admits an expansion theorem, and bisimulation semantics can be used as usual. Our transition systems, however, incorporate timing information, and this enables us to express concurrency: merely adding timing appropriately generalises transition systems to asynchronous transition systems, showing that time gives a link between true concurrency and interleaving. Moreover, we can provide a complete axiomatisation of bisimulation for our algebra; a result that is often problematic in a timed setting. Another advantage of incorporating timing information into the calculus is that it allows a particularly simple definition of action refinement; this we present. The paper concludes with a comparison of the equivalence we present with those in the literature, and an example system specification in our formalism.

Comparing Transition Systems with Independence and Asynchronous Transition Systems

Transition systems with independence and asynchronous transition systems are non-interleaving models for concurrency arising from the same simple idea of decorating transitions with events. They differ for the choice of a derived versus a primitive notion of event which induces considerable differences and makes the two models suitable for different purposes. This opens the problem of investigating their mutual relationships,<br />to which this paper gives a fully comprehensive answer.<br />In details, we characterise the category of extensional asynchronous transitions systems as the largest full subcategory of the category of (labelled) asynchronous transition systems which admits TSI, the category of transition systems with independence, as a coreflective subcategory. In addition, we introduce event-maximal asynchronous transitions systems and we show that their category is equivalent to TSI, so providing an exhaustive characterisation of transition systems with independence in terms of asynchronous transition systems.

Petri nets and bisimulation

Several categorical relationships (adjunctions) between models for concurrency have been established, allowing the translation of concepts and properties from one model to another. A central example is a coreflection between Petri nets and asynchronous transition systems. The purpose of the present paper is to illustrate the use of such relationships by transferring to Petri nets a general concept of bisimulation.

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.