An algebra of high level petri nets

PhD ThesisPetri nets were introduced by C.A. Petri as a theoretical model of concurrency in which the causal relationship between actions, rather than just their temporal ordering, can be represented. As a theoretical model of concurrency, Petri nets have been widely successful. Moreover, Petri nets are popular with practitioners, providing practical tools for the designer and developer of real concurrent and distributed systems. However, it is from this second context that perhaps the most widely voiced criticism of Petri nets comes. It is that Petri nets lack any algebraic structure or modularity, and this results in large, unstructured models of real systems, which are consequently often intractable. Although this is not a criticism of Petri nets per se, but rather of the uses to which Petri nets are put, the criticism is well taken. We attempt to answer this criticism in this work. To do this we return to the view of Petri nets as a model of concurrency and consider how other models of concurrency counter this objection. The foremost examples are then the synchronisation trees of Milner, and the traces of Hoare, (against which such criticism is rarely, if ever, levelled). The difference between the models is clear, and is to be found in the richness of the algebraic characterisations which have been made for synchronisation trees in Milner's Calculus of Communicating Systems (CCS), and for traces in Hoare's Communicating Sequential Processes (CSP). With this in mind we define, in this thesis, a class of high level Petri nets, High Level Petri Boxes, and provide for them a very general algebraic description language, the High Level Petri Box Algebra, with novel ideas for synchronisation, and including both refinement and recursion among its operators. We also begin on the (probably open-ended task of the) algebraic characterisation of High Level Petri Boxes. The major contribution of this thesis is a full behavioural characterisation of the High Level Petri Boxes which form the semantic domain of the algebra. Other contributions are: a very general method of describing communication protocols which extend the synchronisation algebras of Winskel; a recursive operator that preserves finiteness of state (the best possible, given the generality of the algebra); a refinement operator that is syntactic in nature, and for which the recursive construct is a behavioural fix-point; and a notion of behavioural equivalence which is a congruence with respect to a major part of the High Level Petri Box Algebra

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

Finite petri nets as models for recursive causal behaviour

Goltz (1988) discussed whether or not there exist finite Petri nets (with unbounded capacities) modelling the causal behaviour of certain recursive CCS terms. As a representative example, the following term is considered: \ud \ud B=(a.nilb.B)+c.nil. \ud \ud We will show that the answer depends on the chosen notion of behaviour. It was already known that the interleaving behaviour and the branching structure of terms as B can be modelled as long as causality is not taken into account. We now show that also the causal behaviour of B can be modelled as long as the branching structure is not taken into account. However, it is not possible to represent both causal dependencies and the behaviour with respect to choices between alternatives in a finite net. We prove that there exists no finite Petri net modelling B with respect to both pomset trace equivalence and failure equivalence

Read Operators and their Expressiveness in Process Algebras

We study two different ways to enhance PAFAS, a process algebra for modelling asynchronous timed concurrent systems, with non-blocking reading actions. We first add reading in the form of a read-action prefix operator. This operator is very flexible, but its somewhat complex semantics requires two types of transition relations. We also present a read-set prefix operator with a simpler semantics, but with syntactic restrictions. We discuss the expressiveness of read prefixes; in particular, we compare them to read-arcs in Petri nets and justify the simple semantics of the second variant by showing that its processes can be translated into processes of the first with timed-bisimilar behaviour. It is still an open problem whether the first algebra is more expressive than the second; we give a number of laws that are interesting in their own right, and can help to find a backward translation.Comment: In Proceedings EXPRESS 2011, arXiv:1108.407

Spatial Logics for Bigraphs

Bigraphs are emerging as an interesting model for concurrent calculi, like CCS, pi-calculus, and Petri nets. Bigraphs are built orthogonally on two structures: a hierarchical place graph for locations and a link (hyper-)graph for connections. With the aim of describing bigraphical structures, we introduce a general framework for logics whose terms represent arrows in monoidal categories. We then instantiate the framework to bigraphical structures and obtain a logic that is a natural composition of a place graph logic and a link graph logic. We explore the concepts of separation and sharing in these logics and we prove that they generalise some known spatial logics for trees, graphs and tree contexts

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 bunch of events, thus generalizing the principle of true concurrency. While they seem to be 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

On the Model of Computation of Place/Transition Petri Nets

In the last few years, the semantics of Petri nets has been investigated in several different ways. Apart from the classical "token game", one can model the behaviour of Petri nets via non-sequential processes, via unfolding constructions, which provide formal relationships between nets and domains, and via algebraic models, which view Petri nets as essentially algebraic theories whose models are monoidal categories. In this paper we show that these three points of view can be reconciled. More precisely, we introduce the new notion of decorated processes of Petri nets and we show that they induce on nets the same semantics as that of unfolding. In addition, we prove that the decorated processes of a net N can be axiomatized as the arrows of a symmetric monoidal category which, therefore, provides the aforesaid unification

Bisimilarity and Behaviour-Preserving Reconfigurations of Open Petri Nets

We propose a framework for the specification of behaviour-preserving reconfigurations of systems modelled as Petri nets. The framework is based on open nets, a mild generalisation of ordinary Place/Transition nets suited to model open systems which might interact with the surrounding environment and endowed with a colimit-based composition operation. We show that natural notions of bisimilarity over open nets are congruences with respect to the composition operation. The considered behavioural equivalences differ for the choice of the observations, which can be single firings or parallel steps. Additionally, we consider weak forms of such equivalences, arising in the presence of unobservable actions. We also provide an up-to technique for facilitating bisimilarity proofs. The theory is used to identify suitable classes of reconfiguration rules (in the double-pushout approach to rewriting) whose application preserves the observational semantics of the net.Comment: To appear in "Logical Methods in Computer Science", 41 page