Glynn Winskel

Petri nets, event structures and domains, part I

Abstract The general aim of this paper is to find a theory of concurrency combining the approaches of Petri and Scott (and others). In part I we introduce our formalisms. To connect the abstract ideas of events and domains of information, we show how casual nets induce certain kinds of domains where the information points are certain sets of events. This allows translations between the languages of net theory and domain theory. Following the idea that events of causal nets are occurrences, we generalise causal nets to occurrence nets, by adding forwards conflict. Just as infinite flow charts unfold finite ones, so transition nets can be unfolded into occurrence nets. Next we extend the above connections between nets and domains to these new nets. Event structures which are intermediate between nets and domains play an important part in all our work. Finally, as an example of how concepts translate from one formalism to the other, we show how Petris notion of confusion ties up with Kahn and Plotkins concrete domains. In part II we shall continue the job of connecting up notions within net theory and the theory of domains. In particular, we shall examine the idea of states of computations.

Petri Nets, Event Structures and Domains

The general aim of the paper is to find a theory of concurrency combining the approaches of Petri and Scott (and other workers) [Pet 1,2],ESeo ~,3],[Sto]. To connect the abstract ideas of events and domains of information, we show how causal nets induce certain kinds of domains where the information points are certain sets of events. This allows translations between the languages of net theory and domain theory. Following the idea that events of causal nets are occurrences we generalise causal nets to occurrence nets, by adding forwards conflict; just as infinite flow chartsunfold finite ones [Sco 2], so transition nets can be unfolded into occurrence nets. Next we extend the above connections between nets and domains to these new nets. Event structures, which are intermediate between nets and domains play an important part in all our work.

An introduction to event structures

Event structures are models of processes as events constrained by relations of consistency and enabling. These notes are intended to introduce the mathematical theory of event structures, show how they are related to Petri nets and Scott domains, and how they can be used to provide semantics to programming languages for parallel processes as well as languages with higher types.

Bisimulation from Open Maps

An abstract definition of bisimulation is presented. It makes possible a uniform definition of bisimulation across a range of different models for parallel computation presented as categories. As examples, transition systems, synchronisation trees, transition systems with independence (an abstraction from Petri nets), and labelled event structures are considered. On transition systems the abstract definition readily specialises to Milners strong bisimulation. On event structures it explains and leads to a strengthening of the history-preserving bisimulation of Rabinovitch and Traktenbrot and van Glabeek and Goltz. A tie-up with open maps in a (pre)topos, as they appear in the work of Joyal and Moerdijk, brings to light a new model, presheaves on categories of pomsets, into which the usual category of labelled event structures embeds fully and faithfully. As an indication of its promise, this new presheaf model has “refinement” operators. The general approach yields a logic, generalising Hennessy?Milner logic, which is characteristic for the generalised notion of bisimulation.

Event Structure Semantics for CCS and Related Languages

We give denotational semantics to a wide range of parallel programming languages based on the idea of Milner’s CCS [Mil80a], that processes communicate by events of mutual synchronization. Processes are denoted by labeled event structures. Event structures represent concurrency rather directly, as in net theory [Bra80]. The semantics does not simulate concurrency by non-deterministic interleaving. We first define a category E of event structures [NPW79, NPW81, Win80] appropriate to synchronized communication. The category bears a natural relation to a subcategory of trees though an interleaving functor; so results transfer to trees neatly. Then we introduce the concept of a synchronization algebra (S.A.) on labels by adopting an idea of Milner [Mil80b]. An S.A. specifies how two processes synchronize via labels on their events. From each S.A., L, we derive a category EL of labeled event structures with natural operations for composing labeled event structures. In particular the parallel composition L is derived from the product in E. We obtain semantics for a class of CCS-like languages by varying the S.A.. Synchronization algebras are very general so the class is very broad, handling synchrony and asynchrony in a common framework. As a corollary we get an event structure semantics for CCS. When interleaved our semantics is Milner’s synchronization/communication tree semantics [Mil80a]. However our semantics distinguishes more terms as it reflects concurrency. Event structure semantics is at a rather basic level of abstraction but should support all abstract notions of equivalence (see [Mil80a] for examples), including those which take concurrency into account.

Events in security protocols

The events of a security protocol and their causal dependency can play an important role in the analysis of security properties. This insight underlies both strand spaces and the inductive method. But neither of these approaches builds up the events of a protocol in a compositional way, so that there is an informal spring from the protocol to its model. By broadening the models to certain kinds of Petri nets, a restricted form of contextual nets, a compositional event-based semantics is given to an economical, but expressive, language for describing security protocols; so the events and dependency of a wide range of protocols are determined once and for all. The net semantics is formally related to a transition semantics, strand spaces and inductive rules, as well as trace languages and event structures, so unifying a range of approaches, as well as providing conditions under which particular, more limited, models are adequate for the analysis of protocols. The net semantics allows the derivation of general properties and proof principles which are demonstrated in establishing an authentication property, following a diagrammatic style of proof.

Using Information Systems to Solve Recursive Domain Equations Effectively.

This paper aims to make the following main contribution: to show how to use the concrete nature of Scott’s information systems to advantage in solving recursive domain equations. The method is based on the substructure relation between information systems. This essentially makes a complete partial order (cpo) of information systems. Standard domain constructions like function space can be made continuous on this cpo so the solution of recursive domain equations reduces to the more familiar construction of forming the least fixed-point of a continuous function.

Models for concurrency: towards a classification

Models for concurrency can be classified with respect to three relevant parameters: behaviour/ system, interleaving/noninterleaving, linear/branching time. When modelling a process, a choice concerning such parameters corresponds to choosing the level of abstraction of the resulting semantics. In this paper, we move a step towards a classification of models for concurrency based on the parameters above. Formally, we choose a representative of any of the eight classes of models obtained by varying the three parameters, and we study the formal relationships between them using the language of category theory.

A note on model checking the model n-calculus

Abstract This note presents a straightforward algorithm for checking whether or not a state of a labelled transition system satisfies an assertion in the modal ν-calculus and μ-calculus. The algorithm improves on those of Larsen, and Stirling and Walker in that its presentation lays bare the mechanism behind “local model checking”, and leads to a streamlined proof of the correctness and termination of the model-checking algorithm.

Probabilistic event structures and domains

This paper studies how to adjoin probability to event structures, leading to the model of probabilistic event structures. In their simplest form, probabilistic choice is localised to cells, where conflict arises; in which case probabilistic independence coincides with causal independence. An event structure is associated with a domain--that of its configurations ordered by inclusion. In domain theory, probabilistic processes are denoted by continuous valuations on a domain. A key result of this paper is a representation theorem showing how continuous valuations on the domain of a confusion-free event structure correspond to the probabilistic event structures it supports. We explore how to extend probability to event structures which are not confusion-free via two notions of probabilistic runs of a general event structure. Finally, we show how probabilistic correlation and probabilistic event structures with confusion can arise from event structures which are originally confusion-free by using morphisms to rename and hide events.