Uffe Engberg

Fully abstract models for a process language with refinement

We study the use of sets of labelled partial orders (pomsets) as denotational models for process algebras. More specifically, we study their capability to capture degrees of nonsequentiality of processes. We present four full abstractness results. The operational equivalences are based on maximal action-sequences and step-sequences — defined for a very simple process language and its extensions with a refinement combinator (change of atomicity). The denotational models are all expressed as abstractions of a standard association of sets of labelled partial orders with processes.

Completeness results for linear logic on Petri nets

Abstract Completeness is shown for several versions of Girards linear logic with respect to Petri nets as the class of models. One logic considered is the ⊕-free fragment of intuitionistic linear logic without the exponential !. For this fragment Petri nets form a sound and complete model. The strongest logic considered is intuitionistic linear logic, with ⊗,&, ⊕ and the exponential ! (“of course”), and forms of quantification. This logic is shown sound and complete with respect to atomic nets (these include nets in which every transition leads to a nonempty multiset of places), though only once we add extra axioms specific to the Petri-net model. The logic is remarkably expressive, enabling descriptions of the kinds of properties one might wish to show of nets; in particular, negative properties, asserting the impossibility of an assertion, can also be expressed. Unfortunately, with respect to this logic, whether an assertion is true of a finite net becomes undecidable.

Petri nets as models of linear logic

The chief purpose of this paper is to appraise the feasibility of Girards linear logic as a specification language for parallel processes. To this end we propose an interpretation of linear logic in Petri nets, with respect to which we investigate the expressive power of the logic.

Linear Logic on Petri Nets

This article shows how individual Petri nets form models of Girards intuitionistic linear logic. It explores questions of expressiveness and completeness of linear logic with respect to this interpretation. An aim is to use Petri nets to give an understanding of linear logic and give some appraisal of the value of linear logic as a specification logic for Petri nets. This article might serve as a tutorial, providing one in-road into Girards linear logic via Petri nets. With this in mind we have added several exercises and their solutions. We have made no attempt to be exhaustive in our treatment, dedicating our treatment to one semantics of intuitionistic linear logic.

Completeness Results for Linear Logic on Petri Nets

Completeness is shown for several versions of Girards linear logic with respect to Petri nets as the class of models. The strongest logic considered is intuitionistic linear logic, with ⊗, ⊸, &, ⊕ and the exponential ! (“of course”), and forms of quantification. This logic is shown sound and complete with respect to atomic nets (these include nets in which every transition leads to a nonempty multiset of places). The logic is remarkably expressive, enabling descriptions of the kinds of properties one might wish to show of nets; in particular, negative properties, asserting the impossibility of an assertion, can also be expressed.

Failures Semantics for a Simple Process Language with Refinement

We study a suitable semantic theory based on the standard failures preorder [BHR84] for a simple process algebra which includes operators for the refinement of actions by processes. We present a model-theoretic and a behavioural characterization of the largest precongruence associated with failures preorder, ⫇ F c , over the language we consider.

Efficient Simplification of Bisimulation Formulas

The problem of checking or optimally simplifying bisimulation formulas is likely to be computationally very hard. We take a different view at the problem: we set out to define a very fast algorithm, and then see what we can obtain. Sometimes our algorithm can simplify a formula perfectly, sometimes it cannot. However, the algorithm is extremely fast and can, therefore, be added to formula-based bisimulation model checkers at practically no cost. When the formula can be simplified by our algorithm, this can have a dramatic positive effect on the better, but also more time consuming, theorem provers which will finish the job.