H. C. M. Kleijn

A Trace Semantics for Petri Nets

A generalization of the notion of trace is proposed. This enables us to associate with each Petri net a single behavioural object, namely a poset of (generalized) traces. A characterization is given of the trace languages defined by Petri nets. We show that the general event structures of Winskel and the stable event structures can also be characterized in terms of our trace languages. One consequence is that in this framework, stable event structures, general event structures, and Petri nets constitute a strictly ascending chain in terms of expressive power.

Process semantics of general inhibitor nets

We define a causality semantics of Place/Transition nets with weighted inhibitor arcs (PTI-nets). We extend the standard approach to defining the partial order semantics of Place/Transition nets (PT-nets) based on the process semantics given through occurrence nets. To deal with inhibitor arcs at the level of occurrence nets activator arcs (and extra conditions) are used. The properties of the resulting activator occurrence nets are extensively investigated. It is then demonstrated how processes corresponding to step sequences of PTI-nets can be constructed algorithmically, and a non-algorithmic (axiomatic) characterisation is given of all those processes that can be obtained in this way. In addition, a general framework is established allowing to separately discuss behaviour, processes, causality, and their properties before proving that the resulting notions are mutually consistent for the various classes of Petri nets considered. This facilitates an efficient and uniform presentation of our results.

An event structure semantics for general Petri nets

Abstract In this paper we address the following question: What type of event structures are suitable for representing the behaviour of general Petri nets? As a partial answer to this question we define a new class of event structures called local event structures and identify a subclass called UL-event structures . We propose that UL-event structures are appropriate for capturing the behaviour of general Petri nets. Our answer is a partial one in that in the proposed event structure semantics, auto-concurrency is filtered out from the behaviour of Petri nets. It turns out that this limited event structure semantics for Petri nets is nevertheless a non-trivial and conservative extension of the (prime) event structure semantics of 1-safe Petri nets provided in Nielsen et al. (1981). We also show that the strong relationship between prime event structures and 1-safe Petri nets established in a categorical framework in Winskel (1987) can be extended to the present setting, provided we restrict our attention to the subclass of Petri nets whose behaviours do not exhibit any auto-concurrency. Finally, we show that Winskels general and stable event structures can be smoothly related to local event structures and that similarly prime event structures can be related to UL-event structures.

On the generative power of regular pattern grammars

SummaryIt is shown that for every recursively enumerable language L

A study in parallel rewriting systems

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Local Event Structures and Petri Nets

∑* there exists a selective substitution grammar with a regular selector over a binary alphabet that generates L¢5, where ¢∉∑.By requiring additional structural properties of the (already simple) selectors the language generating power is reduced in such a way that the resulting class lies strictly in between the family of EOL languages and the family of context-sensitive languages.For this class of languages some decision problems and normal forms are considered.

Sequential, continuous and parallel grammars

Abstract Selective substitution grammars based on ‘context-free’ productions form a possible framework for the study of ‘grammatically oriented’ formal language theory. Such grammars (with no control governing the composition of derivation steps) are studied in this paper. In particular we study the effect of various conditions on selectors (which define the way that rewriting is performed); those conditions are aimed to formalize the notion of ‘using information about the context’ during the rewriting process. Each of them captures a particular feature of a rewriting according to a context-free grammar or an EOS system (essentially a context-free grammar that can also rewrite terminal symbols). Some of those conditions yield characterizations of the class of context-free languages for other conditions the lower and upper bound on the language generating power are given. Also a natural notion of a class of ‘simple’ rewriting systems is introduced (pattern grammars) and it is demonstrated that they possess surprisingly high language generating power.

Petri net control for grammar systems

In this paper we study systematically three basic classes of grammars incorporating parallel rewriting: Indian parallel grammars, Russian parallel grammars and L systems. In particular by extracting basic characteristics of these systems and combining them we introduce new classes of rewriting systems ( ETOL [ k ] systems, ETOLIP systems and ETOLRP systems) Among others, some results on the combinatorial structure of Indian parallel languages and on the combinatorial structures of the new classes of languages are proved. As far as ETOL systems are concerned we prove that every ETOL language can be generated with a fixed (equal to 8) bounded degree of parallelism.