. We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2-theories. We show that this presentation is equivalent to the well-accepted operational definition proposed by Barendregt et alii---but for the case of circular redexes, for which we propose (and justify formally) a different treatment. The categorical framework allows us to model in a concise way also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures, and to relate term graph rewriting to other rewriting formalisms. R'esum'e. Nous pr'esentons une formulation cat'egorique de la r'e'ecriture des graphes cycliques des termes, bas'ee sur une variante de 2-theorie alg'ebrique. Nous prouvons que cette pr'esentation est 'equivalente `a la d'efinition op'erationnelle propos'ee par Barendregt et d'autres auteurs, mais pas dons le cas des radicaux circulaires, pour lesquels nous proposons (et justifions formellem...

We present a categorical formulation of the rewriting of possibly cyclic term graphs, and the proof that this presentation is equivalent to the well-accepted operational definition proposed in [3] -- but for the case of circular redexes, for which we propose (and justify formally) a different treatment. The categorical framework, based on suitable 2-categories, allows to model also automatic garbage collection and rules for sharing/unsharing and folding/unfolding of structures. Furthermore, it can be used for defining various extensions of term graph rewriting, and for relating it to other rewriting formalisms.

The overall goal of this paper is to investigate the theoretical foundations of algorithmic verification techniques for specifications based on first order linear logic, a logic that can be used to naturally model infinite state systems with internal structured data. The fragment we consider in this paper is based on the linear logic programming language called LO [4] enriched with universally quantified goal formulas. Although LO was originally introduced as a theoretical foundation for extensions of logic programming languages, it can also be viewed as a very general language to specify a wide range of concurrent systems.

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Engberg and Winskel's Petri net semantics of linear logic is re-considered, from the point of view of the logic BI of bunched implications. We first show how BI can be used to overcome a number of difficulties pointed out by Engberg and Winskel, and we argue that it provides a more natural logic for the net semantics. We then briefly consider a more expressive logic based on an extension of BI with classical and modal features.

Object Petri nets are one approach to the investigation of object-orientation and concurrency. We like to relate some work done in the area of the possible semantics for some classes of object Petri nets to the notations used for process calculi in the Milner style and to some algebraic foundations. This algebra reveals some insights to the connections between the reference and the value semantics. We also hope, that this approach helps to unify rewriting systems with the theories on the description of processes of Petri nets and also with the theory of process algebra in a simple way.

Object Petri nets are one approach to the investigation of object-orientation and concurrency. We like to relate some work done in the area of the possible semantics for some classes of object Petri nets to the notations used for process calculi in the Milner style and to some algebraic foundations. This algebra reveals some insights to the connections between the reference and the value semantics. We also hope, that this approach helps to unify rewriting systems with the theories on the description of processes of Petri nets and also with the theory of process algebra in a simple way.

proof. Let us examine why. ; #M 1 :A 2#A 1 2 :A 2 #E. M 2 :A 1 We can make the following inferences. V 1 = #x:A 2 .M # 1 By type preservation and inversion At this point we cannot proceed: we need a derivation of [V 2 /x]M # 1 ## V for some V to complete the derivation of M 1 M 2 ## V . Unfortunately, the induction hypothesis does not tell us anything about [V 2 /x]M # 1 . Basically, we need to extend it so it makes a statement about the result of evaluation ( #x:A 2 .M # 1 ,inthis case). Sticking to the case of linear application for the moment, we call a term M "good" if it evaluates to a "good" value V .AvalueVis "good" if it is a function #x:A 2 .M # 1 and if substituting a "good" value V 2 for x in M # 1 results in a "good" term. Note that this is not a proper definition, since to see if V is "good" we may need to substitute any "good" value V 2 into it, possibly including V itself. We can make this definition inductive if we observe that the value

but ag it to indicate that it may not be exact, but that some of these linear hypotheses may be absorbed if necessary. In other words, in the judgment any of the remaining hypotheses in O need not be consumed in the other branches of the typing derivation. On the other hand, the judgment ; I n O ` 0 M : A indicates the M uses exactly the variables in I O . When we think of the judgment ; I n O ` i M : A as describing an algorithm, we think of , I and M as given, and O and the slack indicator i as part of the result of the computation. The type A may or may not be given|in one case it is synthesized, in the other case checked. This re nes our view as computation being described as the bottom-up construction of a derivation to include parts of the judgment in dierent roles (as input, output, or bidirectional components). In logic programming, which is based on the notion of computation-as-proof-search, these roles of the syntactic constituents of a judgment are called