We present an equational framework for term graph rewriting with cycles. The usual notion of homomorphism is phrased in terms of the notion of bisimulation, which is well-known in process algebra and concurrency theory. Specifically, a homomorphism is a functional bisimulation. We prove that the bisimilarity class of a term graph, partially ordered by functional bisimulation, is a complete lattice. It is shown how Equational Logic induces a notion of copying and substitution on term graphs, or systems of recursion equations, and also suggests the introduction of hidden or nameless nodes in a term graph. Hidden nodes can be used only once. The general framework of term graphs with copying is compared with the more restricted copying facilities embodied in the µ-rule, and translations are given between term graphs and µ-expressions. Using these, a proof system is given for µ-expressions that is complete for the semantics given by infinite tree unwinding. Next, orthogonal term graph rewrite ...

Acyclic Term Graphs are able to represent terms with sharing, and the relationship between Term Graph Rewriting (TGR) and Term Rewrtiting (TR) is now well understood [BvEG + 87, HP91]. During the last years, some researchers considered the extension of TGR to possibly cyclic term graphs, which can represent possibly infinite, rational terms. In [KKSdV94] the authors formalize the classical relationship between TGR and TR as an "adequate mapping" between rewriting systems, and extend it by proving that unraveling is an adequate mapping from cyclic TGR to rational, infinitary term rewriting: In fact, a single graph reduction may correspond to an infinite sequence of term reductions. Using the same notions, we propose a different adequacy result, showing that unraveling is an adequate mapping from cyclic TGR to rational parallel term rewriting, where at each reduction infinitely many rules can be applied in parallel. We also argue that our adequacy result is more natural...

In 1984, Raoult proposed a formalization of graph rewritings using pushouts in the category of graphs and partial functions. This note generalizes his method and formulates algebraic graph structure to introduce a more general framework for graph rewritings and to give a simple proof of existence theorem of pushouts using relational calculus. 1 Introduction There are many researches about graph grammars and graph rewritings using the category theory. The structure of a directed graph is a function from the set E of edges to the product set V 2 V of the source vertices set and destination vertices set. Ehrig[4] characterized the graph grammar and rewriting rules using two pushout squares and pushout complements in the category of graphs. As the category of graphs is considered as a functor category over the category of sets and functions, it becomes a topos and has various useful properties. The existence theorem of pushout complements in a topos including the category of graph was gen...

This paper examines left-linear non-orthogonal term graph rewriting systems that allow asymmetric conflicts between redexes. Using a definition of compatibility of sequences based on Boudol's work on the semantics of term rewriting, it shows that two properties associated with functional languages are true of such graph rewriting systems. First, that a notion of standard computation can be defined and that an associated standardisation theorem can be proved. Second, that a set of events can be associated with a reduction sequence and hence event structures modelling all the possible reduction sequences from a given initial graph can be constructed. 1 Introduction What properties of rewrite systems associated with functional languages carry over to non-orthogonal term graph rewrite systems? This paper concerns itself with two properties of reduction sequences: a notion of standard reduction and the construction of event structures. We show that both are possible. By term graph rewritin...

this paper, but only refer to it for comparison with one of the main streams of related work in the literature. In [BH94], an approach to transformations of expressions in UPAs via transformations of graphs has been presented and proven correct. The approach has been developed with a bias towards VLSI circuit development and the formalisation and drawings reflect this. More or less building on the approach of [BH94], another approach to graphical calculi has been presented in [CL95], where a gentler introduction is given and an attempt is made to somewhat generalise beyond UPAs. Both approaches, however, present the transformation rules as low-level graph manipulation rules and do not resort to any established graph transformation mechanism. As a result, there is only a fixed set of transformation rules that correspond to the basic axioms of the calculus, but no general mechanism to formulate new rules corresponding to proven theorems or special definitions. In this paper we start from a slightly more general definition of diagram as basic data structure for our graphical calculus, and we proceed to give algebraic definitions of rule application and derivation. We cleanly separate the syntax and the semantics of our diagrams and we define correctness of rules on a high level. For reasons of space we do not present any proofs, but concentrate on giving ample motivation and at least a few examples. I gratefully acknowledge the comments of an anonymous referee. 2 Type and Relation Terms

In this paper we study single-pushout transformation in a category of spans, a generalization of the notion of partial morphism in, for instance, [2,4]. As an application, single-pushout transformation in a category of hypergraphs with a special type of partial morphisms, the conformisms, is presented. In particular, we show the existence of the pushout of any pair of conformisms of hypergraphs with the same source hypergraph, and how to construct one such a pushout. Finally, hypergraph rewriting using conformisms is compared to single-pushout hypergraph rewriting by means of a detailed example. We present in this paper an approach to hypergraph rewriting by means of single-pushout transformation in a category of hypergraphs with a special type of partial morphisms, the conformisms, inspired by the homonymous notion in the theory of partial algebras. Although this approach has already been treated by some of us in [1] by means of what we could call "rudeforce methods," in this paper we...

Abstract In graph-based systems there are many methods to compose (possibly different) graphs. However, none of these usual compositions are adequate to naturally express semantics of systems with dynamic topology, i.e., systems whose topology admits successive transformations through its computation. We constructed a categorical semantic domain for graph based systems with dynamic topology using a new way to compose edges of (possible different) graphs. In this context, sequences of different graphs represent successive transformations of system topology during its computation and the edges composition between those graphs, the semantics of the corresponding dynamic system. Then we show how the proposed approach can be used to give semantics to concurrent anticipatory systems.

This note presents a new formalization of graph rewritings which generalizes traditional graph rewritings. Relational notions of graphs and their rewritings are introduced and several properties about graph rewritings are discussed using relational calculus (theory of binary relations). Single pushout approaches to graph rewritings proposed by Raoult and Kennaway are compared with our rewritings of relational (labeled) graph. Moreover a more general sufficient condition for two rewritings to commute and a theorem concerning critical pairs useful to demonstrate the confluency of graph rewriting systems are also given. 1 Introduction There are many researches [1-7,9,13,14,16-18,20-22] on graph grammars and graph rewritings which have a lot of applications including software specification, data bases, analysis of concurrent systems, developmental biology and many others. In these one of the advantages of categorical graph rewritings is to produce a universal reduction which eases theoret...

ED In this section, we define the free rewriting core of DACTL rewriting, i.e. we ignore all issues pertaining both to markings, and (for simplicity) the pattern calculus. In addition, our terminology may appear a little unusual to those familiar with DACTL. Suppose an alphabet of node symbols S = fS; T : : :g to be given. Definition 7.2.1 A term graph (or just graph) G, is a triple (N; oe; ff) where (1) N is a set of nodes, (2) oe is a map N ! S, (3) ff is a map N ! N , Thus oe(x) maps a node to the node symbol that labels it, and ff(x) maps each node to its sequence of successors. We write A(x), the arity of a node, for the domain of ff(x). Note that A(x) is a set of consecutive natural members starting at 1, or empty. We allow ourselves to write x 2 G (instead of x 2 N(G)) etc. Each successor node determines an arc of the graph, and we will refer to arcs using the notation (p k ; c), to indicate that the child c is the k th child of the parent p, i.e. that c = ff(x)[k] f...

. A unifying view of all constructions of pushouts of partial morphisms considered so far in the literature of single-pushout transformation is given in this paper. Pushouts of partial morphisms are studied in an abstract category of spans formed out of two distinguished subcategories of the base category, thus generalizing previous studies in single-pushout transformation. Such spans are single pairs of morphisms, instead of equivalence classes, providing then a notion of transformation which is independent of class representatives. A necessary and sufficient condition for the existence of the pushout of two spans is established which involves properties of the base category, from which the category of spans is derived, as well as properties of the spans themselves. Moreover, a necessary and sufficient condition for single-pushout derivations in a category of spans to subsume doublepushout derivations in the base category is established which only involves properties of the base categ...