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Equational term graph rewriting
 FUNDAMENTA INFORMATICAE
, 1996
"... 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 wellknown in process algebra and concurrency theory. Specifically, a homomorphism is a functional bisimulation. We prove that the bis ..."
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Cited by 79 (8 self)
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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 wellknown 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 µrule, and translations are given between term graphs and µexpressions. Using these, a proof system is given for µexpressions that is complete for the semantics given by infinite tree unwinding. Next, orthogonal term graph rewrite ...
(Cyclic) Term Graph Rewriting is adequate for Rational Parallel Term Rewriting
 CGH
, 1997
"... 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 ..."
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Cited by 20 (6 self)
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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...
Categorical Frameworks for Graph Transformation and
 HLR Systems based on the DPO Approach, Bulletin of the EATCS 102
, 2010
"... Several variants of highlevel replacement (HLR) and adhesive categories have been introduced in the literature as categorical frameworks for graph transformation and HLR systems based on the double pushout (DPO) approach. In addition to HLR, adhesive, and adhesive HLR categories several weak vari ..."
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Cited by 11 (3 self)
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Several variants of highlevel replacement (HLR) and adhesive categories have been introduced in the literature as categorical frameworks for graph transformation and HLR systems based on the double pushout (DPO) approach. In addition to HLR, adhesive, and adhesive HLR categories several weak variants, especially weak adhesive HLR with horizontal and vertical variants, as well as partial variants, including partial map adhesive and partial VK square adhesive categories are reviewed and related to each other. We propose as weakest version the class of vertical weak adhesive HLR categories, shortMadhesive categories, which are still sufficient to obtain most of the main results for graph transformation and HLR systems. The results in this paper are summarized in Fig. 1 showing a hierarchy of all these variants of adhesive, adhesive HLR, andMadhesive categories, which can be considered as different categorical frameworks for graph transformation and HLR systems.
Event Structures and Nonorthogonal term graph rewriting
 Mathematical Structure in Computer Science
, 1996
"... We examine nonorthogonal term graph rewriting systems. We define notions of compatibility and L'evy (or permutation) equivalence for sequences of such rewrites. We define standard sequences and show a standardisation theorem that says that every sequence is L'evy equivalent to a unique st ..."
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Cited by 4 (0 self)
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We examine nonorthogonal term graph rewriting systems. We define notions of compatibility and L'evy (or permutation) equivalence for sequences of such rewrites. We define standard sequences and show a standardisation theorem that says that every sequence is L'evy equivalent to a unique standard sequence. We then define a notion of labelled term graph rewriting (a loose analogy with L'evy labelling in the lambda calculus) and use this together with the standardisation theorem to characterise our definition of L'evy equivalence as a relation between sequences which rewrite the same set of redexes (up to garbage collection). We then define the set of preevents of a sequence and preevent equivalence. After defining irredundant preevents we define a minimisation algorithm which transforms preevents into equivalent irredundant ones. These ideas are used to show that two distinct nonempty preevents cannot belong to the same equivalence class and since we define events to be equivalence...
Algebraic Graph Derivations for Graphical Calculi
 Graph Theoretic Concepts in Computer Science (WG'96), volume 1197 of LNCS
, 1997
"... 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 VL ..."
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Cited by 3 (1 self)
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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 lowlevel 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
Some Properties of NonOrthogonal Term Graph Rewriting
 Electronic Notes in Theoretical Computer Science
, 1995
"... This paper examines leftlinear nonorthogonal 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 langua ..."
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This paper examines leftlinear nonorthogonal 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 nonorthogonal 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...
A Graph Structure Over the Category of Sets and Partial Functions
, 1993
"... 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 th ..."
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Cited by 2 (1 self)
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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...
P.B.: Composition of Transformations: A Framework for Systems with Dynamic Topology
 International Journal of Computing Anticipatory Systems
, 2004
"... Abstract In graphbased 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 computatio ..."
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Abstract In graphbased 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.
Hypergraph Rewriting Using Conformisms
 Electr. Notes in Th. Comp. Sc
, 1995
"... In this paper we study singlepushout transformation in a category of spans, a generalization of the notion of partial morphism in, for instance, [2,4]. As an application, singlepushout transformation in a category of hypergraphs with a special type of partial morphisms, the conformisms, is present ..."
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In this paper we study singlepushout transformation in a category of spans, a generalization of the notion of partial morphism in, for instance, [2,4]. As an application, singlepushout 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 singlepushout hypergraph rewriting by means of a detailed example. We present in this paper an approach to hypergraph rewriting by means of singlepushout 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...