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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 71 (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 ...
Rewriting On Cyclic Structures: Equivalence Between The Operational And The Categorical Description
, 1999
"... . We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2theories. We show that this presentation is equivalent to the wellaccepted operational definition proposed by Barendregt et aliibut for the case of circular redexes, fo ..."
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Cited by 12 (6 self)
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. We present a categorical formulation of the rewriting of possibly cyclic term graphs, based on a variation of algebraic 2theories. We show that this presentation is equivalent to the wellaccepted operational definition proposed by Barendregt et aliibut 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 2theorie 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...
Relating Graph and Term Rewriting via Böhm Models
 in Engineering, Communication and Computing 7
, 1993
"... . Dealing properly with sharing is important for expressing some of the common compiler optimizations, such as common subexpressions elimination, lifting of free expressions and removal of invariants from a loop, as sourcetosource transformations. Graph rewriting is a suitable vehicle to accommoda ..."
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Cited by 8 (4 self)
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. Dealing properly with sharing is important for expressing some of the common compiler optimizations, such as common subexpressions elimination, lifting of free expressions and removal of invariants from a loop, as sourcetosource transformations. Graph rewriting is a suitable vehicle to accommodate these concerns. In [4] we have presented a term model for graph rewriting systems (GRSs) without interfering rules, and shown the partial correctness of the aforementioned optimizations. In this paper we define a different model for GRSs, which allows us to prove total correctness of those optimizations. Differently from [4] we will discard sharing from our observations and introduce more restrictions on the rules. We will introduce the notion of Bohm tree for GRSs, and show that in a system without interfering and nonleft linear rules (orthogonal GRSs), Bohm tree equivalence defines a congruence. Total correctness then follows in a straightforward way from showing that if a program M co...
ABSTRACT MODELS OF TRANSFINITE REDUCTIONS
, 2010
"... We investigate transfinite reductions in abstract reduction systems. To this end, we study two abstract models for transfinite reductions: a metric model generalising the usual metric approach to infinitary term rewriting and a novel partial order model. For both models we distinguish between a we ..."
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Cited by 6 (6 self)
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We investigate transfinite reductions in abstract reduction systems. To this end, we study two abstract models for transfinite reductions: a metric model generalising the usual metric approach to infinitary term rewriting and a novel partial order model. For both models we distinguish between a weak and a strong variant of convergence as known from infinitary term rewriting. Furthermore, we introduce an axiomatic model of reductions that is general enough to cover all of these models of transfinite reductions as well as the ordinary model of finite reductions. It is shown that, in this unifying axiomatic model, many basic relations between termination and confluence properties known from finite reductions still hold. The introduced models are applied to term rewriting but also to term graph rewriting. We can show that for both term rewriting as well as for term graph rewriting the partial order model forms a conservative extension to the metric model.
Rewriting on Cyclic Structures
 Extended abstract in Fixed Points in Computer Science, satellite workshop of MFCS'98
, 1998
"... We present a categorical formulation of the rewriting of possibly cyclic term graphs, and the proof that this presentation is equivalent to the wellaccepted operational definition proposed in [3]  but for the case of circular redexes, for which we propose (and justify formally) a different treatm ..."
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Cited by 4 (3 self)
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We present a categorical formulation of the rewriting of possibly cyclic term graphs, and the proof that this presentation is equivalent to the wellaccepted 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 2categories, 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.
Defining Operational Behavior of Object Specifications by Attributed Graph Transformations
 Fundamenta Informaticae
, 1996
"... . A single pushout approach to the transformation of attributed partial graphs based on categories of partial algebras and partial morphisms is introduced. A sufficient condition for pushouts in these categories is presented. As the synchronization mechanism we use amalgamation of rules and show how ..."
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Cited by 4 (2 self)
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. A single pushout approach to the transformation of attributed partial graphs based on categories of partial algebras and partial morphisms is introduced. A sufficient condition for pushouts in these categories is presented. As the synchronization mechanism we use amalgamation of rules and show how synchronization can be minimized. We point out how the results obtained can be employed in order to define an operational semantics for object specification languages. 1 Introduction Graphs and graph grammars usually yield intuitive descriptions of complex phenomena in computer science. Therefore, numerous approaches to graph grammars have been put forward, among them the logical approach [6], the set theoretic approach [29], and the algebraic approach [9]. Graphbased techniques have for instance been successfully applied in the realm of software engineering development environments [13, 14], for objectoriented languages based on asynchronous communication [22, 24, 20, 21] and in logic p...
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...
Termgraph rewriting via explicit paths
"... Abstract. The notion of path is classical in graph theory but not directly used in the term rewriting community. The main idea of this work is to raise the notion of path to the level of firstorder terms, i.e. paths become part of the terms and not just metainformation about them. These paths are ..."
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Cited by 2 (0 self)
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Abstract. The notion of path is classical in graph theory but not directly used in the term rewriting community. The main idea of this work is to raise the notion of path to the level of firstorder terms, i.e. paths become part of the terms and not just metainformation about them. These paths are represented by words of integers (positive or negative) and are interpreted as relative addresses in terms. In this way, paths can also be seen as a generalization of the classical notion of position for the firstorder terms and are inspired by de Bruijn indexes. In this paper, we define an original framework called Referenced Term Rewriting where paths are used to represent pointers between subterms. Using this approach, any termgraph rewriting systems can be simulated using a term rewritebased environment. 1
DACTL Rewriting is Categorical
, 1991
"... The graphmanipulating core of the general term graph rewriting language DACTL, namely contraction building and redirection, is reexamined from a categorical viewpoint. The essentials of this rather complex twophase operational semantics is recast as a Grothendieck opfibration of a category of grap ..."
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Cited by 1 (1 self)
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The graphmanipulating core of the general term graph rewriting language DACTL, namely contraction building and redirection, is reexamined from a categorical viewpoint. The essentials of this rather complex twophase operational semantics is recast as a Grothendieck opfibration of a category of graph rewrites over a base of rewrite rules. This generalises previous attempts to categorise contractum building and redirection as pushouts and is able to describe more DACTL rewrites than pushout models. The full operational core model conforms to a more restricted version of this construction and is able to successfully cope with examples such as the infamous circular I example a : I[ a ]. 1 INTRODUCTION The general term graph rewriting language DACTL arose as an attempt to provide an intermediate language for graph rewriting based implementations of contemporary programming paradigms such as functional or logic. It features multiple parallel redirections as the chief updating mechanism for...