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Rewriting Logic as a Semantic Framework for Concurrency: a Progress Report
, 1996
"... . This paper surveys the work of many researchers on rewriting logic since it was first introduced in 1990. The main emphasis is on the use of rewriting logic as a semantic framework for concurrency. The goal in this regard is to express as faithfully as possible a very wide range of concurrency mod ..."
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Cited by 83 (23 self)
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. This paper surveys the work of many researchers on rewriting logic since it was first introduced in 1990. The main emphasis is on the use of rewriting logic as a semantic framework for concurrency. The goal in this regard is to express as faithfully as possible a very wide range of concurrency models, each on its own terms, avoiding any encodings or translations. Bringing very different models under a common semantic framework makes easier to understand what different models have in common and how they differ, to find deep connections between them, and to reason across their different formalisms. It becomes also much easier to achieve in a rigorous way the integration and interoperation of different models and languages whose combination offers attractive advantages. The logic and model theory of rewriting logic are also summarized, a number of current research directions are surveyed, and some concluding remarks about future directions are made. Table of Contents 1 In...
The Tile Model
 PROOF, LANGUAGE AND INTERACTION: ESSAYS IN HONOUR OF ROBIN MILNER
, 1996
"... In this paper we introduce a model for a wide class of computational systems, whose behaviour can be described by certain rewriting rules. We gathered our inspiration both from the world of term rewriting, in particular from the rewriting logic framework [Mes92], and of concurrency theory: among the ..."
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Cited by 65 (24 self)
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In this paper we introduce a model for a wide class of computational systems, whose behaviour can be described by certain rewriting rules. We gathered our inspiration both from the world of term rewriting, in particular from the rewriting logic framework [Mes92], and of concurrency theory: among the others, the structured operational semantics [Plo81], the context systems [LX90] and the structured transition systems [CM92] approaches. Our model recollects many properties of these sources: first, it provides a compositional way to describe both the states and the sequences of transitions performed by a given system, stressing their distributed nature. Second, a suitable notion of typed proof allows to take into account also those formalisms relying on the notions of synchronization and sideeffects to determine the actual behaviour of a system. Finally, an equivalence relation over sequences of transitions is defined, equipping the system under analysis with a concurrent semantics, ...
An Algebraic Presentation of Term Graphs, via GSMonoidal Categories
 Applied Categorical Structures
, 1999
"... . We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the wellknown characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particula ..."
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Cited by 38 (25 self)
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. We present a categorical characterisation of term graphs (i.e., finite, directed acyclic graphs labeled over a signature) that parallels the wellknown characterisation of terms as arrows of the algebraic theory of a given signature (i.e., the free Cartesian category generated by it). In particular, we show that term graphs over a signature \Sigma are onetoone with the arrows of the free gsmonoidal category generated by \Sigma. Such a category satisfies all the axioms for Cartesian categories but for the naturality of two transformations (the discharger ! and the duplicator r), providing in this way an abstract and clear relationship between terms and term graphs. In particular, the absence of the naturality of r and ! has a precise interpretation in terms of explicit sharing and of loss of implicit garbage collection, respectively. Keywords: algebraic theories, directed acyclic graphs, gsmonoidal categories, symmetric monoidal categories, term graphs. Mathematical Subject Clas...
A 2Categorical Presentation of Term Graph Rewriting
 CATEGORY THEORY AND COMPUTER SCIENCE, VOLUME 1290 OF LNCS
, 1997
"... It is wellknown that a term rewriting system can be faithfully described by a cartesian 2category, where horizontal arrows represent terms, and cells represent rewriting sequences. In this paper we propose a similar, original 2categorical presentation for term graph rewriting. Building on a re ..."
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Cited by 34 (17 self)
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It is wellknown that a term rewriting system can be faithfully described by a cartesian 2category, where horizontal arrows represent terms, and cells represent rewriting sequences. In this paper we propose a similar, original 2categorical presentation for term graph rewriting. Building on a result presented in [8], which shows that term graphs over a given signature are in onetoone correspondence with arrows of a gsmonoidal category freely generated from the signature, we associate with a term graph rewriting system a gsmonoidal 2category, and show that cells faithfully represent its rewriting sequences. We exploit the categorical framework to relate term graph rewriting and term rewriting, since gsmonoidal (2)categories can be regarded as "weak" cartesian (2)categories, where certain (2)naturality axioms have been dropped.
Research Directions in Rewriting Logic
, 1998
"... Rewriting logic expresses an essential equivalence between logic and computation. System states are in bijective correspondence with formulas, and concurrent computations are in bijective correspondence with proofs. Given this equivalence between computation and logic, a rewriting logic axiom of the ..."
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Cited by 31 (12 self)
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Rewriting logic expresses an essential equivalence between logic and computation. System states are in bijective correspondence with formulas, and concurrent computations are in bijective correspondence with proofs. Given this equivalence between computation and logic, a rewriting logic axiom of the form t \Gamma! t 0 has two readings. Computationally, it means that a fragment of a system 's state that is an instance of the pattern t can change to the corresponding instance of t 0 concurrently with any other state changes; logically, it just means that we can derive the formula t 0 from the formula t. Rewriting logic is entirely neutral about the structure and properties of the formulas/states t. They are entirely userdefinable as an algebraic data type satisfying certain equational axioms. Because of this ecumenical neutrality, rewriting logic has, from a logical viewpoint, good properties as a logical framework, in which many other logics can be naturally represented. And, computationally, it has also good properties as a semantic framework, in which many different system styles and models of concurrent computation and many different languages can be naturally expressed without any distorting encodings. The goal of this paper is to provide a relatively gentle introduction to rewriting logic, and to paint in broad strokes the main research directions that, since its introduction in 1990, have been pursued by a growing number of researchers in Europe, the US, and Japan. Key theoretical developments, as well as the main current applications of rewriting logic as a logical and semantic framework, and the work on formal reasoning to prove properties of specifications are surveyed.
An Inductive View of Graph Transformation
 In Workshop on Algebraic Development Techniques
, 1998
"... . The dynamic behavior of rulebased systems (like term rewriting systems [24], process algebras [27], and so on) can be traditionally determined in two orthogonal ways. Either operationally, in the sense that a way of embedding a rule into a state is devised, stating explicitly how the result i ..."
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Cited by 30 (12 self)
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. The dynamic behavior of rulebased systems (like term rewriting systems [24], process algebras [27], and so on) can be traditionally determined in two orthogonal ways. Either operationally, in the sense that a way of embedding a rule into a state is devised, stating explicitly how the result is built: This is the role played by (the application of) a substitution in term rewriting. Or inductively, showing how to build the class of all possible reductions from a set of basic ones: For term rewriting, this is the usual definition of the rewrite relation as the minimal closure of the rewrite rules. As far as graph transformation is concerned, the operational view is by far more popular: In this paper we lay the basis for the orthogonal view. We first provide an inductive description for graphs as arrows of a freely generated dgsmonoidal category. We then apply 2categorical techniques, already known for term and term graph rewriting [29, 7], recasting in this framework the...
A BiCategorical Axiomatisation of Concurrent Graph Rewriting
, 1999
"... In this paper the concurrent semantics of doublepushout (DPO) graph rewriting, which is classically defined in terms of shiftequivalence classes of graph derivations, is axiomatised via the construction of a free monoidal bicategory. In contrast to a previous attempt based on 2categories, the us ..."
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Cited by 18 (10 self)
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In this paper the concurrent semantics of doublepushout (DPO) graph rewriting, which is classically defined in terms of shiftequivalence classes of graph derivations, is axiomatised via the construction of a free monoidal bicategory. In contrast to a previous attempt based on 2categories, the use of bicategories allows to define rewriting on concrete graphs. Thus, the problem of composition of isomorphism classes of rewriting sequences is avoided. Moreover, as a first step towards the recovery of the full expressive power of the formalism via a purely algebraic description, the concept of disconnected rules is introduced, i.e., rules whose interface graphs are made of disconnected nodes and edges only. It is proved that, under reasonable assumptions, rewriting via disconnected rules enjoys similar concurrency properties like in the classical approach.
A Functorial Semantics for MultiAlgebras and Partial Algebras, With Applications to Syntax
, 2000
"... Multialgebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classica ..."
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Cited by 14 (7 self)
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Multialgebras allow for the modeling of nondeterminism in an algebraic framework by interpreting operators as functions from individual arguments to sets of possible results. We propose a functorial presentation of various categories of multialgebras and partial algebras, analogous to the classical presentation of algebras over a signature as cartesian functors from the algebraic theory over to Set. We introduce two dierent notions of theory over a signature, both having a structure weaker than cartesian, and we consider functors from them to Rel or Pfn, the categories of sets and relations or partial functions, respectively. Next we discuss how the functorial presentation provides guidelines for the choice of syntactical notions for a class of algebras, and as an application we argue that the natural generalization of usual terms are \conditioned terms" for partial algebras, and \term graphs" for multialgebras. Contents 1 Introduction 2 2 A short recap on multialgebras 4 3...
Categorical Term Rewriting: Monads and Modularity
 University of Edinburgh
, 1998
"... Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting syste ..."
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Cited by 11 (6 self)
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Term rewriting systems are widely used throughout computer science as they provide an abstract model of computation while retaining a comparatively simple syntax and semantics. In order to reason within large term rewriting systems, structuring operations are used to build large term rewriting systems from smaller ones. Of particular interest is whether key properties are modular, thatis,ifthe components of a structured term rewriting system satisfy a property, then does the term rewriting system as a whole? A body of literature addresses this problem, but most of the results and proofs depend on strong syntactic conditions and do not easily generalize. Although many specific modularity results are known, a coherent framework which explains the underlying principles behind these results is lacking. This thesis posits that part of the problem is the usual, concrete and syntaxoriented semantics of term rewriting systems, and that a semantics is needed which on the one hand elides unnecessary syntactic details but on the other hand still possesses enough expressive power to model the key concepts arising from