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Rewriting Logic as a Logical and Semantic Framework
, 1993
"... Rewriting logic [72] is proposed as a logical framework in which other logics can be represented, and as a semantic framework for the specification of languages and systems. Using concepts from the theory of general logics [70], representations of an object logic L in a framework logic F are und ..."
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Cited by 147 (52 self)
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Rewriting logic [72] is proposed as a logical framework in which other logics can be represented, and as a semantic framework for the specification of languages and systems. Using concepts from the theory of general logics [70], representations of an object logic L in a framework logic F are understood as mappings L ! F that translate one logic into the other in a conservative way. The ease with which such maps can be defined for a number of quite different logics of interest, including equational logic, Horn logic with equality, linear logic, logics with quantifiers, and any sequent calculus presentation of a logic for a very general notion of "sequent," is discussed in detail. Using the fact that rewriting logic is reflective, it is often possible to reify inside rewriting logic itself a representation map L ! RWLogic for the finitely presentable theories of L. Such a reification takes the form of a map between the abstract data types representing the finitary theories of...
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 82 (22 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 37 (24 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.
Rational Term Rewriting
, 1998
"... . Rational terms (possibly infinite terms with finitely many subterms) can be represented in a finite way via terms, that is, terms over a signature extended with selfinstantiation operators. For example, f ! = f(f(f(: : :))) can be represented as x :f(x) (or also as x :f(f(x)), f(x :f(x)), ..."
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Cited by 21 (12 self)
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. Rational terms (possibly infinite terms with finitely many subterms) can be represented in a finite way via terms, that is, terms over a signature extended with selfinstantiation operators. For example, f ! = f(f(f(: : :))) can be represented as x :f(x) (or also as x :f(f(x)), f(x :f(x)), . . . ). Now, if we reduce a term t to s via a rewriting rule using standard notions of the theory of Term Rewriting Systems, how are the rational terms corresponding to t and to s related? We answer to this question in a satisfactory way, resorting to the definition of infinite parallel rewriting proposed in [7]. We also provide a simple, algebraic description of term rewriting through a variation of Meseguer's Rewriting Logic formalism. 1 Introduction Rational terms are possibly infinite terms with a finite set of subterms. They show up in a natural way in Theoretical Computer Science whenever some finite cyclic structures are of concern (for example data flow diagrams, cyclic te...
Enriched Categories as Models of Computation
 in Proc. Fifth Italian Conference on Theoretical Computer Science, ICTCS'95 , World Scientific
, 1996
"... . In this paper we discuss a general methodology to provide a categorical semantics for a wide class of computational systems, whose behaviour can be described by a suitable set of transition steps. We open our survey presenting some results on the semantics of Petri Nets. Starting from this, we ela ..."
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Cited by 11 (4 self)
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. In this paper we discuss a general methodology to provide a categorical semantics for a wide class of computational systems, whose behaviour can be described by a suitable set of transition steps. We open our survey presenting some results on the semantics of Petri Nets. Starting from this, we elaborate a twosteps procedure allowing for the description of all the sequences of transitions performed by a given system, and equipping them with a suitable equivalence relation. This relation provides the sistem under analisys with a concurrent semantics: equivalence classes denote families of "computationally equivalent" behaviours, corresponding to the execution of the same set of (causally) independent transition steps. 1 Introduction The latest years have seen a wide amount of different approaches to the semantics of computional sistems: a variety that, if only for the comparison between the various formalisms, calls for a unified framework. In this paper we aim to show that enriched ...
Axiomatizing Permutation Equivalence
 Mathematical Structures in Computer Science
, 1994
"... We axiomatize permutation equivalence in term rewriting systems and Klop's orthogonal Combinatory Reduction Systems [Klop 1980]. The axioms for the former ones are provided by the general approach proposed by Meseguer [Meseguer 1992]. The latter ones need extra axioms modeling the interplay between ..."
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Cited by 8 (0 self)
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We axiomatize permutation equivalence in term rewriting systems and Klop's orthogonal Combinatory Reduction Systems [Klop 1980]. The axioms for the former ones are provided by the general approach proposed by Meseguer [Meseguer 1992]. The latter ones need extra axioms modeling the interplay between reductions and the operation of substitution. As a consequence of this work, the definition of permutation equivalence is rid of residual calculi, which are heavy in general. 1 Introduction 1. What does permutation equivalence mean? A wellknown syntactical property of the  calculus is the ChurchRosser theorem. It states that, if a term M reduces into N 1 and N 2 by firing two different redexes, then there exists a term P which is a reduct both of N 1 and N 2 . Graphically: P @ @ @R @ @ @R \Gamma \Gamma \Gamma\Psi \Gamma \Gamma \Gamma\Psi N 1 N 2 \Gamma \Gamma \Gamma\Psi @ @ @R M v oe u ae Actually the ChurchRosser property can be asserted in a stronger way. For this purpose, re...
CCS Semantics via Proved Transition Systems and Rewriting Logic
 In Kirchner and Kirchner [47
, 1998
"... We consider (a slight variant of) the ccs calculus, and we analyze two operational semantics defined in the literature: the first exploits Proved Transition Systems (pts) and the second Rewriting Logic (rl). We show that the interleaving interpretation of both semantics agree, in that they define th ..."
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Cited by 7 (1 self)
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We consider (a slight variant of) the ccs calculus, and we analyze two operational semantics defined in the literature: the first exploits Proved Transition Systems (pts) and the second Rewriting Logic (rl). We show that the interleaving interpretation of both semantics agree, in that they define the same transitions and exhibit the same nondeterministic structure. In addition, we study causality in ccs computations. We recall the treatment via pts, that exhibits the notion of causality presented in the literature, and we show how to recast it in the rl semantics via suitable axioms. 1 Introduction Concurrency is maybe the basic aspect of the operational interpretation of rewriting logic. And as Jos'e Meseguer says in his lecture at concur'96 [20], . . . my main emphasis in this talk will be on rewriting logic as a semantic framework for concurrency. . . . The goal is . . . to express as faithfully as possible each model [of concurrency] on its own terms, avoiding any encodings or tr...
A Causal Semantics for CCS via Rewriting Logic
 Theoretical Computer Science
, 2000
"... We consider two operational semantics for ccs dened in the literature: the rst exploits Proved Transition Systems (pts) and the second Rewriting Logic (rl). We show that the interleaving interpretation of both semantics agree, in that they dene the same transitions and exhibit the same nondeterminis ..."
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Cited by 6 (0 self)
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We consider two operational semantics for ccs dened in the literature: the rst exploits Proved Transition Systems (pts) and the second Rewriting Logic (rl). We show that the interleaving interpretation of both semantics agree, in that they dene the same transitions and exhibit the same nondeterministic structure. In addition, we study causality in ccs computations. We recall its treatment via pts, exhibiting the notion of causality presented in the literature, and we show how to recast it in the rl semantics via suitable axioms. Also in this case, the two semantics agree. Contents 1 Introduction 2 2 Some notions on Process Algebras 3 2.1 The Calculus of Communicating Systems 4 2.2 Proved Transition System 6 2.3 Causality and Concurrency 7 ? Research partly supported by the Italian CNR Progetto Strategico Modelli e Metodi per la Matematica e l'Ingegneria and MURST Progetto Tecniche Formali per la Specica, l'Analisi, la Verica, la Sintesi e la Trasformazione di Sistemi Software. ...