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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 86 (24 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...
CPO models for infinite term rewriting
 in Proc. AMAST'95, LNCS 936
, 1995
"... . Infinite terms in universal algebras are a wellknown topic since the seminal work of the ADJ group [1]. The recent interest in the field of term rewriting (tr) for infinite terms is due to the use of term graph rewriting to implement tr, where terms are represented by graphs: so, a cyclic gra ..."
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Cited by 12 (7 self)
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. Infinite terms in universal algebras are a wellknown topic since the seminal work of the ADJ group [1]. The recent interest in the field of term rewriting (tr) for infinite terms is due to the use of term graph rewriting to implement tr, where terms are represented by graphs: so, a cyclic graph is a finitary description of a possibly infinite term. In this paper we introduce infinite rewriting logic, working on the framework of rewriting logic proposed by Jos'e Meseguer [13, 14]. We provide a simple algebraic presentation of infinite computations, recovering the infinite parallel term rewriting, originally presented by one of the authors ([6]) to extend the classical, settheoretical approach to tr with infinite terms. Moreover, we put all the formalism on firm theoretical bases, providing (for the first time, to the best of our knowledge, for infinitary rewriting systems) a clean algebraic semantics by means of (internal) 2categories. 1 Introduction Term rewriting sy...
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 ...
Compositional Term Rewriting: An Algebraic Proof of Toyama's Theorem
 Rewriting Techniques and Applications, 7th International Conference, number 1103 in Lecture Notes in Computer Science
, 1996
"... This article proposes a compositional semantics for term rewriting systems, i.e. a semantics preserving structuring operations such as the disjoint union. The semantics is based on the categorical construct of a monad, adapting the treatment of universal algebra in category theory to term rewriting ..."
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Cited by 9 (3 self)
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This article proposes a compositional semantics for term rewriting systems, i.e. a semantics preserving structuring operations such as the disjoint union. The semantics is based on the categorical construct of a monad, adapting the treatment of universal algebra in category theory to term rewriting systems. As an example, the preservation of confluence under the disjoint union of two term rewriting systems is shown, obtaining an algebraic proof of Toyama's theorem, generalised slightly to term rewriting systems introducing variables on the righthand side of the rules.
WALDHAUSEN ADDITIVITY: CLASSICAL AND QUASICATEGORICAL
, 2012
"... We give a short proof of classical Waldhausen Additivity, and then prove Waldhausen Additivity for an ∞version of Waldhausen Ktheory. Namely, we prove that Waldhausen Ktheory sends a splitexact sequence of Waldausen quasicategories A → E → B to a stable equivalence of spectra K(E) → K(A) ∨ K( ..."
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Cited by 5 (1 self)
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We give a short proof of classical Waldhausen Additivity, and then prove Waldhausen Additivity for an ∞version of Waldhausen Ktheory. Namely, we prove that Waldhausen Ktheory sends a splitexact sequence of Waldausen quasicategories A → E → B to a stable equivalence of spectra K(E) → K(A) ∨ K(B) under a few mild hypotheses. For example, each cofiber sequence in A of the form A0 → A1 → ∗ is required to have the first map an equivalence. Model structures, presentability, and stability are not needed. In an effort to make the article selfcontained, we provide many details in our proofs, recall all the prerequisites from the theory of quasicategories, and prove some of those as well. For instance, we develop the expected facts about (weak) adjunctions between quasicategories and (weak) adjunctions
Nested Quantification in Graph Transformation Rules
, 2007
"... “Life is really simple, but we insist on making it complicated” –Confucius Preface The document before you contains the resulting documentation of my Master’s Thesis. In the period from September 1st, 2006 to June 28th, 2007 I worked on this project to finalize my study in Computer Science. I would ..."
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Cited by 1 (1 self)
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“Life is really simple, but we insist on making it complicated” –Confucius Preface The document before you contains the resulting documentation of my Master’s Thesis. In the period from September 1st, 2006 to June 28th, 2007 I worked on this project to finalize my study in Computer Science. I would like to thank the members of my graduation committee for their support during this project. A few words for each, in alphabetic order: • Harmen Kastenberg, for pointing me in the right direction when I got stuck during implementation • Jan Kuper, for warning me about Category Theory and helping me understand it and wield its power • Arend Rensink, for supplying the project idea and asking the questions I tried to avoid
Universal Properties of Impure Programming Languages
"... We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums ..."
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We investigate impure, callbyvalue programming languages. Our first language only has variables and letbinding. Its equational theory is a variant of Lambek’s theory of multicategories that omits the commutativity axiom. We demonstrate that type constructions for impure languages — products, sums and functions — can be characterized by universal properties in the setting of ‘premulticategories’, multicategories where the commutativity law may fail. This leads us to new, universal characterizations of two earlier equational theories of impure programming languages: the premonoidal categories of Power and Robinson, and the monadbased models of Moggi. Our analysis thus puts these earlier abstract ideas on a canonical foundation, bringing them to a new, syntactic level. F.3.2 [Semantics of Pro
Nordic Journal of Computing 10(2003), 290–312. REWRITING VIA COINSERTERS
"... Abstract. This paper introduces a semantics for rewriting that is independent of the data being rewritten and which, nevertheless, models key concepts such as substitution which are central to rewriting algorithms. We demonstrate the naturalness of this construction by showing how it mirrors the usu ..."
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Abstract. This paper introduces a semantics for rewriting that is independent of the data being rewritten and which, nevertheless, models key concepts such as substitution which are central to rewriting algorithms. We demonstrate the naturalness of this construction by showing how it mirrors the usual treatment of algebraic theories as coequalizers of monads. We also demonstrate its naturalness by showing how it captures several canonical forms of rewriting.
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"... Abstract. This article proposes a compositional semantics for term rewriting systems, i.e. a semantics preserving structuring operations such as the disjoint union. The semantics is based on the categorical construct of a monad, adapting the treatment of universal algebra in category theory to term ..."
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Abstract. This article proposes a compositional semantics for term rewriting systems, i.e. a semantics preserving structuring operations such as the disjoint union. The semantics is based on the categorical construct of a monad, adapting the treatment of universal algebra in category theory to term rewriting systems. As an example, the preservation of confluence under the disjoint union of two term rewriting systems is shown, obtaining an algebraic proof of Toyama’s theorem, generalised slightly to term rewriting systems introducing variables on the righthand side of the rules. 1
Rewriting the Conditions in Conditional Rewriting
, 2000
"... Category theory has been used to provide a semantics for term rewriting systems at an intermediate level of abstraction between the actual syntax and the relational model. Recently we have developed a semantics for TRSs using monads which generalises the equivalence between algebraic theories and f ..."
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Category theory has been used to provide a semantics for term rewriting systems at an intermediate level of abstraction between the actual syntax and the relational model. Recently we have developed a semantics for TRSs using monads which generalises the equivalence between algebraic theories and finitary monads on the category Set. This semantics underpins the recent categorical proofs of stateoftheart results in modular rewriting. We believe that our methods can be applied to modularity for conditional rewriting where several open problems exist. Any results we achieve here would be highly significant as, for the first time, substantial open problems in rewriting would have been solved using categorical techniques. This paper reports on the first step in this project, namely the construction of a semantics for CTRS using monads.