Results 1  10
of
10
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 ..."
Abstract

Cited by 38 (25 self)
 Add to MetaCart
. 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...
Axiomatic Rewriting Theory I  A Diagrammatic Standardization Theorem
, 2001
"... Machine translation ## calculus interpretation ## calculus Formally, the calculus contains two classes of objects: terms and substitutions. Terms are written in the de Bruijn notation. ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Machine translation ## calculus interpretation ## calculus Formally, the calculus contains two classes of objects: terms and substitutions. Terms are written in the de Bruijn notation.
Aspect Oriented Programming: a language for 2categories
"... AspectOriented Programming (AOP) started ten years ago with the remark that modularization of socalled crosscutting functionalities is a fundamental problem for the engineering of largescale applications. Originating at Xerox PARC, this observation has sparked the development of a new style of pr ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
AspectOriented Programming (AOP) started ten years ago with the remark that modularization of socalled crosscutting functionalities is a fundamental problem for the engineering of largescale applications. Originating at Xerox PARC, this observation has sparked the development of a new style of programming featured that is gradually gaining traction, as it is the case for the related concept of code injection, in the guise of frameworks such as Swing and Google Guice. However, AOP lacks theoretical foundations to clarify this new idea. This paper proposes to put a bridge between AOP and the notion of 2category to enhance the conceptual understanding of AOP. Starting from the connection between the λcalculus and the theory of categories, we propose to see an aspect as a morphism between morphisms—that is as a program that transforms the execution of a program. To make this connection precise, we develop an advised λcalculus that provides an internal language for 2categories and show how it can be used as a base for the definition of the weaving mechanism of a realistic functional AOP language, called MinAML.
Residuals in higherorder rewriting
 Proceedings of Rewriting Techniques and Applications (RTA’03
, 2003
"... Abstract. Residuals have been studied for various forms of rewriting setting. In this article we study residuals in orthogonal Pattern Rewriting Systems (PRSs). First, the rewrite relation is defined by means of a higherorder rewriting logic, and proof terms are defined that witness reductions. The ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
Abstract. Residuals have been studied for various forms of rewriting setting. In this article we study residuals in orthogonal Pattern Rewriting Systems (PRSs). First, the rewrite relation is defined by means of a higherorder rewriting logic, and proof terms are defined that witness reductions. Then, we have the formal machinery to define a residual operator for PRSs, and we will prove that an orthogonal PRS together with the residual operator mentioned above, is a residual system. As a sideeffect, all results of (abstract) residual theory are inherited by orthogonal PRSs, such as confluence, and the notion of permutation equivalence of reductions. 1
Author manuscript, published in "Modularity: AOSD'12 (2012)" A Monadic Interpretation of Execution Levels and Exceptions for AOP
"... has focused on applying aspects in various problem domains and on integration of aspects into fullscale programming languages such as Java. However, aspects are very powerful and the development of a weaving mechanism becomes rapidly a very complex task. While some research efforts [5, 10, 25, 26] ..."
Abstract
 Add to MetaCart
has focused on applying aspects in various problem domains and on integration of aspects into fullscale programming languages such as Java. However, aspects are very powerful and the development of a weaving mechanism becomes rapidly a very complex task. While some research efforts [5, 10, 25, 26] have made significant progress on understanding some of the semantic issues involved, the algebraic explanation of aspect features has never reached the beauty and simplicity of the connection between the λcalculus and Cartesian closed categories that provided a categorical foundation for functional programming. Giving a precise meaning to aspects in AOP is a fairly tangled task because the definition of a single piece of code can have a very rich interaction with the rest of the program, whose effect can come up at anytime during the execution. The main purpose of this paper is to formalize this interaction—in particular for two recent sophisticated weaving definitions of aspects and base computation: (1) execution levels [24] that enable to stratify the computation space in order to prevent from basic infinite recursion; (2) execution levels with exceptions [6] that enable to avoid unexpected catching of exceptions in this stratified space. More precisely, we propose to put a bridge between (weavingbased approach to) AOP and the notion of 2
Categorical Term Rewriting:
, 1997
"... Abstract 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 rewrit ..."
Abstract
 Add to MetaCart
Abstract 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, that is, if the 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 the term structure, such as substitutions, layers, redexes etc. Drawing on the concepts of category theory, such a semantics is proposed, based on the concept of a monad, generalising the very elegant treatment of equational presentations in category theory. The theoretical basis of this work is the theory of enriched monads. It is shown how structuring operations are modelled on the level of monads, and that the semantics is compositional (it preserves the structuring operations). Modularity results can now be obtained directly at the level of combining monads without recourse to the syntax at all. As an application and demonstration of the usefulness of this approach, two modularity results for the disjoint union of two term rewriting systems are proven, the modularity of confluence (Toyama's theorem) and the modularity of strong normalization for a particular class of term rewriting systems (noncollapsing term rewriting systems). The proofs in the categorical setting provide a mild generalisation of these results.
Categorical Term Rewriting:
, 1997
"... Abstract 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 rewrit ..."
Abstract
 Add to MetaCart
Abstract 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, that is, if the 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 the term structure, such as substitutions, layers, redexes etc. Drawing on the concepts of category theory, such a semantics is proposed, based on the concept of a monad, generalising the very elegant treatment of equational presentations in category theory. The theoretical basis of this work is the theory of enriched monads. It is shown how structuring operations are modelled on the level of monads, and that the semantics is compositional (it preserves the structuring operations). Modularity results can now be obtained directly at the level of combining monads without recourse to the syntax at all. As an application and demonstration of the usefulness of this approach, two modularity results for the disjoint union of two term rewriting systems are proven, the modularity of confluence (Toyama's theorem) and the modularity of strong normalization for a particular class of term rewriting systems (noncollapsing term rewriting systems). The proofs in the categorical setting provide a mild generalisation of these results.
Four Equivalent Equivalences of Reductions
, 2002
"... Two coinitial reductions in a term rewriting system are said to be equivalent if they perform the same steps, albeit maybe in a di#erent order. We present four characterisations of such a notion of equivalence, based on permutation, standardisation, labelling and projection, respectively. We prove ..."
Abstract
 Add to MetaCart
Two coinitial reductions in a term rewriting system are said to be equivalent if they perform the same steps, albeit maybe in a di#erent order. We present four characterisations of such a notion of equivalence, based on permutation, standardisation, labelling and projection, respectively. We prove that the characterisations all yield the same notion of equivalence, for the class of firstorder leftlinear term rewriting systems. A crucial role in our development is played by the notion of a proof term. 1
2.2 Residual Theory......................... 4
, 2003
"... Abstract. Residuals have been studied for various forms of rewriting and residual systems have been defined to capture residuals in an abstract setting. In this article we study residuals in orthogonal Pattern Rewriting Systems (PRSs). First, the rewrite relation is defined by means of a higherorde ..."
Abstract
 Add to MetaCart
Abstract. Residuals have been studied for various forms of rewriting and residual systems have been defined to capture residuals in an abstract setting. In this article we study residuals in orthogonal Pattern Rewriting Systems (PRSs). First, the rewrite relation is defined by means of a higherorder rewriting logic, and proof terms are defined that witness reductions. Then, we have the formal machinery to define a residual operator for PRSs, and we will prove that an orthogonal PRS together with the residual operator mentioned above, is a residual system. As a sideeffect, all results of (abstract) residual theory are inherited by orthogonal PRSs, such as confluence, and the notion of