Results 1 
8 of
8
Adjoint Rewriting
, 1995
"... This thesis concerns rewriting in the typed calculus. Traditional categorical models of typed calculus use concepts such as functor, adjunction and algebra to model type constructors and their associated introduction and elimination rules, with the natural categorical equations inherent in these s ..."
Abstract

Cited by 25 (11 self)
 Add to MetaCart
This thesis concerns rewriting in the typed calculus. Traditional categorical models of typed calculus use concepts such as functor, adjunction and algebra to model type constructors and their associated introduction and elimination rules, with the natural categorical equations inherent in these structures providing an equational theory for terms. One then seeks a rewrite relation which, by transforming terms into canonical forms, provides a decision procedure for this equational theory. Unfortunately the rewrite relations which have been proposed, apart from for the most simple of calculi, either generate the full equational theory but contain no decision procedure, or contain a decision procedure but only for a subtheory of that required. Our proposal is to unify the semantics and reduction theory of the typed calculus by generalising the notion of model from categorical structures based on term equality to categorical structures based on term reduction. This is accomplished via...
Confluence Properties of Extensional and NonExtensional lambdaCalculi with Explicit Substitutions (Extended Abstract)
 in Proceedings of the Seventh International Conference on Rewriting Techniques and Applications
, 1996
"... ) Delia Kesner CNRS and LRI, B at 490, Universit e ParisSud  91405 Orsay Cedex, France. email:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and nonextensional #calculi with explicit substitutions, where extensionality is interpreted by #expansion. For ..."
Abstract

Cited by 22 (5 self)
 Add to MetaCart
) Delia Kesner CNRS and LRI, B at 490, Universit e ParisSud  91405 Orsay Cedex, France. email:Delia.Kesner@lri.fr Abstract. This paper studies confluence properties of extensional and nonextensional #calculi with explicit substitutions, where extensionality is interpreted by #expansion. For that, we propose a general scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our general scheme makes it possible to treat at the same time many wellknown calculi such as ## , ## # and ## , or some other new calculi that we propose in this paper. We also show for those calculi not fitting in the general scheme that can be translated to another one fitting the scheme, such as #s , how to reason about confluence properties of their extensional and nonextensional versions. 1 Introduction The #calculus is a convenient framework to study functional programming, where the evaluation process is modeled by #reduction. The...
On the Power of Simple Diagrams
, 1996
"... . In this paper we focus on a set of abstract lemmas that are easy to apply and turn out to be quite valuable in order to establish confluence and/or normalization modularly, especially when adding rewriting rules for extensional equalities to various calculi. We show the usefulness of the lemmas by ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
. In this paper we focus on a set of abstract lemmas that are easy to apply and turn out to be quite valuable in order to establish confluence and/or normalization modularly, especially when adding rewriting rules for extensional equalities to various calculi. We show the usefulness of the lemmas by applying them to various systems, ranging from simply typed lambda calculus to higher order lambda calculi, for which we can establish systematically confluence and/or normalization (or decidability of equality) in a simple way. Many result are new, but we also discuss systems for which our technique allows to provide a much simpler proof than what can be found in the literature. 1 Introduction During a recent investigation of confluence and normalization properties of polymorphic lambda calculus with an expansive version of the # rule, we came across a nice lemma that gives a simple but quite powerful sufficient condition to check the Church Rosser property for a compound rewriting system...
Eta Expansions in System F
 LIENSDMI, Ecole Normale Superieure
, 1996
"... The use of expansionary jrewrite rules in various typed calculi has become increasingly common in recent years as their advantages over contractive jrewrite rules have become apparent. Not only does one obtain the decidability of fijequality, but rewrite relations based on expansions give a natu ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
The use of expansionary jrewrite rules in various typed calculi has become increasingly common in recent years as their advantages over contractive jrewrite rules have become apparent. Not only does one obtain the decidability of fijequality, but rewrite relations based on expansions give a natural interpretation of long fijnormal forms, generalise more easily to other type constructors, retain key properties when combined with other rewrite relations, and are supported by a categorical theory of reduction. This paper extends the initial results concerning the simply typed calculus to System F, that is, we prove strong normalisation and confluence for a rewrite relation consisting of traditional fireductions and jexpansions satisfying certain restrictions. Further, we characterise the second order long fijnormal forms as precisely the normal forms of the restricted rewrite relation. These results are an important step towards showing that jexpansions are compatible with the m...
Reasoning about Redundant Patterns
, 1997
"... The extensional version of the simply typed calculus with product types enriched with layered, wildcard, and product patterns is studied. Extensionality is expressed by the surjective pairing axiom and a generalization of the jconversion to patterns. Two different confluent reduction systems, ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
The extensional version of the simply typed calculus with product types enriched with layered, wildcard, and product patterns is studied. Extensionality is expressed by the surjective pairing axiom and a generalization of the jconversion to patterns. Two different confluent reduction systems, called lwp \Gamma! and lw \Gamma! respectively, are obtained by turning the extensional axioms as expansion rules, and then adding some restrictions to these expansions to avoid reduction loops. It is shown that only layered and wildcard patterns are redundant in lw \Gamma!, while product patterns are unnecessary in lwp \Gamma!. Confluence of both reduction systems is proven by the composition of modular properties of the systems' extensional and nonextensional parts. Recursion is also added to both systems by keeping the modularity of the confluence property. 1 Introduction Patternmatching in function definitions is one of the most popular features of functional languages...
On Lazy Commutation
, 2009
"... We investigate combinatorial commutation properties for reordering a sequence of two kinds of steps, and for separating wellfoundedness of unions of relations. To that end, we develop the notion of a constricting sequence. These results can be applied, for example, to generic path orderings used i ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
We investigate combinatorial commutation properties for reordering a sequence of two kinds of steps, and for separating wellfoundedness of unions of relations. To that end, we develop the notion of a constricting sequence. These results can be applied, for example, to generic path orderings used in termination proofs.
Rewriting with Extensional Polymorphic λcalculus
, 1996
"... We provide a confluent and strongly normalizing rewriting system, based on expansion rules, for the extensional second order typed lambda calculus with product and unit types: this system corresponds to the Intuitionistic Positive Calculus with implication, conjunction, quantification over propositi ..."
Abstract
 Add to MetaCart
We provide a confluent and strongly normalizing rewriting system, based on expansion rules, for the extensional second order typed lambda calculus with product and unit types: this system corresponds to the Intuitionistic Positive Calculus with implication, conjunction, quantification over proposition and the constant True. This result is an important step towards a new theory of reduction based on expansion rules, and gives a natural interpretation to the notion of second order jlong normal forms used in higher order resolution and unification, that are here just the normal forms of our reduction system.
Author manuscript, published in "23rd International Conference on Rewriting Techniques and Applications (RTA'12) (2012)" An Abstract Factorization Theorem for Explicit Substitutions
, 2013
"... We study a simple form of standardization, here called factorization, for explicit substitutions calculi, i.e. lambdacalculi where betareduction is decomposed in various rules. These calculi, despite being nonterminating and nonorthogonal, have a key feature: each rule terminates when considered ..."
Abstract
 Add to MetaCart
We study a simple form of standardization, here called factorization, for explicit substitutions calculi, i.e. lambdacalculi where betareduction is decomposed in various rules. These calculi, despite being nonterminating and nonorthogonal, have a key feature: each rule terminates when considered separately. It is wellknown that the study of rewriting properties simplifies in presence of termination (e.g. confluence reduces to local confluence). This remark is exploited to develop an abstract theorem deducing factorization from some axioms on local diagrams. The theorem is then applied to some explicit substitution calculi related to ProofNets. We show how to recover standardization by levels, we model both callbyname and callbyvalue calculi and we characterize linear head reduction via a factorization theorem for a linear calculus of substitutions.