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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 ..."
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Cited by 25 (11 self)
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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 ..."
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Cited by 22 (5 self)
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) 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...
Confluence of Extensional and NonExtensional λcalculi with Explicit Substitutions
 Theoretical Computer Science
"... This paper studies confluence of extensional and nonextensional calculi with explicit substitutions, where extensionality is interpreted by jexpansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. O ..."
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Cited by 12 (2 self)
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This paper studies confluence of extensional and nonextensional calculi with explicit substitutions, where extensionality is interpreted by jexpansion. For that, we propose a scheme for explicit substitutions which describes those abstract properties that are sufficient to guarantee confluence. Our method makes it possible to treat at the same time many wellknown calculi such as oe , oe * , OE , s , AE , f , d and dn . Keywords: functional programming, calculi, explicit substitutions, confluence, extensionality. 1 Introduction The calculus is a convenient framework to study functional programming, where the evaluation process is modeled by fireduction. The main mechanism used to perform fireduction is substitution, which consists of the replacement of formal parameters by actual arguments. The correctness of substitution is guaranteed by a systematic renaming of bound variables, inconvenient which can be simply avoided in the calculus `a la de Bruijn by using natur...
Combining Algebraic Rewriting, Extensional Lambda Calculi, and Fixpoints
"... It is well known that confluence and strong normalization are preserved when combining algebraic rewriting systems with the simply typed lambda calculus. It is equally well known that confluence fails when adding either the usual contraction rule for #, or recursion together with the usual contract ..."
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Cited by 8 (3 self)
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It is well known that confluence and strong normalization are preserved when combining algebraic rewriting systems with the simply typed lambda calculus. It is equally well known that confluence fails when adding either the usual contraction rule for #, or recursion together with the usual contraction rule for surjective pairing. We show that confluence and strong normalization are modular properties for the combination of algebraic rewriting systems with typed lambda calculi enriched with expansive extensional rules for # and surjective pairing. We also show how to preserve confluence in a modular way when adding fixpoints to di#erent rewriting systems. This result is also obtained by a simple translation technique allowing to simulate bounded recursion. 1 Introduction Confluence and strong normalization for the combination of lambda calculus and algebraic rewriting systems have been the object of many studies [BT88, JO91, BTG94, HM90], where the modularity of these properties is s...
A Brief History of Rewriting With Extensionality
, 1996
"... A A ×B oo # 1 // # 2 B That is: h = #f, g# = ## 1 # h, # 2 # h# Case C A // in 1 << f y y y y y y y y y y y y y y y y y y y A +B OO [f,g] h OO B o ..."
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Cited by 8 (0 self)
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<F NaN><F NaN> A A ×B<F NaN><F NaN> oo # 1<F NaN><F NaN> // # 2 B That is: h = #f, g# = ## 1 # h, # 2 # h# Case C A<F NaN><F NaN> // in 1<F NaN><F NaN> << f y y y y y y y y y y y y y y y y y y y A +B<F NaN><F NaN> OO [f,g] h<F NaN><F NaN> OO B<F NaN><F NaN> oo in 2<F NaN><F NaN> bb g E E E E E E E E E E E E E E E E E E E That is: h = [f, g] = [h # in 1 , h # in 2 ] Roberto Di Cosmo Glasgow, September 96 4 From equations to rewriting Two choices to orient
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 ..."
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Cited by 6 (0 self)
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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 Layered, Wildcard and Product Patterns
 ALP '94, volume 850 of LNCS
, 1994
"... We study the extensional version of the simply typed calculus with product types and fixpoints enriched with layered, wildcard and product patterns. Extensionality is expressed by the surjective pairing axiom and a generalization of the jconversion to patterns. We obtain a confluent reduction syst ..."
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Cited by 3 (2 self)
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We study the extensional version of the simply typed calculus with product types and fixpoints enriched with layered, wildcard and product patterns. Extensionality is expressed by the surjective pairing axiom and a generalization of the jconversion to patterns. We obtain a confluent reduction system by turning the extensional axioms as expansion rules, and then adding some restrictions to these expansions in order to avoid reduction loops. Confluence is proved by composition of modular properties of the extensional and nonextensional subsystems of the reduction calculus. 1 Introduction Patternmatching function definitions is one of the most popular features of functional languages, allowing to specify the behavior of functions by cases, according to the form of their arguments. Lefthand sides of function definitions are usually expressed using Layered, Wildcard and Product Patterns (LWPP), as for example the following Caml Light [ea93] program where the function cons new pair ta...
Simulating etaExpansions with betaReductions in the SecondOrder Polymorphic Calculus
, 1996
"... . We introduce an approach to simulating jexpansions with fireductions in the secondorder polymorphic calculus. This generalizes the work of Di Cosmo and Delia Kesner which simulates jexpansions with fireductions in simply typed settings, positively solving the conjecture on whether the simu ..."
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Cited by 1 (1 self)
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. We introduce an approach to simulating jexpansions with fireductions in the secondorder polymorphic calculus. This generalizes the work of Di Cosmo and Delia Kesner which simulates jexpansions with fireductions in simply typed settings, positively solving the conjecture on whether the simulation technique can be extended to polymorphic settings. We then present a modular proof that the secondorder polymorphic calculus with an expansive version of jreduction is strong normalizing and confluent. The simulation is also promising to provide modular proofs showing that other rewriting systems are also strongly normalizing after expanded with certain versions of jexpansion. 1 Introduction and Related Work jconversion presents an approach to studying extensional equalities for terms. Given an jequality x:M (x) = j M , where x has no free occurrences in M ; one can either say x:M (x) jcontracts (! j ) to M , or M jexpands (! j ) to x:M (x); the former is usually adopted ...
EtaExpansions III  F omega
, 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 simultaneously a decision procedure for fij equality and a procedure for the calculat ..."
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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 simultaneously a decision procedure for fij equality and a procedure for the calculation of the long fijnormal form of a term, but rewrite relations using expansions retain key properties when combined with first order rewrite systems, generalise more easily to other type constructors and are supported by a categorical theory of reduction. However, until now jcontractions have remained the only possibility in the more powerful type systems of the cube. In this paper we begin to rectify this situation by extending the techniques previously developed to a higher order polymorphic calculus called F ! , where reduction no longer occurs only at the level of terms but also at the level of types. 1 Introduction Extensional equality for terms of the simply typed calculus req...
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 ..."
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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.