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43
Confluence properties of Weak and Strong Calculi of Explicit Substitutions
 JOURNAL OF THE ACM
, 1996
"... Categorical combinators [12, 21, 43] and more recently oecalculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the calculus. We reintroduce here the ingredients of these calculi in a selfcontained and stepwise way, with a special emphasis on confluence prope ..."
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Cited by 120 (7 self)
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Categorical combinators [12, 21, 43] and more recently oecalculus [1, 23], have been introduced to provide an explicit treatment of substitutions in the calculus. We reintroduce here the ingredients of these calculi in a selfcontained and stepwise way, with a special emphasis on confluence properties. The main new results of the paper w.r.t. [12, 21, 1, 23] are the following: 1. We present a confluent weak calculus of substitutions, where no variable clashes can be feared. 2. We solve a conjecture raised in [1]: oecalculus is not confluent (it is confluent on ground terms only). This unfortunate result is "repaired" by presenting a confluent version of oecalculus, named the Envcalculus in [23], called here the confluent oecalculus.
Combinatory Reduction Systems: introduction and survey
 THEORETICAL COMPUTER SCIENCE
, 1993
"... Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual firstorder format of term rewriting with the presence of bound variables as in pure λcalculus and various typed calculi. Bound variables are also present in many other rewrite systems, such as systems with simpl ..."
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Cited by 84 (9 self)
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Combinatory Reduction Systems, or CRSs for short, were designed to combine the usual firstorder format of term rewriting with the presence of bound variables as in pure λcalculus and various typed calculi. Bound variables are also present in many other rewrite systems, such as systems with simplification rules for proof normalization. The original idea of CRSs is due to Aczel, who introduced a restricted class of CRSs and, under the assumption of orthogonality, proved confluence. Orthogonality means that the rules are nonambiguous (no overlap leading to a critical pair) and leftlinear (no global comparison of terms necessary). We introduce the class of orthogonal CRSs, illustrated with many examples, discuss its expressive power, and give an outline of a short proof of confluence. This proof is a direct generalization of Aczel's original proof, which is close to the wellknown confluence proof for λcalculus by Tait and MartinLof. There is a wellknown connection between the para...
Inductive Data Type Systems
 THEORETICAL COMPUTER SCIENCE
, 1997
"... In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λcalculus enriched by patternmatching definitions following a certain format, called the “General Schema”, w ..."
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Cited by 44 (10 self)
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In a previous work (“Abstract Data Type Systems”, TCS 173(2), 1997), the last two authors presented a combined language made of a (strongly normalizing) algebraic rewrite system and a typed λcalculus enriched by patternmatching definitions following a certain format, called the “General Schema”, which generalizes the usual recursor definitions for natural numbers and similar “basic inductive types”. This combined language was shown to be strongly normalizing. The purpose of this paper is to reformulate and extend the General Schema in order to make it easily extensible, to capture a more general class of inductive types, called “strictly positive”, and to ease the strong normalization proof of the resulting system. This result provides a computation model for the combination of an algebraic specification language based on abstract data types and of a strongly typed functional language with strictly positive inductive types.
On Girard’s “Candidats de Réductibilité
 Logic and Computer Science
, 1990
"... Abstract: We attempt to elucidate the conditions required on Girard’s candidates of reducibility (in French, “candidats de reductibilité”) in order to establish certain properties of various typed lambda calculi, such as strong normalization and ChurchRosser property. We present two generalizations ..."
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Cited by 33 (5 self)
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Abstract: We attempt to elucidate the conditions required on Girard’s candidates of reducibility (in French, “candidats de reductibilité”) in order to establish certain properties of various typed lambda calculi, such as strong normalization and ChurchRosser property. We present two generalizations of the candidates of reducibility, an untyped version in the line of Tait and Mitchell, and a typed version which is an adaptation of Girard’s original method. As an application of this general result, we give two proofs of strong normalization for the secondorder polymorphic lambda calculus under ⌘reduction (and thus underreduction). We present two sets of conditions for the typed version of the candidates. The first set consists of conditions similar to those used by Stenlund (basically the typed version of Tait’s conditions), and the second set consists of Girard’s original conditions. We also compare these conditions, and prove that Girard’s conditions are stronger than Tait’s conditions. We give a new proof of the ChurchRosser theorem for bothreduction and ⌘reduction, using the modified version of Girard’s method. We also compare various proofs that have appeared in the literature (see section 11). We conclude by sketching the extension of the above results to Girard’s higherorder polymorphic calculus F!, and in appendix 1, to F! with product types. i 1
Matching Power
 Proceedings of RTA’2001, Lecture Notes in Computer Science, Utrecht (The Netherlands
, 2001
"... www.loria.fr/{~cirstea,~ckirchne,~lliquori} Abstract. In this paper we give a simple and uniform presentation of the rewriting calculus, also called Rho Calculus. In addition to its simplicity, this formulation explicitly allows us to encode complex structures such as lists, sets, and objects. We pr ..."
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Cited by 30 (20 self)
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www.loria.fr/{~cirstea,~ckirchne,~lliquori} Abstract. In this paper we give a simple and uniform presentation of the rewriting calculus, also called Rho Calculus. In addition to its simplicity, this formulation explicitly allows us to encode complex structures such as lists, sets, and objects. We provide extensive examples of the calculus, and we focus on its ability to represent some object oriented calculi, namely the Lambda Calculus of Objects of Fisher, Honsell, and Mitchell, and the Object Calculus of Abadi and Cardelli. Furthermore, the calculus allows us to get object oriented constructions unreachable in other calculi. In summa, we intend to show that because of its matching ability, the Rho Calculus represents a lingua franca to naturally encode many paradigms of computations. This enlightens the capabilities of the rewriting calculus based language ELAN to be used as a logical as well as powerful semantical framework. 1
The Calculus of Algebraic Constructions
 In Proc. of the 10th Int. Conf. on Rewriting Techniques and Applications, LNCS 1631
, 1999
"... Abstract. In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by hi ..."
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Cited by 27 (10 self)
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Abstract. In a previous work, we proved that an important part of the Calculus of Inductive Constructions (CIC), the basis of the Coq proof assistant, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higherorder rewrite rules. In this paper, we prove that almost all CIC can be seen as a CAC, and that it can be further extended with nonstrictly positive types and inductiverecursive types together with nonfree constructors and patternmatching on defined symbols. 1.
Modularity of Strong Normalization and Confluence in the algebraiclambdacube
, 1994
"... In this paper we present the algebraiccube, an extension of Barendregt's cube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraiccube, provided that the firstorder rewrite rules are nonduplicating and the hig ..."
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Cited by 25 (7 self)
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In this paper we present the algebraiccube, an extension of Barendregt's cube with first and higherorder algebraic rewriting. We show that strong normalization is a modular property of all systems in the algebraiccube, provided that the firstorder rewrite rules are nonduplicating and the higherorder rules satisfy the general schema of Jouannaud and Okada. This result is proven for the algebraic extension of the Calculus of Constructions, which contains all the systems of the algebraiccube. We also prove that local confluence is a modular property of all the systems in the algebraiccube, provided that the higherorder rules do not introduce critical pairs. This property and the strong normalization result imply the modularity of confluence. 1 Introduction Many different computational models have been developed and studied by theoretical computer scientists. One of the main motivations for the development This research was partially supported by ESPRIT Basic Research Act...
Rewriting calculus with(out) types
 Proceedings of the fourth workshop on rewriting logic and applications
, 2002
"... The last few years have seen the development of a new calculus which can be considered as an outcome of the last decade of various researches on (higher order) term rewriting systems, and lambda calculi. In the Rewriting Calculus (or Rho Calculus, ρCal), algebraic rules are considered as sophisticat ..."
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Cited by 22 (13 self)
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The last few years have seen the development of a new calculus which can be considered as an outcome of the last decade of various researches on (higher order) term rewriting systems, and lambda calculi. In the Rewriting Calculus (or Rho Calculus, ρCal), algebraic rules are considered as sophisticated forms of “lambda terms with patterns”, and rule applications as lambda applications with pattern matching facilities. The calculus can be customized to work modulo sophisticated theories, like commutativity, associativity, associativitycommutativity, etc. This allows us to encode complex structures such as list, sets, and more generally objects. The calculus can either be presented “à la Curry ” or “à la Church ” without sacrificing readability and without complicating too much the metatheory. Many static type systems can be easily pluggedin on top of the calculus in the spirit of the rich typeoriented literature. The Rewriting Calculus could represent a lingua franca to encode many paradigms of computations together with a formal basis used to build powerful theorem provers based on lambda calculus and efficient rewriting, and a step towards new proof engines based on the CurryHoward isomorphism. 1
Combining HigherOrder and FirstOrder Computation Using ρcalculus: Towards a Semantics of ELAN
 In Frontiers of Combining Systems 2
, 1999
"... The ρcalculus permits to express in a uniform and simple way firstorder rewriting, λcalculus and nondeterministic computations as well as their combination. In this paper, we present the main components of the ρcalculus and we give a full firstorder presentation of this rewriting calculus using ..."
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Cited by 20 (9 self)
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The ρcalculus permits to express in a uniform and simple way firstorder rewriting, λcalculus and nondeterministic computations as well as their combination. In this paper, we present the main components of the ρcalculus and we give a full firstorder presentation of this rewriting calculus using an explicit substitution setting, called ρσ, that generalizes the λσcalculus. The basic properties of the nonexplicit and explicit substitution versions are presented. We then detail how to use the ρcalculus to give an operational semantics to the rewrite rules of the ELAN language. 1
Adding algebraic rewriting to the untyped lambda calculus
 Information and Computation
, 1992
"... We investigate the system obtained by adding an algebraic rewriting system R to an untyped lambda calculus in which terms are formed using the function symbols from R as constants. On certain classes of terms, called here "stable", we prove that the resulting calculus is confluent if R is confluent, ..."
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Cited by 20 (0 self)
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We investigate the system obtained by adding an algebraic rewriting system R to an untyped lambda calculus in which terms are formed using the function symbols from R as constants. On certain classes of terms, called here "stable", we prove that the resulting calculus is confluent if R is confluent, and terminating if R is terminating. The termination result has the corresponding theorems for several typed calculi as corollaries. The proof of the confluence result suggests a general method for proving confluence of typed β reduction plus rewriting; we sketch the application to the polymorphic lambda calculus.