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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 83 (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...
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 25 (9 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.
Inductive types in the calculus of algebraic constructions
 FUNDAMENTA INFORMATICAE 65(12) (2005) 61–86 JOURNAL VERSION OF TLCA’03
, 2005
"... In a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), the basis of the proof assistant Coq, can be seen as a Calculus of Algebraic Constructions (CAC), an extension of the Calculus of Constructions with functions and predicates defined by higherorder rewrit ..."
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Cited by 15 (4 self)
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In a previous work, we proved that almost all of the Calculus of Inductive Constructions (CIC), the basis of the proof assistant Coq, 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 CIC as a whole can be seen as a CAC, and that it can be extended with nonstrictly positive types and inductiverecursive types together with nonfree constructors and patternmatching on defined symbols.
Termination and confluence of higherorder rewrite systems
 In Proc. RTA ’00, volume 1833 of LNCS
, 2000
"... Abstract: In the last twenty years, several approaches to higherorder rewriting have been proposed, among which Klop’s Combinatory Rewrite Systems (CRSs), Nipkow’s Higherorder Rewrite Systems (HRSs) and Jouannaud and Okada’s higherorder algebraic specification languages, of which only the last on ..."
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Cited by 13 (7 self)
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Abstract: In the last twenty years, several approaches to higherorder rewriting have been proposed, among which Klop’s Combinatory Rewrite Systems (CRSs), Nipkow’s Higherorder Rewrite Systems (HRSs) and Jouannaud and Okada’s higherorder algebraic specification languages, of which only the last one considers typed terms. The later approach has been extended by Jouannaud, Okada and the present author into Inductive Data Type Systems (IDTSs). In this paper, we extend IDTSs with the CRS higherorder patternmatching mechanism, resulting in simplytyped CRSs. Then, we show how the termination criterion developed for IDTSs with firstorder patternmatching, called the General Schema, can be extended so as to prove the strong normalization of IDTSs with higherorder patternmatching. Next, we compare the unified approach with HRSs. We first prove that the extended General Schema can also be applied to HRSs. Second, we show how Nipkow’s higherorder critical pair analysis technique for proving local confluence can be applied to IDTSs. 1
HigherOrder Lazy Narrowing Calculus: A Computation Model for a Higherorder Functional Logic Language
 In Proceedings of Sixth International Joint Conference, ALP '97  HOA '97, LNCS 1298
, 1997
"... this paper we present a computation model for a higherorder functional and logic programming. Although investigations of computation models for higherorder functional logic languages are under way[13, 9, 8, 20, 22], implemented functional logic languages like KLEAF[6] and Babel[18] among others, a ..."
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Cited by 8 (4 self)
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this paper we present a computation model for a higherorder functional and logic programming. Although investigations of computation models for higherorder functional logic languages are under way[13, 9, 8, 20, 22], implemented functional logic languages like KLEAF[6] and Babel[18] among others, are all based on firstorder models of computation. Firstorder narrowing has been used as basic computation mechanism. The lack of higherorderness is exemplified by the following prototypical program
On the confluence of λcalculus with conditional rewriting
"... Abstract. The confluence of untyped λcalculus with unconditional rewriting has already been studied in various directions. In this paper, we investigate the confluence of λcalculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic ..."
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Abstract. The confluence of untyped λcalculus with unconditional rewriting has already been studied in various directions. In this paper, we investigate the confluence of λcalculus with conditional rewriting and provide general results in two directions. First, when conditional rules are algebraic. This extends results of Müller and Dougherty for unconditional rewriting. Two cases are considered, whether betareduction is allowed or not in the evaluation of conditions. Moreover, Dougherty’s result is improved from the assumption of strongly normalizing βreduction to weakly normalizing βreduction. We also provide examples showing that outside these conditions, modularity of confluence is difficult to achieve. Second, we go beyond the algebraic framework and get new confluence results using a restricted notion of orthogonality that takes advantage of the conditional part of rewrite rules. 1
An Object Calculus with Algebraic Rewriting
"... . In trying to use Abadi and Cardelli's object calculi as a foundation for a programming language the addition of algebraic data types arises naturally. This paper defines such an extension, shows a motivating example, and explores the new calculi by establishing properties such as ChurchRosse ..."
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. In trying to use Abadi and Cardelli's object calculi as a foundation for a programming language the addition of algebraic data types arises naturally. This paper defines such an extension, shows a motivating example, and explores the new calculi by establishing properties such as ChurchRosser, subject reduction and uniqueness of types. Keywords: Object calculi; Rewriting; Combined calculi; Type systems. 1 Introduction Abadi and Cardelli [2] presented a simple object calculus, the &calculus, which can be used to model object oriented languages in the same way as functional languages are modelled by the calculus. The untyped &calculus supports method update. The firstorder (typed) version of the calculus, extended with a subtyping relation, models also object subsumption. The &calculus is computationally complete (the calculus can be encoded as shown in [2]). However, it lacks expressiveness for the representation of data types: data types can be encoded (e.g. through encoding...
Type Systems for the ObjectOriented Paradigm
, 1999
"... In this thesis we study some objectoriented mechanisms from the type system perspective. Our starting point is the axiomatic model of Fisher, Honsell and Mitchell, the Lambda Calculus of Objects. Following the path of recent researches on the topic, we show how some objectbased language features c ..."
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In this thesis we study some objectoriented mechanisms from the type system perspective. Our starting point is the axiomatic model of Fisher, Honsell and Mitchell, the Lambda Calculus of Objects. Following the path of recent researches on the topic, we show how some objectbased language features can be happily modeled within some functional calculi, proved sound with respect to the messagenotunderstood error. Then, we approach the more complicated problem of modeling class constructs, presenting two proposals and discussing their respective qualities and drawbacks, together with some future directions we would like to follow to try to better understand classbased languages.