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11
Cut Rules and Explicit Substitutions
, 2000
"... this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the l ..."
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Cited by 16 (0 self)
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this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call Nsystems, are symbolic logics generally given via introduction and elimination rules for the logical connectives which operate on the right, i.e., they manipulate the succedent formula. Examples are Gentzen's NJ and NK (Gentzen 1935). Logical deduction systems are given via leftintroduction and rightintroduction rules for the logical connectives. Although others have called these systems "sequent calculi", we call them Lsystems to avoid confusion with other systems given in sequent style. Examples are Gentzen's LK and LJ (Gentzen 1935). In this paper we are primarily interested in Lsystems. The advantage of Nsystems is that they seem closer to actual reasoning, while Lsystems on the other hand seem to have an easier proof theory. Lsystems are often extended with a "cut" rule as part of showing that for a given Lsystem and Nsystem, the derivations of each system can be encoded in the other. For example, NK proves the same as LK + cut (Gentzen 1935). Proof Normalization. A system is consistent when it is impossible to prove false, i.e., derive absurdity from zero assumptions. A system is analytic (has the analycity property) when there is an e#ective method to decompose any conclusion sequent into simpler premise sequents from which the conclusion can be obtained by some rule in the system such that the conclusion is derivable i# the premises are derivable (Maenpaa 1993). To achieve the goals of consistency and analycity, it has been customary to consider
Comparing and Implementing Calculi of Explicit Substitutions with Eta Reduction
 Annals of Pure and Applied Logic
, 2005
"... The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that e ..."
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Cited by 10 (8 self)
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The past decade has seen an explosion of work on calculi of explicit substitutions. Numerous work has illustrated the usefulness of these calculi for practical notions like the implementation of typed functional programming languages and higher order proof assistants. It has also been shown that eta reduction is useful for adapting substitution calculi for practical problems like higher order uni cation. This paper concentrates on rewrite rules for eta reduction in three dierent styles of explicit substitution calculi: , se and the suspension calculus. Both and se when extended with eta reduction, have proved useful for solving higher order uni cation. We enlarge the suspension calculus with an adequate etareduction which we show to preserve termination and conuence of the associated substitution calculus and to correspond to the etareductions of the other two calculi. We prove that and se as well as and the suspension calculus are non comparable while se is more adequate than the suspension calculus in simulating one step of betacontraction.
A Theory of Explicit Substitutions with Safe and Full Composition
 Logical Methods in Computer Science
"... Vol. 5 (3:1) 2009, pp. 1–29 ..."
Efficiency of LambdaCalculi With Explicit Substitutions
, 1996
"... We introduce a criterion of efficiency to simulate fireduction in calculi of explicit substitutions and we apply it to several calculi: oe, oe * , AE, s, t and u. The latter is presented here for the first time and may be considered as an efficient variant of s. The results of this paper imply tha ..."
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Cited by 2 (1 self)
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We introduce a criterion of efficiency to simulate fireduction in calculi of explicit substitutions and we apply it to several calculi: oe, oe * , AE, s, t and u. The latter is presented here for the first time and may be considered as an efficient variant of s. The results of this paper imply that calculi `a la s are usually more efficient at simulating fireduction than calculi in the oestyle. In fact, we prove that t is more efficient than AE and that u is more efficient than AE, oe * and s. We also give counterexamples to show that all other comparisons are impossible. 1 Introduction The classical calculus (cf. [2]) deals with substitution in an implicit way. This means that the computations to perform substitution are usually described with operators which do not belong to the language of the calculus. There has however been an interest in formalising substitution explicitly in order to provide a theoretical framework for the implementation of functional programming langua...
Pure Type Systems with de Bruijn indices
"... Nowadays, type theory has many applications and is used in many different disciplines. Within computer science, logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. A general framework called Pure Type Systems (PTSs for short) ..."
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Cited by 1 (1 self)
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Nowadays, type theory has many applications and is used in many different disciplines. Within computer science, logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. A general framework called Pure Type Systems (PTSs for short) has been introduced independently by Terlouw and Berardi in 1988 and 1989, in order to provide a unified formalism in which many type systems can be represented. In particular, PTSs allow the representation of the simple theory of types, the polymophic theory of types, the dependent theory of types and various other wellknown type systems such as the Edinburgh Logical Frameworks LF and the Automath system. Pure Type Systems are usually presented using variable names. In this article, we present a formulation of PTSs with de Bruijn indices. De Bruijn indices [6] avoid the problems caused by variable names during the implementation of type systems. We show that PTSs with variable names and PTSs with de Bruijn indices are isomorphic. This isomorphism enables us to answer questions about PTSs with de Bruijn indices including confluence, termination (strong normalisation) and safety (subject reduction).
A de Bruijn notation for higherorder rewriting (Extended Abstract)
 In Proceedings of the 11th International Conference on Rewriting Techniques and Applications (RTA'00
, 2000
"... ) Eduardo Bonelli 1;2 , Delia Kesner 2 , Alejandro R'ios 1 1 Departamento de Computaci'on  Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabell'on I, Ciudad Universitaria (1428), Buenos Aires, Argentina. febonelli,riosg@dc.uba.ar 2 LRI (UMR 8623)  ..."
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) Eduardo Bonelli 1;2 , Delia Kesner 2 , Alejandro R'ios 1 1 Departamento de Computaci'on  Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Pabell'on I, Ciudad Universitaria (1428), Buenos Aires, Argentina. febonelli,riosg@dc.uba.ar 2 LRI (UMR 8623)  Bat 490, Universit'e de ParisSud, 91405 Orsay Cedex, France. kesner@lri.fr Abstract. We propose a formalism for higherorder rewriting in de Bruijn notation. This notation not only is used for terms (as usually done in the literature) but also for metaterms, which are the syntactical objects used to express general higherorder rewrite systems. We give formal translations from higherorder rewriting with names to higherorder rewriting with de Bruijn indices, and viceversa. These translations can be viewed as an interface in programming languages based on higherorder rewrite systems, and they are also used to show some properties, namely, that both formalisms are operationally equivalent, and th...
Explicit Substitutions a la de Bruijn: the local . . .
 ELECTRONIC NOTES IN THEORETICAL COMPUTER SCIENCE 85 NO. 7 (2003)
, 2003
"... Kamareddine and Nederpelt [9], resp. Kamareddine and Ríos [11] gave two calculi of explicit of substitutions highly inuenced by de Bruijn's notation of the calculus. These calculi added to the explosive pool of work on explicit substitution in the past 15 years. As far as we know, calculi of e ..."
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Kamareddine and Nederpelt [9], resp. Kamareddine and Ríos [11] gave two calculi of explicit of substitutions highly inuenced by de Bruijn's notation of the calculus. These calculi added to the explosive pool of work on explicit substitution in the past 15 years. As far as we know, calculi of explicit substitutions: a) are unable to handle local substitutions, and b) have answered (positively or negatively) the question of the termination of the underlying calculus of substitutions. The exception to a) is the calculus of [9] where substitution is handled both locally and globally. However, the calculus of [9] does not satisfy properties like conuence and termination. The exception to b) is the s e calculus [11] for which termination of the s e calculus, the underlying calculus of substitutions, remains unsolved. This paper has two aims: (i) To provide a calculus a la de Bruijn which deals with local substitution and whose underlying calculus of substitutions is terminating and conuent.
Higher Order Unification via ...Style of Explicit Substitution
"... A higher order unification (HOU) method based on the ...style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...style of explicit substitution. Correctness and completeness properties of the proposed approach ..."
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A higher order unification (HOU) method based on the ...style of explicit substitution is proposed. The method is based on the treatment introduced by Dowek, Hardin and Kirchner in [DHK95] using the ...style of explicit substitution. Correctness and completeness properties of the proposed approach are shown and advantages of the method, inherited from the qualities of the ... calculus, are pointed out.
Eciency of calculi with explicit substitutions
, 1996
"... We introduce a criterion of eciency to simulate reduction in calculi of explicit substitutions and we apply it to several calculi: , , , s, t and u. The latter is presented here for the rst time and may be considered as an ecient variant of s. The results of this paper imply that calculi a la s are ..."
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We introduce a criterion of eciency to simulate reduction in calculi of explicit substitutions and we apply it to several calculi: , , , s, t and u. The latter is presented here for the rst time and may be considered as an ecient variant of s. The results of this paper imply that calculi a la s are usually more ecient at simulatingreduction than calculi in the style. In fact, we prove that t is more ecient than and that u is more ecient than , and s. We also give counterexamples to show that all other comparisons are impossible. 1