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Strong Normalization of Proof Nets Modulo Structural Congruences
 Proc of RTA, volume 1631 of LNCS
, 1999
"... . This paper proposes a notion of reduction for the proof nets of Linear Logic modulo an equivalence relation on the contraction links, that essentially amounts to consider the contraction as an associative commutative binary operator that can float freely in and out of proof net boxes. The need for ..."
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. This paper proposes a notion of reduction for the proof nets of Linear Logic modulo an equivalence relation on the contraction links, that essentially amounts to consider the contraction as an associative commutative binary operator that can float freely in and out of proof net boxes. The need for such a system comes, on one side, from the desire to make proof nets an even more parallel syntax for Linear Logic, and on the other side from the application of proof nets to lcalculus with or without explicit substitutions, which needs a notion of reduction more flexible than those present in the literature. The main result of the paper is that this relaxed notion of rewriting is still strongly normalizing. Keywords: Proof Nets. Linear Logic. Strong Normalization. 1 Introduction In his seminal paper [6], Girard proposed proof nets as a parallel syntax for Linear Logic, where uninteresting permutations in the order of application of logical rules are desequentialised and collapsed. Neve...
Resource operators for λcalculus
 INFORM. AND COMPUT
, 2007
"... We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proofnets. We show the operational behaviour of the calculus and some of its fundamental properties ..."
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We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proofnets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simplytyped terms, step by step simulation of βreduction and full composition.
Dependent Types and Explicit Substitutions
, 1999
"... We present a dependenttype system for a #calculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization. ..."
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We present a dependenttype system for a #calculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.
A Proof of Weak Termination of Typed λσCalculi
, 1998
"... We show that reducing any simplytyped oeterm (resp. oe*) by applying the rules in oe (resp. oe *) eagerly always terminates, by a translation to the simplytyped calculus. This holds even with term and substitution metavariables. In fact, every reduction terminates provided that (fi)redexes ar ..."
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We show that reducing any simplytyped oeterm (resp. oe*) by applying the rules in oe (resp. oe *) eagerly always terminates, by a translation to the simplytyped calculus. This holds even with term and substitution metavariables. In fact, every reduction terminates provided that (fi)redexes are only contracted under socalled safe contexts; and in oe, resp. oe *normal forms, all contexts around terms of sort T are safe. The result is then extended to secondorder type systems.
Ptime Completeness of Light Linear Logic and its Nondeterministic Extension
, 2004
"... In CSL’99 Roversi pointed out that the Turing machine encoding of Girard’s seminal paper ”Light Linear Logic ” has a flaw. Moreover he presented a working version of the encoding in Light Affine Logic, but not in Light Linear Logic. In this paper we present a working version of the encoding in Light ..."
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In CSL’99 Roversi pointed out that the Turing machine encoding of Girard’s seminal paper ”Light Linear Logic ” has a flaw. Moreover he presented a working version of the encoding in Light Affine Logic, but not in Light Linear Logic. In this paper we present a working version of the encoding in Light Linear Logic. The idea of the encoding is based on a remark of Girard’s tutorial paper on Linear Logic. The encoding is also an example which shows usefulness of additive connectives. Moreover we also consider a nondeterministic extension of Light Linear Logic. We show that the extended system is NPcomplete in the same meaning as Pcompleteness of Light Linear Logic.
Operated by Universities Space Research Association
"... CÉSAR MUÑOZ∗ Abstract. We present a dependenttype system for a λcalculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak ..."
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CÉSAR MUÑOZ∗ Abstract. We present a dependenttype system for a λcalculus with explicit substitutions. In this system, metavariables, as well as substitutions, are firstclass objects. We show that the system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.
Under consideration for publication in Math. Struct. in Comp. Science Natural Deduction via Graphs:
, 2006
"... We introduce the formalism of deduction graphs as a generalization of both GentzenPrawitz style natural deduction and Fitch style flag deduction. The advantage of this formalism is that subproofs can be shared, like in flag deductions (and unlike natural deduction), but also that the linearisation ..."
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We introduce the formalism of deduction graphs as a generalization of both GentzenPrawitz style natural deduction and Fitch style flag deduction. The advantage of this formalism is that subproofs can be shared, like in flag deductions (and unlike natural deduction), but also that the linearisation used in flag deductions is avoided. Our deduction graphs have both nodes and boxes, which are collections of nodes that also form a node themselves. This is reminiscent of the bigraphs of Milner, where the link graph describes the nodes and edges and the place graph describes the nesting of nodes. In the paper we give a precise definition of deduction graphs and we give examples to illustrate them. Furthermore we analyse their computational behaviour by studying the process of cutelimination and by defining translations from deduction graphs to simply typed lambda terms. From a slight variation of this translation we conclude that the process of cutelimination is strongly normalising. The translation to simple type theory removes quite a lot of structure and we therefore also propose a translation to a context calculus with lets, that faithfully captures the structure of deduction graphs. The proof nets of linear logic also present a graphlike presentation of natural deduction. We point out some similarities of the two formalisms. Key words: natural deduction, cutelimination, lambda calculus with letbinding, typed lambda calculus.
Dependent Types with Explicit Substitutions: A metatheoretical development
, 1997
"... We present a theory of dependent types with explicit substitutions. We follow a metatheoretical approach where open expressions expressions with metavariables are firstclass objects. The system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normal ..."
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We present a theory of dependent types with explicit substitutions. We follow a metatheoretical approach where open expressions expressions with metavariables are firstclass objects. The system enjoys properties like type uniqueness, subject reduction, soundness, confluence and weak normalization.
International Journal of Foundations of Computer Science cfl World Scientific Publishing Company Light Affine Logic as a Programming Language: a First Contribution
"... Revised (revised date) Communicated by Editor's name ABSTRACT This work is about an experimental paradigmatic functional language for programming with PTIME functions. The language is designed from Intuitionistic Light Affine Logic. It can be typed automatically by a type inference algorithm t ..."
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Revised (revised date) Communicated by Editor's name ABSTRACT This work is about an experimental paradigmatic functional language for programming with PTIME functions. The language is designed from Intuitionistic Light Affine Logic. It can be typed automatically by a type inference algorithm that deduces polymorphic types `a la ML. Keywords: Functional programming languages, PTIME computations, Light Affine Logic, Automatic type inference.
Explicit Pure Type Systems for the λCube
, 2004
"... Pure type systems are a general formalism allowing to represent many type systems – in particular, Barendregt’s λcube, including Girard’s system F, dependent types, and the calculus of constructions. We built a variant of pure type systems by adding a cut rule associated to an explicit substitution ..."
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Pure type systems are a general formalism allowing to represent many type systems – in particular, Barendregt’s λcube, including Girard’s system F, dependent types, and the calculus of constructions. We built a variant of pure type systems by adding a cut rule associated to an explicit substitution in the syntax, according to the CurryHowardde Bruijn correspondence. The addition of the cut requires the addition of a new rule for substitutions, with which we are able to show type correctness and subject reduction for all explicit systems. Moreover, we proved that the explicit λcube obtained this way is strongly normalizing.