The First-Order Hypothetical Logic of Proofs

The Propositional Logic of Proofs (LP) is a modal logic in which the modality □A is revisited as [​[t]​]​A , t being an expression that bears witness to the validity of A . It enjoys arithmetical soundness and completeness, can realize all S4 theorems and is capable of reflecting its own proofs ( ⊢A implies ⊢[​[t]​]A , for some t ). A presentation of first-order LP has recently been proposed, FOLP, which enjoys arithmetical soundness and has an exact provability semantics. A key notion in this presentation is how free variables are dealt with in a formula of the form [​[t]​]​A(i) . We revisit this notion in the setting of a Natural Deduction presentation and propose a Curry–Howard correspondence for FOLP. A term assignment is provided and a proof of strong normalization is given.Fil: Steren, Gabriela. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Bonelli, Eduardo Augusto. Universidad Nacional de Quilmes. Departamento de Ciencia y Tecnología; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

A calculus of multiary sequent terms

Multiary sequent terms were originally introduced as a tool for proving termination of permutative conversions in cut-free sequent calculus. This work develops the language of multiary sequent terms into a term calculus for the computational (Curry-Howard) interpretation of a fragment of sequent calculus with cuts and cut-elimination rules. The system, named generalised multiary lambda-calculus, is a rich extension of the lambda-calculus where the computational content of the sequent calculus format is explained through an enlarged form of the application constructor. Such constructor exhibits the features of multiarity (the ability of forming lists of arguments) and generality (the ability of prescribing a kind of continuation). The system integrates in a modular way the multiary lambda-calculus and an isomorphic copy of the lambda-calculus with generalised application LambdaJ (in particular, natural deduction is captured internally up to isomorphism). In addition, the system: (i) comes with permutative conversion rules, whose role is to eliminate the new features of application; (ii) is equipped with reduction rules --- either the mu-rule, typical of the multiary setting, or rules for cut-elimination, which enlarge the ordinary beta-rule. This paper establishes the meta-theory of the system, with emphasis on the role of the mu-rule, and including a study of the interaction of reduction and permutative conversions.Fundação para a Ciência e a Tecnologia (FCT

Permutative conversions in intuitionistic multiary sequent calculi with cuts

This work presents an extension with cuts of Schwichtenberg's multiary sequent calculus. We identify a set of permutative conversions on it, prove their termination and confluence and establish the permutability theorem. We present our sequent calculus as the typing system of the {\em generalised multiary $\lambda$-calculus} lambda-Jm, a new calculus introduced in this work. Lambda-Jm corresponds to an extension of $\lambda$-calculus with a notion of {\em generalised multiary application}, which may be seen as a function applied to a list of arguments and then explicitly substituted in another term. Proof-theoretically the corresponding typing rule encompasses, in a modular way, generalised eliminations of von Plato and Herbelin's head cuts.Fundação para a Ciência e a Tecnologia (FCT)

Some properties of the lambda-mu-and-or-calculus

International audienceIn this paper, we present the lambda-mu-and-or-calculus which at the typed level corresponds to the full classical propositional natural deduction system. Church- Rosser property of this system is proved using the standardization and the finiteness developments theorem. We defi ne also the leftmost reduction and prove that it is a winning strateg

Permutability in proof terms for intuitionistic sequent calculus with cuts

This paper gives a comprehensive and coherent view on permutability in the intuitionistic sequent calculus with cuts. Specifically we show that, once permutability is packaged into appropriate global reduction procedures, it organizes the internal structure of the system and determines fragments with computational interest, both for the computation-as-proof-normalization and the computation-as-proof-search paradigms. The vehicle of the study is a lambda-calculus of multiary proof terms with generalized application, previously developed by the authors (the paper argues this system represents the simplest fragment of ordinary sequent calculus that does not fall into mere natural deduction). We start by adapting to our setting the concept of normal proof, developed by Mints, Dyckhoff, and Pinto, and by defining natural proofs, so that a proof is normal iff it is natural and cut-free. Natural proofs form a subsystem with a transparent Curry-Howard interpretation (a kind of formal vector notation for lambda-terms with vectors consisting of lists of lists of arguments), while searching for normal proofs corresponds to a slight relaxation of focusing (in the sense of LJT). Next, we define a process of permutative conversion to natural form, and show that its combination with cut elimination gives a concept of normalization for the sequent calculus. We derive a systematic picture of the full system comprehending a rich set of reduction procedures (cut elimination, flattening, permutative conversion, normalization, focalization), organizing the relevant subsystems and the important subclasses of cut-free, normal, and focused proofs.Partially financed by FCT through project UID/MAT/00013/2013, and by COST action CA15123 EUTYPES. The first and the last authors were partially financed by Fundação para a Ciência e a Tecnologia (FCT) through project UID/MAT/00013/2013. The first author got financial support by the COST action CA15123 EUTYPES.info:eu-repo/semantics/publishedVersio

Decidability for Non-Standard Conversions in Typed Lambda-Calculi

This thesis studies the decidability of conversions in typed lambda-calculi, along with the algorithms allowing for this decidability. Our study takes in consideration conversions going beyond the traditional beta, eta, or permutative conversions (also called commutative conversions). To decide these conversions, two classes of algorithms compete, the algorithms based on rewriting, here the goal is to decompose and orient the conversion so as to obtain a convergent system, these algorithms then boil down to rewrite the terms until they reach an irreducible forms; and the "reduction free" algorithms where the conversion is decided recursively by a detour via a meta-language. Throughout this thesis, we strive to explain the latter thanks to the former