A Lambda-calculus Structure Isomorphic to Gentzen-style Sequent Calculus Structure

International audienceWe consider a lambda-calculus for which applicative terms have no longer the form (...((u u_1) u_2) ... u_n) but the form (u [u_1 ; ... ; u_n]), for which [u_1 ; ... ; u_n] is a list of terms. While the structure of the usual lambda-calculus is isomorphic to the structure of natural deduction, this new structure is isomorphic to the structure of Gentzen-style sequent calculus. To express the basis of the isomorphism, we consider intuitionistic logic with the implication as sole connective. However we do not consider Gentzen's calculus LJ, but a calculus LJT which leads to restrict the notion of cut-free proofs in LJ. We need also to explicitly consider, in a simply typed version of this lambda-calculus, a substitution operator and a list concatenation operator. By this way, each elementary step of cut-elimination exactly matches with a beta-reduction, a substitution propagation step or a concatenation computation step. Though it is possible to extend the isomorphism to classical logic and to other connectives, we do not treat of it in this paper

Forcing-based cut-elimination for Gentzen-style intuitionistic sequent calculus

International audienceWe give a simple intuitionistic completeness proof of Kripke semantics for intuitionistic logic with implication and universal quantification with respect to cut-free intuitionistic sequent calculus. The Kripke semantics is ``simplified'' in the way that the domain remains constant. The proof has been formalised in the Coq proof assistant and by combining soundness with completeness, we obtain an executable cut-elimination procedure. The proof easily extends to the case of the absurdity connective using Kripke models with exploding nodes à la Veldman

Forcing-based cut-elimination for Gentzen-style intuitionistic sequent calculus

International audienceWe give a simple intuitionistic completeness proof of Kripke semantics for intuitionistic logic with implication and universal quantification with respect to cut-free intuitionistic sequent calculus. The Kripke semantics is ``simplified'' in the way that the domain remains constant. The proof has been formalised in the Coq proof assistant and by combining soundness with completeness, we obtain an executable cut-elimination procedure. The proof easily extends to the case of the absurdity connective using Kripke models with exploding nodes à la Veldman

What is the meaning of proofs? A Fregean distinction in proof-theoretic semantics

The origins of proof-theoretic semantics lie in the question of what constitutes the meaning of the logical connectives and its response: the rules of inference that govern the use of the connective. However, what if we go a step further and ask about the meaning of a proof as a whole? In this paper we address this question and lay out a framework to distinguish sense and denotation of proofs. Two questions are central here. First of all, if we have two (syntactically) different derivations, does this always lead to a difference, firstly, in sense, and secondly, in denotation? The other question is about the relation between different kinds of proof systems (here: natural deduction vs. sequent calculi) with respect to this distinction. Do the different forms of representing a proof necessarily correspond to a difference in how the inferential steps are given? In our framework it will be possible to identify denotation as well as sense of proofs not only within one proof system but also between different kinds of proof systems. Thus, we give an account to distinguish a mere syntactic divergence from a divergence in meaning and a divergence in meaning from a divergence of proof objects analogous to Frege's distinction for singular terms and sentences.Comment: Post-peer-review, pre-copyedit version of article, published version available open access under DOI: 10.1007/s10992-020-09577-

Justification Logic as a foundation for certifying mobile computation

We explore an intuitionistic fragment of Artëmov's . Justification Logic as a type system for a programming language for . mobile units. Such units consist of both a code and a certificate component. Our language, the . Certifying Mobile Calculus, caters for code and certificate development in a unified theory. In the same way that mobile code is constructed out of code components and extant type systems track local resource usage to ensure the mobile nature of these components, our system . additionally ensures correct . certificate construction out of certificate components. We present proofs of type safety and strong normalization for a run-time system based on an abstract machine.Fil: Bonelli, Eduardo Augusto. Universidad Nacional de Quilmes. Departamento de Ciencia y Tecnología; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Feller, Federico. Universidad Nacional de La Plata. Facultad de Informática. Laboratorio de Investigación y Formación en Informática Avanzada; Argentin

Call-by-Value Lambda-calculus and LJQ

Accepté pour publication dans J. Logic Comput. ; 24 pagesLJQ is a focused sequent calculus for intuitionistic logic, with a simple restriction on the first premiss of the usual left introduction rule for implication. In a previous paper we discussed its history (going back to about 1950, or beyond) and presented its basic theory and some applications; here we discuss in detail its relation to call-by-value reduction in lambda calculus, establishing a connection between LJQ and the CBV calculus Lambda_C of Moggi. In particular, we present an equational correspondence between these two calculi forming a bijection between the two sets of normal terms, and allowing reductions in each to be simulated by reductions in the other

Justification Logic as a foundation for certifying mobile computation

We explore an intuitionistic fragment of Artëmov's Justification Logic as a type system for a programming language for mobile units. Such units consist of both a code and a certificate component. Our language, the Certifying Mobile Calculus, caters for code and certificate development in a unified theory. In the same way that mobile code is constructed out of code components and extant type systems track local resource usage to ensure the mobile nature of these components, our system additionally ensures correct certificate construction out of certificate components. We present proofs of type safety and strong normalization for a run-time system based on an abstract machine.Facultad de Informátic

A Focused Sequent Calculus Framework for Proof Search in Pure Type Systems

Basic proof-search tactics in logic and type theory can be seen as the root-first applications of rules in an appropriate sequent calculus, preferably without the redundancies generated by permutation of rules. This paper addresses the issues of defining such sequent calculi for Pure Type Systems (PTS, which were originally presented in natural deduction style) and then organizing their rules for effective proof-search. We introduce the idea of Pure Type Sequent Calculus with meta-variables (PTSCalpha), by enriching the syntax of a permutation-free sequent calculus for propositional logic due to Herbelin, which is strongly related to natural deduction and already well adapted to proof-search. The operational semantics is adapted from Herbelin's and is defined by a system of local rewrite rules as in cut-elimination, using explicit substitutions. We prove confluence for this system. Restricting our attention to PTSC, a type system for the ground terms of this system, we obtain the Subject Reduction property and show that each PTSC is logically equivalent to its corresponding PTS, and the former is strongly normalising iff the latter is. We show how to make the logical rules of PTSC into a syntax-directed system PS for proof-search, by incorporating the conversion rules as in syntax-directed presentations of the PTS rules for type-checking. Finally, we consider how to use the explicitly scoped meta-variables of PTSCalpha to represent partial proof-terms, and use them to analyse interactive proof construction. This sets up a framework PE in which we are able to study proof-search strategies, type inhabitant enumeration and (higher-order) unification