Natural deduction for intuitionistic linear logic

AbstractThe paper deals with two versions of the fragment with unit, tensor, linear implication and storage operator (the exponential!) of intuitionistic linear logic. The first version, ILL, appears in a paper by Benton, Bierman, Hyland and de Paiva; the second one, ILL+, is described in this paper. ILL has a contraction rule and an introduction rule !I for the exponential; in ILL+, instead of a contraction rule, multiple occurrences of labels for assumptions are permitted under certain conditions; moreover, there is a different introduction rule for the exponential, !I+, which is closer in spirit to the necessitation rule for the normalizable version of S4 discussed by Prawitz in his monograph “Natural Deduction”.It is relatively easy to adapt Prawitz's treatment of natural deduction for intuitionistic logic to ILL+; in particular one can formulate a notion of strong validity (as in Prawitz's “Ideas and Results in Proof Theory”) permitting a proof of strong normalization.The conversion rules for ILL explicitly mentioned in the paper by Benton et al. do not suffice for normal forms with subformula property, but we can show that this can be remedied by addition of a special permutation conversion plus some “satellite” permutation conversions.Some discussion of the categorical models which might correspond to ILL+ is given

Counting proofs in propositional logic

We give a procedure for counting the number of different proofs of a formula in various sorts of propositional logic. This number is either an integer (that may be 0 if the formula is not provable) or infinite

Algebraic totality, towards completeness

Finiteness spaces constitute a categorical model of Linear Logic (LL) whose objects can be seen as linearly topologised spaces, (a class of topological vector spaces introduced by Lefschetz in 1942) and morphisms as continuous linear maps. First, we recall definitions of finiteness spaces and describe their basic properties deduced from the general theory of linearly topologised spaces. Then we give an interpretation of LL based on linear algebra. Second, thanks to separation properties, we can introduce an algebraic notion of totality candidate in the framework of linearly topologised spaces: a totality candidate is a closed affine subspace which does not contain 0. We show that finiteness spaces with totality candidates constitute a model of classical LL. Finally, we give a barycentric simply typed lambda-calculus, with booleans ${\mathcal{B}}$ and a conditional operator, which can be interpreted in this model. We prove completeness at type ${\mathcal{B}}^n\to{\mathcal{B}}$ for every n by an algebraic method

A robust semantics hides fewer errors

In this paper we explore how formal models are interpreted and to what degree meaning is captured in the formal semantics and to what degree it remains in the informal interpretation of the semantics. By applying a robust approach to the definition of refinement and semantics, favoured by the event-based community, to state-based theory we are able to move some aspects from the informal interpretation into the formal semantics

Computational interpretations of analysis via products of selection functions

We show that the computational interpretation of full comprehension via two wellknown functional interpretations (dialectica and modified realizability) corresponds to two closely related infinite products of selection functions

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

Computational Interpretations of Classical Linear Logic

Abstract. We survey several computational interpretations of classical linear logic based on two-player one-move games. The moves of the games are higher-order functionals in the language of finite types. All interpretations discussed treat the exponential-free fragment of linear logic in a common way. They only differ in how much advantage one of the players has in the exponentials games. We dis-cuss how the several choices for the interpretation of the modalities correspond to various well-known functional interpretations of intuitionistic logic, including Gödel’s Dialectica interpretation and Kreisel’s modified realizability.

Abstract Datatypes for Real Numbers in Type Theory

Abstract. We propose an abstract datatype for a closed interval of real numbers to type theory, providing a representation-independent approach to programming with real numbers. The abstract datatype requires only function types and a natural numbers type for its formulation, and so can be added to any type theory that extends Gödel’s System datatype is equivalent in power to programming intensionally with representations of real numbers. We also consider representing arbitrary real numbers using a mantissa-exponent representation in which the mantissa is taken from the abstract interval.