We introduce a Curry–Howard correspondence for a large class of intermediate logics characterized by intuitionistic proofs with non-nested applications of rules for classical disjunctive tautologies (1-depth intermediate proofs). The resulting calculus, we call it λ∥, is a strongly normalizing parallel extension of the simply typed λ-calculus. Although simple, the λ∥ reduction rules can model arbitrary process network topologies, and encode interesting parallel programs ranging from numeric computation to algorithms on graphs.

Along the lines of Abramsky’s “Proofs-as-Processes” program, we present an interpretation of multiplicative linear logic as typing system for concurrent functional programming. In particular, we study a linear multiple-conclusion natural deduction system and show it is isomorphic to a simple and natural extension of λ-calculus with parallelism and communication primitives, called λpar. We shall prove that λpar satisfies all the desirable properties for a typed programming language: subject reduction, progress, strong normalization and confluence.

Abstract Herbrand’s theorem is one of the most fundamental insights in logic. From the syntactic point of view, it suggests a compact representation of proofs in classical first- and higher-order logics by recording the information of which instances have been chosen for which quantifiers. This compact representation is known in the literature as Miller’s expansion tree proof. It is inherently analytic and hence corresponds to a cut-free sequent calculus proof. Recently several extensions of such proof representations to proofs with cuts have been proposed. These extensions are based on graphical formalisms similar to proof nets and are limited to prenex formulas. In this paper, we present a new syntactic approach that directly extends Miller’s expansion trees by cuts and also covers non-prenex formulas. We describe a cut-elimination procedure for our expansion trees with cut that is based on the natural reduction steps and shows that it is weakly normalizing.

Propositional Gödel logic G extends intuitionistic logic with the non-constructive principle of linearity (A → B) ∨ (B → A). We introduce a Curry-Howard correspondence for G and show that a simple natural deduction calculus can be used as a typing system. The resulting functional language extends the simply typed λ-calculus via a synchronous communication mechanism between parallel processes, which increases its expressive power. The normalization proof employs original termination arguments and proof transformations implementing forms of code mobility. Our results provide a computational interpretation of G, thus proving A. Avron's 1991 thesis.

Abstract We present abstract complexity results about Coquand and Hyland–Ong game semantics, that will lead to new bounds on the length of first-order cut-elimination, normalization, interaction between expansion trees and any other dialogical process game semantics can model and apply to. In particular, we provide a novel method to bound the length of interactions between visible strategies and to measure precisely the tower of exponentials defining the worst-case complexity. Our study improves the old estimates on average by several exponentials.

In Interactive realizability for second-order Heyting Arithmetic with EM1 and SK1 (the excluded middle and Skolem axioms restricted to Σ10-formulas), realizers are written in a classical version of Girard's System F. Since the usual reducibility semantics does not apply to such a system, we introduce a constructive forcing/reducibility semantics: though realizers are not computable functionals in the sense of Girard, they can be forced to be computable. We apply this semantics to show how to extract witnesses for realizable Π20-formulas. In particular, a constructive and efficient method is introduced. It is based on a new ‘(state-extending-continuation)-passing-style translation’ whose properties are described with the constructive forcing/reducibility semantics.

We introduce a realizability semantics based on interactive learning for full second-order Heyting arithmetic with excluded middle and Skolem axioms over Σ10-formulas. Realizers are written in a classical version of Girard's System $\mathsf{F}$ and can be viewed as programs that learn by interacting with the environment. We show that the realizers of any Π20-formula represent terminating learning processes whose outcomes are numerical witnesses for the existential quantifier of the formula.

We present a new Curry-Howard correspondence for HA + EM_1, constructive Heyting Arithmetic with the excluded middle on \Sigma^0_1-formulas. We add to the lambda calculus an operator ||_a which represents, from the viewpoint of programming, an exception operator with a delimited scope, and from the viewpoint of logic, a restricted version of the excluded middle. We motivate the restriction of the excluded middle by its use in proof mining; we introduce new techniques to prove strong normalization for HA + EM_1 and the witness property for simply existential statements. One may consider our results as an application of the ideas of Interactive realizability, which we have adapted to the new setting and used to prove our main theorems.