Strong Normalization for HA + EM1 by Non-Deterministic Choice

We study the strong normalization of a new Curry-Howard correspondence for HA + EM1, constructive Heyting Arithmetic with the excluded middle on Sigma01-formulas. The proof-term language of HA + EM1 consists in the lambda calculus plus 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 give a strong normalization proof for the system based on a technique of "non-deterministic immersion".Comment: In Proceedings COS 2013, arXiv:1309.092

On Natural Deduction in Classical First-Order Logic: Curry-Howard Correspondence, Strong Normalization and Herbrand's Theorem

International audienceWe present a new Curry-Howard correspondence for classical first-order natural deduction. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for raising and catching multiple exceptions, and from the viewpoint of logic, the excluded middle over arbitrary prenex formulas. The machinery will allow to extend the idea of learning -- originally developed in Arithmetic -- to pure logic. We prove that our typed calculus is strongly normalizing and show that proof terms for simply existential statements reduce to a list of individual terms forming a Herbrand disjunction. A by-product of our approach is a natural-deduction proof and a computational interpretation of Herbrand's Theorem

Interactive Learning-Based Realizability for Heyting Arithmetic with EM1

We apply to the semantics of Arithmetic the idea of ``finite approximation'' used to provide computational interpretations of Herbrand's Theorem, and we interpret classical proofs as constructive proofs (with constructive rules for $\vee, \exists$) over a suitable structure \StructureN for the language of natural numbers and maps of G\"odel's system \SystemT. We introduce a new Realizability semantics we call ``Interactive learning-based Realizability'', for Heyting Arithmetic plus \EM_1 (Excluded middle axiom restricted to $\Sigma^0_1$ formulas). Individuals of \StructureN evolve with time, and realizers may ``interact'' with them, by influencing their evolution. We build our semantics over Avigad's fixed point result, but the same semantics may be defined over different constructive interpretations of classical arithmetic (Berardi and de' Liguoro use continuations). Our notion of realizability extends intuitionistic realizability and differs from it only in the atomic case: we interpret atomic realizers as ``learning agents''

On Natural Deduction for Herbrand Constructive Logics II: Curry-Howard Correspondence for Markov\u27s Principle in First-Order Logic and Arithmetic

Intuitionistic first-order logic extended with a restricted form of Markov\u27s principle is constructive and admits a Curry-Howard correspondence, as shown by Herbelin. We provide a simpler proof of that result and then we study intuitionistic first-order logic extended with unrestricted Markov\u27s principle. Starting from classical natural deduction, we restrict the excluded middle and we obtain a natural deduction system and a parallel Curry-Howard isomorphism for the logic. We show that proof terms for existentially quantified formulas reduce to a list of individual terms representing all possible witnesses. As corollary, we derive that the logic is Herbrand constructive: whenever it proves any existential formula, it proves also an Herbrand disjunction for the formula. Finally, using the techniques just introduced, we also provide a new computational interpretation of Arithmetic with Markov\u27s principle

Interactive Realizability and the elimination of Skolem functions in Peano Arithmetic

We present a new syntactical proof that first-order Peano Arithmetic with Skolem axioms is conservative over Peano Arithmetic alone for arithmetical formulas. This result - which shows that the Excluded Middle principle can be used to eliminate Skolem functions - has been previously proved by other techniques, among them the epsilon substitution method and forcing. In our proof, we employ Interactive Realizability, a computational semantics for Peano Arithmetic which extends Kreisel's modified realizability to the classical case.Comment: In Proceedings CL&C 2012, arXiv:1210.289

Learning, realizability and games in classical arithmetic

PhDAbstract. In this dissertation we provide mathematical evidence that the concept of learning can be used to give a new and intuitive computational semantics of classical proofs in various fragments of Predicative Arithmetic. First, we extend Kreisel modi ed realizability to a classical fragment of rst order Arithmetic, Heyting Arithmetic plus EM1 (Excluded middle axiom restricted to 0 1 formulas). We introduce a new realizability semantics we call \Interactive Learning-Based Realizability". Our realizers are self-correcting programs, which learn from their errors and evolve through time, thanks to their ability of perpetually questioning, testing and extending their knowledge. Remarkably, that capability is entirely due to classical principles when they are applied on top of intuitionistic logic. Secondly, we extend the class of learning based realizers to a classical version PCFClass of PCF and, then, compare the resulting notion of realizability with Coquand game semantics and prove a full soundness and completeness result. In particular, we show there is a one-to-one correspondence between realizers and recursive winning strategies in the 1-Backtracking version of Tarski games. Third, we provide a complete and fully detailed constructive analysis of learning as it arises in learning based realizability for HA+EM1, Avigad's update procedures and epsilon substitution method for Peano Arithmetic PA. We present new constructive techniques to bound the length of learning processes and we apply them to reprove - by means of our theory - the classic result of G odel that provably total functions of PA can be represented in G odel's system T. Last, we give an axiomatization of the kind of learning that is needed to computationally interpret Predicative classical second order Arithmetic. Our work is an extension of Avigad's and generalizes the concept of update procedure to the trans nite case. Trans- nite update procedures have to learn values of trans nite sequences of non computable functions in order to extract witnesses from classical proofs

Mathematical Logic: Proof theory, Constructive Mathematics

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit

On an Intuitionistic Logic for Pragmatics

We reconsider the pragmatic interpretation of intuitionistic logic [21] regarded as a logic of assertions and their justications and its relations with classical logic. We recall an extension of this approach to a logic dealing with assertions and obligations, related by a notion of causal implication [14, 45]. We focus on the extension to co-intuitionistic logic, seen as a logic of hypotheses [8, 9, 13] and on polarized bi-intuitionistic logic as a logic of assertions and conjectures: looking at the S4 modal translation, we give a denition of a system AHL of bi-intuitionistic logic that correctly represents the duality between intuitionistic and co-intuitionistic logic, correcting a mistake in previous work [7, 10]. A computational interpretation of cointuitionism as a distributed calculus of coroutines is then used to give an operational interpretation of subtraction.Work on linear co-intuitionism is then recalled, a linear calculus of co-intuitionistic coroutines is dened and a probabilistic interpretation of linear co-intuitionism is given as in [9]. Also we remark that by extending the language of intuitionistic logic we can express the notion of expectation, an assertion that in all situations the truth of p is possible and that in a logic of expectations the law of double negation holds. Similarly, extending co-intuitionistic logic, we can express the notion of conjecture that p, dened as a hypothesis that in some situation the truth of p is epistemically necessary