Federico Aschieri

Learning based realizability for HA + EM1 and 1-Backtracking games: Soundness and completeness

Abstract We prove a soundness and completeness result for Aschieri and Berardiʼs learning based realizability for Heyting Arithmetic plus Excluded Middle over semi-decidable statements with respect to 1-Backtracking Coquand game semantics. First, we prove that learning based realizability is sound with respect to 1-Backtracking Coquand game semantics. In particular, any realizer of an implication-and-negation-free arithmetical formula embodies a winning recursive strategy for the 1-Backtracking version of Tarski games. We also give examples of realizers and winning strategy extraction for some classical proofs. Secondly, we extend our notion of realizability to a total recursive learning based realizability and show that the notion is complete with respect to 1-Backtracking Coquand game semantics.

A New Use of Friedman’s Translation: Interactive Realizability

Friedman’s translation is a well-known transformation of formulas. The Friedman translation has two properties: i) it validates intuitionistic theorems – if a formula is intuitionistically provable, then so it is its Friedman translation; ii) it is suitable for program extraction from classical proofs – the intuitionistic provability of the Friedman translation of the negative translation of a for-all-exist-formula implies the intuitionistic provability of the formula itself. However, the Friedman translation does not validate classical principles, like the Excluded Middle. Here, we define a restricted Friedman translation which both validates the Excluded Middle and Skolem axiom schemata restricted to Σ1-formulas and it is also suitable for program extraction from classical proofs using such principles: the intuitionistic provability of the restricted Friedman translation of a for-all-exist-formula implies the intuitionistic provability of the formula itself. Then we introduce a learning-based Realizability Semantics for Heyting Arithmetic with all finite types, extended with the two previous axiom schemata. We call this semantics “Interactive Realizability”, and we characterize it as the composition of our restricted Friedman translation with Kreisel modified realizability. As a corollary, we show that Interactive Realizability is, in a sense, “axiom-driven”, while the other Realizability Semantics for Classical Arithmetic, like the semantics of Krivine, are “goal-driven”.

A constructive analysis of learning in Peano Arithmetic

Abstract We give a constructive analysis of learning as it arises in various computational interpretations of classical Peano Arithmetic, such as Aschieri and Berardi learning based realizability, Avigad’s update procedures and epsilon substitution method. In particular, we show how to compute in Godel’s system T upper bounds on the length of learning processes, which are themselves represented in T through learning based realizability. The result is achieved by the introduction of a new non standard model of Godel’s T , whose new basic objects are pairs of non standard natural numbers (convergent sequences of natural numbers) and moduli of convergence, where the latter are objects giving constructive information about the former. As a foundational corollary, we obtain that that learning based realizability is a constructive interpretation of Heyting Arithmetic plus excluded middle over Σ 1 0 formulas (for which it was designed) and of all Peano Arithmetic when combined with Godel’s double negation translation. As a byproduct of our approach, we also obtain a new proof of Avigad’s theorem for update procedures and thus of the termination of the epsilon substitution method for PA .

Non-determinism, Non-termination and the Strong Normalization of System T

We consider a de’Liguoro-Piperno-style extension of the pure lambda calculus with a non-deterministic choice operator as well as a non-deterministic iterator construct, with the aim of studying its normalization properties. We provide a simple characterization of non-strongly normalizable terms by means of the so called “zoom-in” perpetual reduction strategy. We then show that this characterization implies the strong normalization of the simply typed version of the calculus. As straightforward corollary of these results we obtain a new proof of strong normalization of Godel’s System T by a simple translation of this latter system into the former.

Interactive Realizability for Classical Peano Arithmetic with Skolem Axioms

Interactive realizability is a computational semantics of classical Arithmetic. It is based on interactive learning and was originally designed to interpret excluded middle and Skolem axioms for simple existential formulas. A realizer represents a proof/construction depending on some state, which is an approximation of some Skolem functions. The realizer interacts with the environment, which may provide a counter-proof, a counterexample invalidating the current construction of the realizer. But the realizer is always able to turn such a negative outcome into a positive information, which consists in some new piece of knowledge learned about the mentioned Skolem functions. The aim of this work is to extend Interactive realizability to a system which includes classical first-order Peano Arithmetic with Skolem axioms. For witness extraction, the learning capabilities of realizers will be exploited according to the paradigm of learning by levels. In particular, realizers of atomic formulas will be update procedures in the sense of Avigad and thus will be understood as stratified-learning algorithms.

Interactive Learning Based Realizability and 1-Backtracking Games

We prove that interactive learning based classical realizability (introduced by Aschieri and Berardi for first order arithmetic) is sound with respect to Coquand game semantics. In particular, any realizer of an implication-and-negation-free arithmetical formula embodies a winning recursive strategy for the 1-Backtracking version of Tarski games. We also give examples of realizer and winning strategy extraction for some classical proofs. We also sketch some ongoing work about how to extend our notion of realizability in order to obtain completeness with respect to Coquand semantics, when it is restricted to 1-Backtracking games.

On Natural Deduction for Herbrand Constructive Logics I: Curry-Howard Correspondence for Dummett's Logic LC

Dummetts logic LC is intuitionistic logic extended with Dummetts axiom: for every two statements the first implies the second or the second implies the first. We present a natural deduction and a Curry-Howard correspondence for first-order and second-order Dummetts logic. We add to the lambda calculus an operator which represents, from the viewpoint of programming, a mechanism for representing parallel computations and communication between them, and from the point of view logic, Dummetts axiom. We prove that our typed calculus is normalizing and show that proof terms for existentially quantified formulas reduce to a list of individual terms forming an Herbrand disjunction.

On natural deduction in classical first-order logic: Curry–Howard correspondence, strong normalization and Herbrand's theorem

Abstract We 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 an Herbrand disjunction. A by-product of our approach is a natural-deduction proof and a computational interpretation of Herbrands Theorem.

Realizability and Strong Normalization for a Curry-Howard Interpretation of HA + EM1

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.

Gödel logic: From natural deduction to parallel computation

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. Avrons 1991 thesis.