Interactive Realizability for second-order Heyting arithmetic with EM1 and SK1

Abstract

Link to this article: http://journals.cambridge.org/abstract_S0960129513000455 How to cite this article: FEDERICO ASCHIERI (2014). Interactive Realizability for second-order Heyting arithmetic with EM1 and SK1.We introduce a realizability semantics based on interactive learning for full second-order Heyting arithmetic with excluded middle and Skolem axioms over Σ 0 1 -formulas. Realizers are written in a classical version of Girard's System F and can be viewed as programs that learn by interacting with the environment. We show that the realizers of any Π 0 2 -formula represent terminating learning processes whose outcomes are numerical witnesses for the existential quantifier of the formula.

Bar recursion over finite partial functions

“…On top of this, due to the fact that it often avoids unnecessary backtracking, we believe that our symmetric recursor will often give rise to programs that are more natural and easier to understand from an algorithmic perspective. In particular, we conjecture that there are close links between the symmetric bar recursive interpretation of choice over sequences law of excluded middle for Σ 1 -formulas (of which the instance of AC N here is an example) and the learning procedures for PA + EM 1 described in [1].…”

“…The basic formal theory we work over is fully extensional 1 Heyting arithmetic E-HA ω in all finite types (and its classical variant E-PA ω ), whose quantifier-free fragment is Gödel's system T of primitive recursive functionals (see [14,21] for full details). The finite types T are typically defined using the following inductive rules…”

“…Spector approached this by tackling a stronger problem, namely to solve the underlying system of equations ϕf = n f n = ε n p qf = p(ε n p) (1) in f , p and n. We call the equations (1) Spector's equations, and the issue of solving them the countable choice problem. It is clear that a solution Spector's equations is also a realizer for DNS, even independent of the formula B, and thus to extend the ND interpretation to classical analysis it suffices to find a general solution to the countable choice problem.…”

“…To summarise, so far in this section we have outlined Spector's well-known reduction of the problem of realizing the extension of the ND interpretation of classical analysis to that of solving the set of equations (1). We then recounted his standard bar recursive solution to these equations, and followed this with a novel solution using a new, symmetric variant of bar recursion.…”

Game Semantics and the Geometry of Backtracking: A New Complexity Analysis of Interaction

“…In Arithmetic the landscape is already partially studied, thanks to the correspondence between the excluded middle EM n over formulas with n quantifiers and the concept of learning. In [4] it was shown, by means of Interactive realizability [5], that every arithmetical proof using only EM 1 as classical principle represents a winning recursive strategy for the related 1-backtracking game in the sense of [11]. In [11] was proved -in a less refined way -that any arithmetical formula derivable using EM n as a classical principle, has a winning recursive strategy in n-backtracking games.…”