Interactive Realizability, Monads and Witness Extraction

In this dissertation we collect some results about “interactive realizability”, a realizability semantics that extends the Brouwer-Heyting-Kolmogorov interpretation to (sub-)classical logic, more precisely to first-order intuitionistic arithmetic (Heyting Arithmetic, HA) extended by the law of the excluded middle restricted to Σ1 formulas (EM1), a system motivated by its interest in proof mining. These results are three interconnected works, listed below. • We describe the interactive interpretation of a classical proof involving real numbers. The statement we prove is a simple but non-trivial fact about points in the real plane. The proof employs EM1 to deduce properties of the ordering on the real numbers, which is undecidable and thus problematic from a constructive point of view. • We present a new set of reductions for derivations in natural deduction that can extract witnesses from closed derivations of simply existential formulas in HA + EM1. The reduction we present are inspired by the informal idea of learning by making falsifiable hypothesis and checking them, and by the interactive realizability interpretation. We extract the witnesses directly from derivations in HA + EM1 by reduction, without encoding derivations by a realizability interpretation. • We give a new presentation of interactive realizability with a more explicit syntax. We express interactive realizers by means of an abstract framework that applies the monadic approach used in functional programming to modified realizability, in order to obtain less strict notions of realizability that are suitable to classical logic. In particular we use a combination of the state and exception monads in order to capture the learning-from-mistakes nature of interactive realizers.