It is shown that all the provably total functions of Basic Arithmetic , a theory introduced by Ruitenburg based on Predicate Basic Calculus, are primitive recursive. Along the proof a new kind of primitive recursive realizability to which is sound, is introduced. This realizability is similar to Kleene’s recursive realizability, except that recursive functions are restricted to primitive recurs...

For input/ouput (i/o) operators, an equivalence is shown between realizability by state space systems and the existence of analytic constraints on higher order derivatives of i/o signals. This provides a precise characterization of realizability, extending to the general analytic case previous work that dealt with the equivalence between algebraic realizability and algebraic i/o equations.

Introduced by Dal Lago and Hofmann, quantitative realizability is a technique used to define models for logics based on Multiplicative Linear Logic. A particularity is that functions are interpreted as bounded time computable functions. It has been used to give new and uniform proofs of soundness of several type systems with respect to certain time complexity classes. We propose a reformulation...

We study the problem of realizing a given graph as an α-complex of a set of points in the plane. We study the realizability problem for trees and 2-trees. In the case of 2-trees, we confine our attention to the realizability of graphs as the α-complex minus faces of dimension two; in other words, realizability of the graph in terms of the 1-skeleton of the α-complex of the point set. We obtain ...

In the proof, we used both finite-ness of H and the realizability assumptions in a crucial way. But we already saw one problem where the theorem does not apply, namely learning linear separators, where |H| =∞. Of course it is more common than not that realizability is not satisfied, so we’d also like some sort of guarantee in the absence of realizability. In the next few lectures we will build ...

Early investigators of realizability were interested in metamathematical questions. In keeping with the traditions of the time they concentrated on interpretations of one formal system in another. They considered an ad hoc collection of increasingly ingenious interpretations to establish consistency, independence and conservativity results. van Oosten’s contribution to the Workshop (see van Oos...