Gödel regarded the Dialectica interpretation as giving constructive content to intuitionism, which otherwise failed to meet reasonable conditions of constructivity. He founded his theory of primitive recursive functions, in which the interpretation is given, on the concept of computable function of finite type. I will 1) criticize this foundation, 2) propose a quite different one, and 3) note t...

Safety standards (e.g., ISO 26262) define safety life-cycles to be adopted for the development of safety-critical systems. Professionals (i.e., safety engineers, safety managers, and, more broadly safety cultureaware personnel) who are responsible of the development of such systems can be, in turn, considered as safety-critical systems. Coursemodules aimed at forming such professionals are crit...

Following the approaches and motivations given in recent works about constructive interpretation of description logics, we introduce the constructive description logic KALC. This logic is based on a Kripke-style semantics inspired by the Kripke semantics for Intuitionistic first order logic. In the paper we present the main features of our semantics and we study its relations with other approac...

These are lecture notes for a tutorial seminar which I gave at a satellite seminar of “Computability and Complexity in Analysis 2004” in Kyoto. The main message of the notes is that computable mathematics is the realizability interpretation of constructive mathematics. The presentation is targeted at an audience which is familiar with computable mathematics but less so with constructive mathema...

But it is beyond the power of human capacity to frame and retain distinct ideas of all the particular things we meet with: every bird and beast men saw; every tree and plant that affected the senses, could not find a place in the most capacious understanding. During the Middle Ages, much of the debate about universals concerned the question whether or not they could be considered to " exist ". ...

Realizability theory is not just a fundamental tool in logic and computability. It also has direct application to the design and implementation of programs, since it can produce code interfaces for the data structure corresponding to a mathematical theory. Our tool, called RZ, serves as a bridge between the worlds of constructive mathematics and programming. By using the realizability interpret...

According to Gödel’s original Dialectica Interpretation from 1958 a proposition is a pair of types X and Y together with a decidable relation R ⊆ X×Y between them. For our purposes we assume that types are subsets of N and (constructive) functionals between them are total recursive functions between sets of natural numbers. Entailment between propositions (X, Y,R) and (U, V, S) is given by a pa...

We study a realisability interpretation for interleaved inductive and coinductive definitions and discuss its application to program extraction in constructive analysis. A speciality of this interpretation is that realisers are given by terms that correspond directly to programs in a lazy functional programming language such as Haskell.