Transfinite Update Procedures for Predicative Systems of Analysis

We present a simple-to-state, abstract computational problem, whose solution implies the 1-consistency of various systems of predicative Analysis and offers a way of extracting witnesses from classical proofs. In order to state the problem, we formulate the concept of transfinite update procedure, which extends Avigad\u27s notion of update procedure to the transfinite and can be seen as an axiomatization of learning as it implicitly appears in various computational interpretations of predicative Analysis. We give iterative and bar recursive solutions to the problem

Learning, realizability and games in classical arithmetic

PhDAbstract. In this dissertation we provide mathematical evidence that the concept of learning can be used to give a new and intuitive computational semantics of classical proofs in various fragments of Predicative Arithmetic. First, we extend Kreisel modi ed realizability to a classical fragment of rst order Arithmetic, Heyting Arithmetic plus EM1 (Excluded middle axiom restricted to 0 1 formulas). We introduce a new realizability semantics we call \Interactive Learning-Based Realizability". Our realizers are self-correcting programs, which learn from their errors and evolve through time, thanks to their ability of perpetually questioning, testing and extending their knowledge. Remarkably, that capability is entirely due to classical principles when they are applied on top of intuitionistic logic. Secondly, we extend the class of learning based realizers to a classical version PCFClass of PCF and, then, compare the resulting notion of realizability with Coquand game semantics and prove a full soundness and completeness result. In particular, we show there is a one-to-one correspondence between realizers and recursive winning strategies in the 1-Backtracking version of Tarski games. Third, we provide a complete and fully detailed constructive analysis of learning as it arises in learning based realizability for HA+EM1, Avigad's update procedures and epsilon substitution method for Peano Arithmetic PA. We present new constructive techniques to bound the length of learning processes and we apply them to reprove - by means of our theory - the classic result of G odel that provably total functions of PA can be represented in G odel's system T. Last, we give an axiomatization of the kind of learning that is needed to computationally interpret Predicative classical second order Arithmetic. Our work is an extension of Avigad's and generalizes the concept of update procedure to the trans nite case. Trans- nite update procedures have to learn values of trans nite sequences of non computable functions in order to extract witnesses from classical proofs

Mathematical Logic: Proof theory, Constructive Mathematics

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit

A proof of strong normalisation using domain theory

Ulrich Berger presented a powerful proof of strong normalisation using domains, in particular it simplifies significantly Tait's proof of strong normalisation of Spector's bar recursion. The main contribution of this paper is to show that, using ideas from intersection types and Martin-Lof's domain interpretation of type theory one can in turn simplify further U. Berger's argument. We build a domain model for an untyped programming language where U. Berger has an interpretation only for typed terms or alternatively has an interpretation for untyped terms but need an extra condition to deduce strong normalisation. As a main application, we show that Martin-L\"{o}f dependent type theory extended with a program for Spector double negation shift.Comment: 16 page

Non-principal ultrafilters, program extraction and higher order reverse mathematics

We investigate the strength of the existence of a non-principal ultrafilter over fragments of higher order arithmetic. Let U be the statement that a non-principal ultrafilter exists and let ACA_0^{\omega} be the higher order extension of ACA_0. We show that ACA_0^{\omega}+U is \Pi^1_2-conservative over ACA_0^{\omega} and thus that ACA_0^{\omega}+\U is conservative over PA. Moreover, we provide a program extraction method and show that from a proof of a strictly \Pi^1_2 statement \forall f \exists g A(f,g) in ACA_0^{\omega}+U a realizing term in G\"odel's system T can be extracted. This means that one can extract a term t, such that A(f,t(f))

On Bar Recursive Interpretations of Analysis.

PhDThis dissertation concerns the computational interpretation of analysis via proof interpretations, and examines the variants of bar recursion that have been used to interpret the axiom of choice. It consists of an applied and a theoretical component. The applied part contains a series of case studies which address the issue of understanding the meaning and behaviour of bar recursive programs extracted from proofs in analysis. Taking as a starting point recent work of Escardo and Oliva on the product of selection functions, solutions to Godel's functional interpretation of several well known theorems of mathematics are given, and the semantics of the extracted programs described. In particular, new game-theoretic computational interpretations are found for weak Konig's lemma for 01 -trees and for the minimal-bad-sequence argument. On the theoretical side several new definability results which relate various modes of bar recursion are established. First, a hierarchy of fragments of system T based on finite bar recursion are defined, and it is shown that these fragments are in one-to-one correspondence with the usual fragments based on primitive recursion. Secondly, it is shown that the so called `special' variant of Spector's bar recursion actually defines the general one. Finally, it is proved that modified bar recursion (in the form of the implicitly controlled product of selection functions), open recursion, update recursion and the Berardi-Bezem- Coquand realizer for countable choice are all primitive recursively equivalent in the model of continuous functionals.EPSR

Computability and analysis: the legacy of Alan Turing

We discuss the legacy of Alan Turing and his impact on computability and analysis.Comment: 49 page