In this article, we study the lattice of Lawvere-Tierney topologies on Hyland’s effective topos. For purpose, introduce a new computability-theoretic reducibility notion, which is common extension notions Turing and generalized Weihrauch reducibility. Based work by Lee van Oosten [Ann. Pure Appl. Logic 164 (2013), pp. 866-883], utilize notion for providing concrete description As an application...

Turing’s beautiful capture of the concept of computability by the “Turing machine” linked computability to a device with explicit steps of operations and use of resources. This invention led in a most natural way to build the foundations for computational complexity.

The Connes Embedding Problem (CEP) asks whether every separable II1 factor embeds into an ultrapower of the hyperfinite II1 factor. We show that the CEP is equivalent to the computability of the universal theory of every type II1 von Neumann algebra. We also derive some further computability-theoretic consequences of the CEP.

This thesis presents a series of theoretical results and practical realisations about the theory of computation in distributive categories. Distributive categories have been proposed as a foundational tool for Computer Science in the last years, starting from the papers of R.F.C. Walters. We shall focus on two major topics: distributive computability, i.e., a generalized theory of computability...

In this paper, we first briefly survey automated termination proof methods for higher-order calculi. We then concentrate on the higher-order recursive path ordering, for which we provide an improved definition, the Computability Path Ordering. This new definition appears indeed to capture the essence of computability arguments à la Tait and Girard, therefore explaining the name of the improved ...

While event handling is a key element in modern interactive programming, it is unfortunate that its theoretical foundation is rather weak. To solve this problem, we propose to adopt a game-logical approach of computability logic [2] to event handling. keywords: event handling, game semantics, interaction, computability logic.

concepts that cut across different areas of mathematics. Logical formality per se does not facilitate the finding of arguments from given assumptions to a particular conclusion. However, strategic considerations can be formulated (for natural deduction calculi) and help to bridge the gap between assumptions and conclusion, suggesting at least a very rough structure of arguments. These logical s...

0. INTRODUCTION. Computability is perhaps the most significant and distinctive notion modern logic has introduced; in the guise of decidability and effective calculability it has a venerable history within philosophy and mathematics. Now it is also the basic theoretical concept for computer science, artificial intelligence and cognitive science. This essay discusses, at its heart, methodologica...