Uniformity, universality, and computability theory

Uniformity, Universality, and Computability Theory

We prove a number of results motivated by global questions of uniformity in computability theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened ...

Uniformity, Universality, and Recursion Theory

We prove a number of results motivated by global questions of uniformity in recursion theory, and some longstanding open questions about universality of countable Borel equivalence relations. Our main technical tool is a class of games for constructing functions on free products of countable groups. These games show the existence of refinements of Martin’s ultrafilter on Turing invariant sets t...

Computability Theory

Computability Theory

formalisms of computability

Computability Theory

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...