Recursively invariant β-recursion theory

A representation of recursively enumerable sets through Horn formulas in higher recursion theory

We extend a classical result in ordinary recursion theory to higher recursion theory, namely that every recursively enumerable set can be represented in any model A by some Horn theory, where A can be any model of a higher recursion theory, like primitive set recursion, αrecursion, or β -recursion. We also prove that, under suitable conditions, a set defined through a Horn theory in a set A is ...

Ordinal Recursion Theory

Quantum recursion theory

Incompleteness and undecidability theorems have to be revised in view of quantum information and computation theory.

Recursion Theory I

This document presents the formalization of introductory material from recursion theory — definitions and basic properties of primitive recursive functions, Cantor pairing function and computably enumerable sets (including a proof of existence of a one-complete computably enumerable set and a proof of the Rice’s theorem).

Recursion Theory and Undecidability

In the second half of the class, we will explore limits on computation. These are questions of the form “What types of things can be computed at all?” “What types of things can be computed efficiently?” “How fast can problem XYZ be solved?” We already saw some positive examples in class: problems we could solve, and solve efficiently. For instance, we saw that we can sort n numbers in time O(n ...