$\beta $-recursion theory

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 ...

Recursion Theory and Symbolic Dynamics

A set P ⊆ {0,1}N may be viewed as a mass problem, i.e., a decision problem with more than one solution. By definition, the solutions of P are the elements of P . A mass problem is said to be solvable if at least one of its solutions is recursive. A mass problem P is said to be Muchnik reducible to a mass problem Q if for each solution of Q there exists a solution of P which is Turing reducible ...