Turing or Cantor: That is the Question

TL;DR

This paper explores the foundational links between Turing's work and Cantor's set theory, proposing new complexity classes and measures for undecidable problems, extending classical computation theory.

Contribution

It introduces new complexity classes for undecidable problems, proposes measures of problem undecidability, and extends Turing's models to super-Turing computation.

Findings

Defined three new complexity classes: U-complete, D-complete, H-complete.

Proposed a measure of undecidability based on input data probability distribution.

Answered negatively the equivalent of P ≠ NP question for U-complete class.

Abstract

Alan Turing is considered as a founder of current computer science together with Kurt Godel, Alonzo Church and John von Neumann. In this paper multiple new research results are presented. It is demonstrated that there would not be Alan Turing's achievements without earlier seminal contributions by Georg Cantor in the set theory and foundations of mathematics. It is proposed to introduce the measure of undecidability of problems unsolvable by Turing machines based on probability distribution of its input data, i.e., to provide the degree of unsolvabilty based on the number of undecidable instances of input data versus decidable ones. It is proposed as well to extend the Turing's work on infinite logics and Oracle machines to a whole class of super-Turing models of computation. Next, the three new complexity classes for TM undecidable problems have been defined: U-complete (Universal complete), D-complete (Diagonalization complete) and H-complete (Hypercomputation complete) classes. The above has never been defined explicitly before by other scientists, and has been inspired by Cook/Levin NP-complete class for intractable problems. Finally, an equivalent to famous P is not equal to NP unanswered question for NP-complete class, has been answered negatively for U-complete class of complexity for undecidable problems.