Turing or Cantor: That is the Question
For theoretical computer scientists, this paper proposes new complexity classes and a measure for undecidable problems, but the claims are unsubstantiated and the significance is unclear.
This paper argues that Turing's work was influenced by Cantor, proposes a probabilistic measure of undecidability, extends Turing's work to super-Turing models, defines three new complexity classes for undecidable problems (U-complete, D-complete, H-complete), and claims to have negatively answered an equivalent of P≠NP for the U-complete class. No concrete results or numbers are provided.
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.