Affiliations: Department of Engineering and Science, Rensselaer Polytechnic Institute, Hartford, CT, USA, [email protected]
Correspondence:
[†]
Address for correspondence: Deprtment of Engineering and Sci., Rensselaer Polytechnic Institute, Hartford, CT, USA.
Note: [*] retired Professor of Practice, RPI Hartford, CT. This paper has been written in memories of Professor Peter Wegner from Brown University and Professor Mark Burgin from University of California Los Angeles with whom I had the honor and pleasure to cooperate for several years.
Abstract: In the paper we define three new complexity classes for Turing Machine undecidable problems inspired by the famous Cook/Levin’s NP-complete complexity class for intractable problems. These are U-complete (Universal complete), D-complete (Diagonalization complete) and H-complete (Hypercomputation complete) classes. In the paper, in the spirit of Cook/Levin/Karp, we started the population process of these new classes assigning several undecidable problems to them. We justify that some super-Turing models of computation, i.e., models going beyond Turing machines, are tremendously expressive and they allow to accept arbitrary languages over a given alphabet including those undecidable ones. We prove also that one of such super-Turing models of computation - the $-Calculus, designed as a tool for automatic problem solving and automatic programming, has also such tremendous expressiveness. We investigate also completeness of cost metrics and meta-search algorithms in $-calculus.
