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On Completeness of Cost Metrics and Meta-Search Algorithms in $-Calculus

Eberbach Eugene

Fundamenta Informaticae

2022

Vol. 188 nr 2

63--90

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.

On Hypercomputation, Universal and Diagonalization Complete Problems

Eberbach E.

Fundamenta Informaticae

2015

Vol. 139, nr 4

329--346

In the paper we define the class of hypercomputation complete and hard undecidable problems. We prove that typical unsolvable problems are decidable in infinity. We hypothesize that all undecidable problems can be reduced in infinity to each other, i.e., they are H-complete (hypercomputation complete). We propose also two other classes: U-complete (universal complete) and D-complete (diagonalization complete) that use asymptotic reduction and allow separate non- RE and RE non-recursive undecidable classes.