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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.

Universality for Turing Machines, Inductive Turing Machines and Evolutionary Algorithms

Burgin M., Eberbach E.

Fundamenta Informaticae

2009

Vol. 91, nr 1

53-77

The aim of this paper is the development of foundations for evolutionary computations. To achieve this goal, a mathematical model of evolutionary automata is introduced and studied. The main classes of evolutionary automata considered in this paper are evolutionary Turing machines and evolutionary inductive Turing machines. Various subclasses and modes of evolutionary computation are defined. Problems of existence of universal objects in these classes are explored. Relations between Turing machines, inductive Turing machines, evolutionary Turing machines, and evolutionary inductive Turing machines are investigated.