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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.

One of the roots of evolutionary computation was the idea of Turing about unorganized machines. The goal of this work is the development of foundations for evolutionary computations, connecting Turing's ideas and the contemporary state of art in evolutionary computations. To achieve this goal, we develop a general approach to evolutionary processes in the computational context, building mathematical models of computational systems, functioning of which is based on evolutionary processes, and studying properties of such systems. Operations with evolutionary machines are described and it is explored when definite classes of evolutionary machines are closed with respect to basic operations with these machines. We also study such properties as linguistic and functional equivalence of evolutionary machines and their classes, as well as computational power of evolutionary machines and their classes, comparing of evolutionary machines to conventional automata, such as finite automata or Turing machines.