Eugene Eberbach

Eugene Eberbach

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.

Eugene Eberbach

Evolution by natural selection, which is one of the most compelling themes of modern science, brought forth evolutionary algorithms and evolutionary computation, applying mechanisms of evolution in nature to various problems solved by computers. In this paper we concentrate on evolutionary automata that constitute an analogous model of evolutionary computation compared to well-known evolutionary algorithms. Evolutionary automata provide a more complete dual model of evolutionary computation, similar like abstract automata (e.g., Turing machines) form a more formal and precise model compared to recursive algorithms and their subset - evolutionary algorithms. An evolutionary automaton is an automaton that evolves performing evolutionary computation perhaps using an infinite number of generations. This model allows for a direct modeling evolution of evolution, and leads to tremendous expressiveness of evolutionary automata and evolutionary computation. This also gives the hint to the power of natural evolution that is self-evolving by interactive feedback with the environment.

Mark Burgin, Eugene Eberbach

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.