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

Abstract

In the paper we define three new complexity classes for Turing Machineundecidable problems inspired by the famous Cook/Levin's NP-complete complexityclass for intractable problems. These are U-complete (Universal complete),D-complete (Diagonalization complete) and H-complete (Hypercomputationcomplete) classes. In the paper, in the spirit of Cook/Levin/Karp, we startedthe population process of these new classes assigning several undecidableproblems to them. We justify that some super-Turing models of computation,i.e., models going beyond Turing machines, are tremendously expressive and theyallow to accept arbitrary languages over a given alphabet including thoseundecidable ones. We prove also that one of such super-Turing models ofcomputation - the \$-Calculus, designed as a tool for automatic problem solvingand automatic programming, has also such tremendous expressiveness. Weinvestigate also completeness of cost metrics and meta-search algorithms in\$-calculus