Feasible Computability and Resource Bounded Topology

• Published in: Information and computation Vol. 116; no. 2; pp. 214 - 230
• Main Author: Yamakami, T.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.02.1995; Elsevier
• Subjects: Applied sciences; Automata. Abstract machines. Turing machines; Computer science; control theory; systems; Exact sciences and technology; Theoretical computing; Hierarchy; Computation time; Set theory; Turing machine; Computability; Polynomial time
• ISSN: 0890-5401; 1090-2651
• DOI: 10.1006/inco.1995.1015

Townsend introduced a resource-bounded extension of polynomial-time computable functions on strings to type-two functionals, and studied a type-two version of the Meyer-Stockmeyer polynomial hierarchy which is founded on polynomial-time computable functionals of type two. A functional of type two is polynomial-time computable if it is computed by a deterministic function-oracle Turing machine whose runtime is bounded by a polynomial that does not depend on the choice of oracle functions. Townsend also introduced a boldface polynomial hierarchy of type two by a relativization method, and gave a "topological" characterization of the first level of this hierarchy. We reformulate Townsend′s topological notions associated with time-bounded computations of function-oracle Turing machines, and further extend his "topological" characterization to all levels of the boldface polynomial hierarchy of type two. This leads to a polynomialized version of descriptive set theory.