Type-2 complexity theory

The notion of type-2 computability occurs naturally in many practical and theoretical settings in computer science. For examples, machine learning, programing languages, databases enquiry, complexity-theoretic problem reductions, and so on, are immediate applications of type-2 computation. However, there is no satisfactory type-2 complexity theory to characterize the computational cost of these widely ranged applications. Thus, the purpose of this thesis is to give a theoretical framework for analyzing the complexity of type-2 computation.; We use the Oracle Turing Machine (OTM) as our standard formalism for type-2 computation. The best way to characterize the computational cost of type-2 computation is to give a robust notion of type-2 complexity classes. In order to do so, we first study the induced topologies determined by type-2 continuous functionals of type (N → N) × N $\rightharpoonup$ N. Then, based on the compact sets in the induced topologies, we define a type-2 almost-everywhere relation $\le *2$ over type-2 continuous functionals. The type-2 almost-everywhere relation $\le *2$ provides an analogous notion of asymptotic approach for complexity analysis in type-2. We also specify a clocking scheme for OTMs based on a class of computable functions called Type-2 Time Bounds ( T₂TB). With the tools we developed, each type-2 time bound β ∈ T₂TB determines a type-2 complexity class C(β). We also define a type-2 big-O notation—O(β)—which would be a useful tool for type-2 algorithm analysis.; To justify our notion of type-2 complexity classes, we prove the Union Theorem, the Gap Theorem, the Compression Theorem, and the Speed-up Theorem in type-2 along the lines of classical complexity theory. Most of the theorems we proved are very different from their type-1 counterparts. We thus learn that the structure of type-2 complexity classes is not as sturdy as the structure in type-1; they are very sensitive to the topological constraint. With theses complexity results, we have a reasonable outlook for a general type-2 complexity theory.