Bounded Queries In Recursion Theory (progress In Computer Science And Applied Logic)

One of the major concerns of theoretical computer science is the classifi­ cation of problems in terms of how hard they are. The natural measure of difficulty of a function is the amount of time needed to compute it (as a function of the length of the input). Other resources, such as space, have also been considered. In recursion theory, by contrast, a function is considered to be easy to compute if there exists some algorithm that computes it. We wish to classify functions that are hard, i.e., not computable, in a quantitative way. We cannot use time or space, since the functions are not even computable. We cannot use Turing degree, since this notion is not quantitative. Hence we need a new notion of complexity-much like time or spac~that is quantitative and yet in some way captures the level of difficulty (such as the Turing degree) of a function. Front Matter....Pages i-xiii Front Matter....Pages 1-1 Basic Concepts....Pages 3-42 Bounded Queries and the Halting Set....Pages 43-52 Definitions and Questions....Pages 53-73 Front Matter....Pages 75-75 The Complexity of C n A ....Pages 77-141 # n A and Other Functions....Pages 143-182 Front Matter....Pages 183-183 The Complexity of ODD n A and MOD m n A ....Pages 185-216 Q Versus QC....Pages 217-253 Separating and Collapsing Classes....Pages 255-269 Front Matter....Pages 271-271 Nondeterministic Complexity....Pages 273-293 The Literature on Bounded Queries....Pages 295-324 Back Matter....Pages 325-353