One of the major concerns of theoretical computer science is the classifiยญ 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 spac̃that is quantitative and yet in some way captures the level of difficulty (such as the Turing degree) of a function
CONTENT
A: Getting Your Feet Wet -- 1 Basic Concepts -- 2 Bounded Queries and the Halting Set -- 3 Definitions and Questions -- B: The Complexity of Functions -- 4 The Complexity of CnA -- 5 #nA and Other Functions -- C: The Complexity of Sets -- 6 The Complexity of ODDnA and MODmnA -- 7 Q Versus QC -- 8 Separating and Collapsing Classes -- D: Miscellaneous -- 9 Nondeterministic Complexity -- 10 The Literature on Bounded Queries -- References
