Are Recursion Theoretic Arguments Useful in Complexity Theory?

Abstract

Recursion theory is that area of mathematical logic where one studies the qualitative aspects of computability. In complexity theory, which is part of computer science, one studies in addition quantitative aspects of computations. A number of open problems about the structure of NP where one can prove that even under the assumption PI≠NP recursion theoretic arguments will not suffice. The chapter presents polynomial time approximation schemes for some strongly NP-complete problems that arise—for example, in robotics. The chapter presents a survey for a proof of optimal lower bounds for two tapes versus one on deterministic and nondeterministic Turing machines. Results that show a substantial superiority of nondeterminism over determinism resp. co-nondeterminism over nondeterminism for one-tape Turing machines are given in the chapter. The chapter presents the proof of the desired result as the construction of a winning strategy for a two-person game. © 1986, Elsevier B.V. All rights reserved.