Relativization of the theory of computational complexity

Abstract

Blum''s machine-independent treatment of the complexity of partial recursive functions is extended to relative algorithms (as represented by Turing machines with oracles). We prove relativizations of several results of Blum complexity theory, such as the compression theorem. A recursive relatedness theorem is proved, showing that any two relative complexity measures are related by a fixed recursive function. This theorem allows us to obtain proofs of results for all measures from proofs for a particular measure. We study complexity-determined reducibilities, the parallel notion to complexity classes for the relativized case. Truth-table and primitive recursive reducibilities are reducibilities of this type, while other commonly-studied reducibilities are not. We formalize the concept of a set the computation of function (by causing a saving in resource when used as an oracle in the computation of the function). Basic properties of the helping relation are proved, including non-transitivity and bounds on the amount of help certain sets can provide. Several independence results (results about sets that don''t help each other''s computation) are proved; they are subrecursive analogs to degrees-of-unsolvability theorems, with similar proofs using diagonalization and priority arguments. In particular, we discuss the existence of a universally-helped set, obtaining partial results in both directions. The deepest result is a finite-injury priority argument (without an apparent recursive bound on the number of injuries) which produces sets preserving an arbitrary lower bound on the complexity of a set. Our methods of proof include proof for a simple measure (e.g. space) and appeal to recursive relatedness, diagonalization and priority techniques, and heavy use of arguments about the domain of convergence of partial recursive functions in order to define total recursive functions.