Uwe Schöning

A low and a high hierarchy within NP

A low and a high hierarchy within NP are defined. The definition is similar to the jump hierarchies below the degree of the halting problem. For this purpose a complexity theoretic counterpart of the jump operator in recursion theory is defined. Some elementary properties of these hierarchies are investigated. The high hierarchy is, in some sense, a hierarchy of generalized NP-completeness notions.

A uniform approach to obtain diagonal sets in complexity classes

Abstract A uniform method for constructing sets which diagonalize over certain complexity classes while preserving other complexity properties is given. We obtain some known results as well as some new ones as corollaries of our main theorem. The new results concern the complexity classes P, NP, co-NP, PSPACE, APT (almost polynomial time), R (random polynomial time), and the polynomial hierarchy.

The polynomial-time hierarchy and sparse oracles

Questions about the polynomial-time hierarchy are studied. In particular, the questions, “Does the polynomial-time hierarchy collapse?” and “Is the union of the hierarchy equal to PSPACE?” are considered, along with others comparing the union of the hierarchy with certain probabilistic classes. In each case it is shown that the answer is “yes” if and only if <italic>for every</italic> sparse set <italic>S</italic>, the answer is “yes” when the classes are relativized to <italic>S</italic> if and only if <italic>there exists</italic> a sparse set <italic>S</italic> such that the answer is “yes” when the classes are relativized to <italic>S</italic>. Thus, in each case the question is answered if it is answered for any arbitrary sparse oracle set. Long and Selman first proved that the polynomial-time hierarchy collapses if and only if for every sparse set <italic>S</italic>, the hierarchy relative to <italic>S</italic> collapses. This result is re-proved here by a different technique.

On Circuit-Size Complexity and the Low Hierarchy in NP

Let A be a set having polynomial size circuits. If A is also known to be in NP, then we may conclude that the graph of the polynomial size circuits for A is actually in

Immunity, relativizations, and nondeterminism

\Pi _2^p

Robust algorithms: a different approach to oracles

. Using this observation, we show that sets in NP which have polynomial size ciruits are in

On small generators

L_3^p

Minimal pairs for P

, the third level of the low hierarchy in NP. By a similar technique, we are able to show that some other intuitively low sets in NP are in

On bounded query machines

L_2^p

Sparse Oracles And Uniform Complexity Classes

, and even in a certain refinement of