Polynomial Depth, Highness and Lowness for E

On Boolean Lowness and Boolean Highness

The concepts of lowness and highness originate from recursion theory and were introduced into the complexity theory by Sch. oning (Lecture Notes in Computer Science, Vol. 211, Springer, Berlin, 1985). Informally, a set is low (high resp.) for a relativizable class K of languages if it does not add (adds maximal resp.) power to K when used as an oracle. In this paper, we introduce the notions of...

Generalized Lowness and Highness and Probabilisti Complexity Classes

Sparse Oracles, Lowness, and Highness

The polynomial-time hierarchy has been studied extensively since it lies between the class P of languages accepted deterministically in polynomial time and the class PSPACE of languages accepted (deterministically or nondeterministically) in polynomial space. Since the P =? NP problem is still unsolved, it is not known whether the hierarchy is nontrivial, although it is known that there is a re...

Randomness and lowness notions via open covers

One of the main lines of research in algorithmic randomness is that of lowness notions. Given a randomness notion R, we ask for which sequences A does relativization to A leave R unchanged (i.e., RA = R)? Such sequences are call low for R. This question extends to a pair of randomness notions R and S , where S is weaker: for which A is S A still weaker than R? In the last few years, many result...

On the Structure of Low Sets

Over a decade ago, Schh oning introduced the concept of lowness into structural complexity theory. Since then a large body of results has been obtained classifying various complexity classes according to their lowness properties. In this paper we highlight some of the more recent advances on selected topics in the area. Among the lowness properties we consider are polynomial-size circuit comple...