Abstract

Balc'azar, Book, and Schoning introduced the extended low hierarchy based on the \Sigmalevels of the polynomial-time hierarchy as follows: for k 1, level k of the extended low hierarchy is the set EL P;\Sigma k = fA j \Sigma P k (A) ` \Sigma P k\Gamma1 (A \Phi SAT)g. Allender and Hemachandra and Long and Sheu introduced refinements of the extended low hierarchy based on the \Delta and \Theta-levels, respectively, of the polynomial-time hierarchy: for k 2, EL P;\Delta k = fA j \Delta P k (A) ` \Delta P k\Gamma1 (A \Phi SAT)g and EL P;\Theta k = fA j \Theta P k (A) ` \Theta P k\Gamma1 (A \Phi SAT)g. In this paper we show that the extended low hierarchy is properly infinite by showing, for k 2, that EL P;\Sigma k ` / EL P;\Theta k+1 ` / EL P;\Delta k+1 ` / EL P;\Sigma k+1 . Our proofs use the circuit lower bound techniques of Hastad and Ko. As corollaries to our constructions, we obtain, for k 2, oracle sets B k , C k , and D k , such that PH(B k ) = \Sigma P k (B k )...

Citations