The Extended Low Hierarchy Is an In niteHierarchy 1

Balc azar, Book, and Sch oning introduced the extended low hierarchy based on the levels of the polynomial-time hierarchy as follows: for k 1, level k of the extended low hierarchy is the set ELP; k = fA j Pk (A) Pk 1(A SAT)g. Allender and Hemachandra and Long and Sheu introduced re nements of the extended low hierarchy based on the and -levels, respectively, of the polynomial-time hierarchy: for k 2, ELP; k = fA j Pk (A) Pk 1(A SAT)g and ELP; k = fA j Pk (A) Pk 1(A SAT)g. In this paper we show that the extended low hierarchy is properly in nite by showing, for k 2, that ELP; k / ELP; k+1 / ELP; k+1 / ELP; k+1. Our proofs use the circuit lower bound techniques of H astad and Ko. As corollaries to our constructions, we obtain, for k 2, oracle sets Bk, Ck, and Dk, such that PH(Bk) = Pk (Bk) / Pk (Bk), PH(Ck) = Pk (Ck) / Pk (Ck), and PH(Dk) = Pk (Dk) / Pk 1(Dk).