Some connections between bounded query classes and nonuniform complexity

It is shown that if there is a polynomial-time algorithm that tests k(n)=O(log n) points for membership in a set A by making only k(n)-1 adaptive queries to an oracle set X, then A belongs to NP/poly intersection co-NP/poly (if k(n)=O(1) then A belong to P/poly). In particular, k(n)=O(log n) queries to an NP -complete set (k(n)=O(1) queries to an NP-hard set) are more powerful than k(n)-1 queries, unless the polynomial hierarchy collapses. Similarly, if there is a small circuit that tests k(n) points for membership in A by making only k(n)-1 adaptive queries to a set X, then there is a correspondingly small circuit that decides membership in A without an oracle. An investigation is conducted of the quantitatively stronger assumption that there is a polynomial-time algorithm that tests 2^k strings for membership in A by making only k queries to an oracle X, and qualitatively stronger conclusions about the structure of A are derived: A cannot be self-reducible unless A∈P, and A cannot be NP-hard unless P=NP. Similar results hold for counting classes. In addition, relationships between bounded-query computations, lowness, and the p-degrees are investigated