On computing Boolean connectives of characteristic functions

This paper is a study of the existence of polynomial time Boolean connective functions for languages. A language L has an AND function if there is a polynomial time f such that f(x, y) ∈ L ⇐⇒ x ∈ L and y ∈ L. L has an OR function if there is a polynomial time g such that g(x, y) ∈ L ⇐⇒ x ∈ L or y ∈ L. While all NP complete sets have these functions, Graph Isomorphism, which is probably not complete, is also shown to have both AND and OR functions. The results in this paper characterize the complete sets for the classes DP and PSAT(O(log n)) in terms of AND and OR, and relate these functions to the structure of the Boolean hierarchy and the query hierarchies. Also, this paper shows that the complete sets for the levels of the Boolean hierarchy above the second level cannot have AND or OR unless the polynomial hierarchy collapses. Finally, most of the structural properties of the Boolean hierarchy and query hierarchies are shown to depend only on the existence of AND and OR functions for the NP complete sets.