On Computing Boolean Connectives of Characteristic Functions

Abstract

We study 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 $f(x,y) \in L \Longleftrightarrow x \in L$ and $y \in L$. $L$ has an OR function if there is a polynomial time $g$ such that $g(x,y) \in L \Longleftrightarrow x \in L$ or $y \in L$. While all NP-complete sets have these functions, we show that Graph Isomorphism, which is probably not complete, also has them. We characterize the complete sets for the classes $D^{P}$ and $P^{NP[O(log n)]}$ in terms of AND and OR, and we relate these functions to the structure of the Boolean hierarchy and the query hierarchies. We show that the sets that are complete for levels above the second level of the Boolean hierarchy do not have AND and OR unless the polynomial hierarchy collapses. We show that most of the structural properties of the Boolean hierarchy and query hierarchies depend only on the existence of AND and OR functions for NP-complete sets.