Bounded Query Functions with Limited Output Bits

This paper explores the difference between parallel and serial queries to an NP-complete oracle, SAT, from the perspective of functions with a limited number of output bits. For polynomial-time bounded query language classes, which can be considered as functions with 1-bit output, previous work has shown that 2 serial queries to SAT is equivalent to 3 parallel queries to SAT. In contrast, for function classes with no limit on the number of output bits, previous work has shown that there exists a function that can be computed in polynomial time using 3 parallel queries to SAT, but cannot be computed using 2 serial queries to SAT, unless P = NP. The results in this paper show that there exists a function with 2-bit output that can be computed using 3 parallel queries to SAT, but cannot be computed using 2 serial queries to SAT, unless the Polynomial Hierarchy collapses.