A Structural Theorem that Depends Quantitatively on the Complexity of SAT

We consider the class of functions that can be computed with a bounded number of queries to a given oracle A, in a bounded amount of time. We ask whether additional queries allow us to compute strictly more functions within the same time bound. For almost all oracles, the answer to that question is yes; thus "number of queries" is an important measure of the complexity of a function. We define a new notion of computability by a set of functions that seems to capture the information-theoretic aspects of oracle computations. We also consider the notion of terseness of an oracle, which originally appeared in [BGGO86]. Combining these two notions yields a theorem about the number of oracle queries that suffice to compute a function. An important class of problems is Δ $_\text{2}^\text{P}$ (i.e., P ^NP , the class of all decision problems that can be solved in polynomial time with an otherwise unbounded number of queries to a SAT oracle). It is known that an unbounded number of queries to a Δ $_\text{2}^\text{P}$ complete oracle do not allow us to solve more decision problems than we can solve with just a single query to that oracle. However, we show that additional queries to a Δ $_\text{2}^\text{P}$ complete oracle do allow us to compute strictly more functions, if we make appropriate assumptions about the complexity of SAT.