Bounded queries to SAT and the Boolean hierarchy

We study classes of sets and functions computable by algorithms that make a limited number of queries to an oracle We distinguish between queries made in parallel (each question being independent of the answers to the others, as in a truth-table reduction) and queries made in serial (each question being permitted to depend on the answers to the previous questions, as in a Turing reduction)
We define computability by a set of functions, and we show that it captures the information-theoretic aspects of computability by a fixed number of queries to an oracle Using that concept, we prove a very powerful result, the Nonspeedup Theorem, which states that 2$\sp{n}$ parallel queries to any fixed nonrecursive oracle cannot be answered by an algorithm that makes only n queries to any oracle whatsoever This is the tightest general result possible A corollary is the intuitively obvious, but nontrivial result that additional parallel queries to an oracle allow us to compute additional functions; the same is true of serial queries
We show that if k + 1 parallel queries to the oracle A can be answered by an algorithm that makes only k serial queries to any oracle B, then n parallel queries to the oracle A can be answered by an algorithm that makes only O(log n) parallel queries to a third oracle C
We also consider polynomial time bounded algorithms that make a fixed number of queries to an oracle It has been shown that the Nonspeedup Theorem does not apply in the polynomial time bounded framework However, we prove a Weak Nonspeedup Theorem, which states that if 2$\sp{k}$ parallel queries to the oracle A can be answered by an algorithm that makes only k serial queries to the oracle B, then any n parallel queries to the oracle A can be answered by an algorithm that makes only 2$\sp{k}$ $-$ 1 of the same queries to A A corollary is that if A is NP-hard and P $\not=$ NP, then extra parallel queries to A allow us to compute extra functions in polynomial time; the same is true of serial queries

Query-limited reducibilities

The frontier of knowledge about the structural properties of sparse sets is explored. A collection of topics that are related to the issue of how hard or easy sparse sets is surveyed. The strongest currently known results, together with the open problems that the results leave, are presented. >

How hard are sparse sets

This paper is concerned with three basic questions about sparse sets: (1) With respect to what types of reductions might NP have hard or complete sparse sets? (2) If a set A reduces to a sparse set, does it follow that A is reducible to some sparse set that is "simple" relative to A? (3) With respect to what types of reductions might NP have hard or complete sets of low instance complexity, and, relatedly, what is the structure of the class of sets with low instance complexity? .pp With respect to the first and third questions, intuitively one would expect that even with respect to flexible reductions NP is unlikely to have complete sets whose information content is low. With respect to the second question, one might intuitively feel that the structure imposed on a set by the fact that it reduces to a sparse set makes it plausible that we can indeed find a simple sparse set that can masquerade as the original sparse set. These two intuitions are in many ways certified by the current literature, and by the results of this paper.

Reductions to sets of low information content

We prove that if P # NP, then there exists a set in NP that is polynomial time bounded truth-table reducible (in short, <Ft,-reducible) to no sparse set. In other words, we prove that no sparse <[tt-hard set exists for NP unless P = NP. By using the technique proving this result, we investigate intractability of several number theoretic decision problems, i.e., decision problems defined naturally from number theoretic problems. We show that for those number theoretic decision problems, if it is not in P, then it is _<Ft,-reducible to no sparse set.

On polynomial time bounded truth-table reducibility of NP sets to sparse sets

We study sets that are truth-table reducible to sparse sets in polynomial time. The principal results are as follows: (1) For every integer $k > 0$, there is a set L and a sparse set S such that $L...

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Papers

On sets truth-table reducible to sparse sets