On sets truth-table reducible to sparse sets
01 Oct 1988-SIAM Journal on Computing (Society for Industrial and Applied Mathematics)-Vol. 17, Iss: 5, pp 903-919
TL;DR: This work studies sets that are truth-table reducible to sparse sets in polynomial time and results show that for every integer k > 0, there is a set L and a sparse set S such that $L...
Abstract: 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 $L...
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TL;DR: The complexity of decision problems that can be solved by a polynomial-time Turing machine that makes a bounded number of queries to an NP oracle is studied and the Boolean hierarchy and the bounded query hierarchies either stand or collapse together are studied.
142 citations
Cites background from "On sets truth-table reducible to sp..."
...(In [11] serial queries are called adaptive, and parallel queries are called nonadaptive....
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01 Jun 1988
TL;DR: The Nonspeedup Theorem is proved, which states that 2$\sp{n}$ parallel queries to any fixed nonrecursive oracle cannot be answered by an algorithm that makes only n queries toAny oracle whatsoever.
Abstract: 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 $
ot=$ NP, then extra parallel queries to A allow us to compute extra functions in polynomial time; the same is true of serial queries
89 citations
22 Jun 1992
TL;DR: The frontier of knowledge about the structural properties of sparse sets is explored and the strongest currently known results, together with the open problems that the results leave, are presented.
Abstract: 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. >
70 citations
Book•
01 Dec 1993
TL;DR: In this article, it was shown 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.
Abstract: 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.
69 citations
01 Apr 1990
References
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TL;DR: It is shown that the class of sets of small generalized Kolmogorov complexity is exactly theclass of sets which are P-isomorphic to a tally language.
Abstract: P-printable sets arise naturally in the.studies of generalized Kolmogorov complexity and data compression, as well as in other areas. We present new characterizations of the P-printable sets and present necessary and sufficient conditions for the existence of sparse sets in P that are not P-printable. As a corollary to one of our results, we show that the class of sets of small generalized Kolmogorov complexity is exactly the class of sets which are P-isomorphic to a tally language.
125 citations
TL;DR: It is proved that the polynomial-time hierarchy collapses if and only if for every sparse set S, the hierarchy relative to S collapses and the question is answered if it is answered for any arbitrary sparse oracle set.
Abstract: Questions about the polynomial-time hierarchy are studied. In particular, the questions, “Does the polynomial-time hierarchy collapse?” and “Is the union of the hierarchy equal to PSPACE?” are considered, along with others comparing the union of the hierarchy with certain probabilistic classes. In each case it is shown that the answer is “yes” if and only if for every sparse set S, the answer is “yes” when the classes are relativized to S if and only if there exists a sparse set S such that the answer is “yes” when the classes are relativized to S. Thus, in each case the question is answered if it is answered for any arbitrary sparse oracle set.Long and Selman first proved that the polynomial-time hierarchy collapses if and only if for every sparse set S, the hierarchy relative to S collapses. This result is re-proved here by a different technique.
102 citations
TL;DR: The class of sets with small generalized Kolmogorov complexity is properly included in the class of “self-p-printable” sets and its results are established.
Abstract: We study the class of sets with small generalized Kolmogorov complexity. The following results are established: 1. A set has small generalized Kolmogorov complexity if and only if it is “semi-isomorphic” to a tally set. 2. The class of sets with small generalized Kolmogorov complexity is properly included in the class of “self-p-printable” sets. 3. The class of self-p-printable sets is properly included in the class of sets with “selfproducible circuits”. 4. A set S has self-producible circuits if and only if there is a tally set T such that P(T)=P(S). 5. If a set S has self-producible circuits, then NP(S)=NPB(S), where NPB( ) is the restriction of NP( ) studied by Book, Long, and Selman [4]. 6. If a set S is such that NP(S) =NPB(S), then NP(S)\( \subseteq\)P(S⊕SAT).
64 citations