## Abstract

Let A(x) be the characteristic function of A. Consider the function C _{k}^{A}(x1,...,xk) = A(x1)... A(xk). We show that if C _{k}^{A} can be computed in polynomial time with fewer than k queries to some set X then A ∈ P/poly. A generalization of this result has applications to bounded query classes, circuits, and enumerability. In particular we obtain the following. (1) Assuming ∑_{3}^{p} ∏_{3}^{p}, there are functions computable using f(n) + 1 queries to SAT that are not computable using f(n) queries to SAT, for f(n) = O(log n). (2) If C_{k}^{A}, restricted to length n inputs, can be computed by an unbounded fanin oracle circuit of size s(n) and depth d(n), with k - 1 queries to some set X, then A can be computed with an unbounded fanin (non-oracle) circuit of size n^{O(k)}s(n) and depth d(n) + O(1). (3) Assuming that PH ≠ ∑_{4}^{p} ∩ _{4} ^{p}, and ε < 1, #SAT is not 2^{n/ε}-enumerable.

Original language | English |
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Pages (from-to) | 104-139 |

Number of pages | 36 |

Journal | Information and Computation |

Volume | 186 |

Issue number | 1 |

DOIs | |

State | Published - 10 Oct 2003 |

### Bibliographical note

Funding Information:This paper is dedicated to the memory of Ronald V. Book, 1937–1997. ∗Corresponding author. Fax: +1-301-405-6707. E-mail addresses: [email protected] (W. Gasarch), [email protected] (A. Amir), [email protected] (R. Beigel). 1Research performed at University of Maryland and Georgia Tech. Supported in part by NSF Grants CCR-8803641 and CCR-96101709. 2Research performed at Johns Hopkins, Yale University, The University of Maryland, Lehigh University, and DIMACS. Supported in part by the National Science Foundation under Grants CCR–8958528, CCR–9415410, and CCR-9877150; in part by DIMACS, an NSF Science and Technology Center funded under Contract STC-91-19999 and by the NJ Commission on Science and Technology; and in part by the Human–Computer Interaction Laboratory under NASA Grant NAG 52895. 3Supported in part by NSF Grants CCR-8803641, CCR-9020079, CCR-9301339, and CCR-9732692.

### Funding

This paper is dedicated to the memory of Ronald V. Book, 1937–1997. ∗Corresponding author. Fax: +1-301-405-6707. E-mail addresses: [email protected] (W. Gasarch), [email protected] (A. Amir), [email protected] (R. Beigel). 1Research performed at University of Maryland and Georgia Tech. Supported in part by NSF Grants CCR-8803641 and CCR-96101709. 2Research performed at Johns Hopkins, Yale University, The University of Maryland, Lehigh University, and DIMACS. Supported in part by the National Science Foundation under Grants CCR–8958528, CCR–9415410, and CCR-9877150; in part by DIMACS, an NSF Science and Technology Center funded under Contract STC-91-19999 and by the NJ Commission on Science and Technology; and in part by the Human–Computer Interaction Laboratory under NASA Grant NAG 52895. 3Supported in part by NSF Grants CCR-8803641, CCR-9020079, CCR-9301339, and CCR-9732692.

Funders | Funder number |
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Human–Computer Interaction Laboratory | |

NJ Commission on Science and Technology | |

NSF Science and Technology Center | STC-91-19999 |

National Science Foundation | CCR–8958528, CCR-96101709, CCR-9877150, CCR–9415410, CCR-8803641 |

Directorate for Computer and Information Science and Engineering | 8958528, 9732692, 8803641, 9301339, 9415410, 9877150, 9020079 |

National Aeronautics and Space Administration | CCR-9732692, CCR-9020079, NAG 52895, CCR-9301339 |