## Abstract

It is shown that if there is a polynomial-time algorithm that tests k(n) = O(log n) points for membership in a set A by making only k(n)-1 adaptive queries to an oracle set X, then A belongs to NP/poly intersection co-NP/poly (if k(n) = O(1) then A belong to P/poly). In particular, k(n) = O(log n) queries to an NP-complete set (k(n) = O(1) queries to an NP-hard set) are more powerful than k(n)-1 queries, unless the polynomial hierarchy collapses. Similarly, if there is a small circuit that tests k(n) points for membership in A by making only k(n)-1 adaptive queries to a set X, then there is a correspondingly small circuit that decides membership in A without an oracle. An investigation is conducted of the quantitatively stronger assumption that there is a polynomial-time algorithm that tests 2^{k} strings for membership in A by making only k queries to an oracle X, and qualitatively stronger conclusions about the structure of A are derived: A cannot be self-reducible unless A ε P, and A cannot be NP-hard unless P = NP. Similar results hold for counting classes. In addition, relationships between bounded-query computations, lowness, and the p-degrees are investigated.

Original language | English |
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Title of host publication | Proc Fifth Annu Struct Complexity Theor |

Publisher | Publ by IEEE |

Pages | 232-243 |

Number of pages | 12 |

ISBN (Print) | 0818620722 |

State | Published - 1990 |

Externally published | Yes |

Event | Proceedings of the Fifth Annual Structure in Complexity Theory Conference - Barcelona, Spain Duration: 8 Jul 1990 → 11 Jul 1990 |

### Publication series

Name | Proc Fifth Annu Struct Complexity Theor |
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### Conference

Conference | Proceedings of the Fifth Annual Structure in Complexity Theory Conference |
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City | Barcelona, Spain |

Period | 8/07/90 → 11/07/90 |

### 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: gasarch@cs.umd.edu (W. Gasarch), amir@cs.biu.ac.il (A. Amir), beigel@cis.edu (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.