This paper is dedicated to the memory of Ronald V. Book, 1937-1997.

We study 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. Depending on whether we allow some queries to depend on the results of other queries, we obtain two (probably) different hierarchies. We present several results relating the bounded NP query hierarchies to each other and to the Boolean hierarchy. We also consider the similarly-defined hierarchies of functions that can be computed by a polynomial-time Turing machine that makes a bounded number of queries to an NP oracle. We present relations among these two hierarchies and the Boolean hierarchy. In particular we show for all k that there are functions computable with 2 k parallel queries to an NP set that are not computable in polynomial time with k serial queries to any oracle, unless P = NP. As a corollary k + 1 parallel queries to an NP set allow us to compute more functions than are computable with only k parallel queries to a...

A set A is p-terse (p-superterse) if, for all q, it is not possible to answer q queries to A by making only q \Gamma 1 queries to A (any set X). Formally, let PF A q-tt be the class of functions reducible to A via a polynomial-time truthtable reduction of norm q, and let PF A q-T be the class of functions reducible to A via a polynomial-time Turing reduction that makes at most q queries. A set A is p-terse if PF A q-tt 6` PF A (q\Gamma1)-T for all constants q. A is p-superterse if PF A q-tt 6` PF X q-T for all constants q and sets X . We show that all NP-hard sets (under p tt -reductions) are p-superterse, unless it is possible to distinguish uniquely satisfiable formulas from satisfiable formulas in polynomial time. Consequently, all NP-complete sets are psuperterse unless P = UP (one-way functions fail to exist), R = NP (there exist randomized polynomial-time algorithms for all problems in NP), and the polynomial-time hierarchy collapses. This mostly solves the main open...

We study the fine structure of the classification of sets of natural numbers A according to the number of queries which are needed to compute the n-fold characteristic function of A. A complete characterization is obtained, relating the question to finite combinatorics. In order to obtain an explicit description we consider several interesting combinatorial problems. 1 Introduction In the theory of bounded queries, we measure the complexity of a function by the number of queries to an oracle which are needed to compute it. The field has developed in various directions, both in complexity theory and in recursion theory; see Gasarch [21] for a recent survey. One of the original concerns is the classification of sets A of natural numbers by their "query complexity," i.e., according to the number of oracle queries that are needed to compute the n-fold characteristic function F A n = x 1 ; : : : ; x n : (ØA (x 1 ); : : : ; ØA (x n )). In [3, 8] a set A is called verbose iff F A n is com...

Let A be any nonrecursive set. We define a hierarchy of sets (and a corresponding hierarchy of degrees) that are reducible to A based on bounding the number of queries to A that an oracle machine can make. When A is the halting problem K our hierarchy of sets interleaves with the difference hierarchy Current address: Department of Computer Science, Yale University, 51 Prospect Street, P.O. Box 2158 Yale Station, New Haven, CT 06520. Supported in part by NSF grant CCR-8808949. Part of this work was completed while this author was a student at Stanford University supported by fellowships from the National Science Foundation and from the Fannie and John Hertz Foundation. y Supported in part by NSF grant CCR-8803641. z Part of this work was completed while this author was on sabbatical leave at the University of California, Berkeley. on the r.e. sets in a logarithmic way; this follows from a tradeoff between the number of parallel queries and the number of serial queries needed to...

Do self-reducible sets inherently lack immunity from deterministic polynomial time? Though this is unlikely to be true in general, in this paper we prove that sufficiently strong self-reducibility precludes sufficiently strong immunity from deterministic polynomial time. In particular, we prove that NT is not P balanced immune. However, we prove that NT, a class whose sets have very strong self-reducibility properties, is P bi-immune relative to a generic oracle. Thus, the previous result cannot be relativizably extended to bi-immunity. We also prove that NP and \PhiP are both P balanced immune relative to a random oracle; the former provides the strongest known relativized separation of NP from P. 1 Introduction Relativization results have a long but increasingly checkered history. Today, there are conflicting views as to the extent to which relativization results give insight into the structure of feasible computation [HCRR90,All90,HCC + 92,For94], even as oracle construction has ...

) Martin Kummer and Frank Stephan ? Universitat Karlsruhe, Institut fur Logik, Komplexitat und Deduktionssysteme, D-76128 Karlsruhe, Germany. fkummer; fstephang@ira.uka.de Abstract. The notion of frequency computation concerns approximative computations of n distinct parallel queries to a set A. A is called (m; n)-recursive if there is an algorithm which answers any n distinct parallel queries to A such that at least m answers are correct. This paper gives natural combinatorial characterizations of the fundamental inclusion problem, namely the question for which choices of the parameters m; n; m 0 ; n 0 , every (m;n)-recursive set is (m 0 ; n 0 )-recursive. We also characterize the inclusion problem restricted to recursively enumerable sets and the inclusion problem for the polynomial-time bounded version of frequency computation. Furthermore, using these characterizations we obtain many explicit inclusions and noninclusions. 1 Introduction Frequency computation is a classic...

We show that there exists a 2-membership comparable set that is not btt-reducible to any pselective set. This is a rare example of an unconditional separation in computational complexity.

It is known that infinite binary sequences of constant Kolmogorov complexity are exactly the recursive ones. Such a kind of statement no longer holds in the presence of resource bounds. Contrary to what intuition might suggest, there are sequences of constant, polynomial-time bounded Kolmogorov complexity that are not polynomial-time computable. This motivates the study of several resource-bounded variants in search for a characterization, similar in spirit, of the polynomial-time computable sequences. We propose some definitions, based on Kobayashi's notion of compressibility, and compare them to both the standard resource-bounded Kolmogorov complexity of infinite strings, and the uniform complexity. Some nontrivial coincidences and disagreements are proved. The resource-unbounded case is also considered. Partially supported by the E.U. through the HCM network CHRX--CT93--0415 (COLORET), by DGICYT, project number PB92--0709, and by the Accion Integrada Hispano-Alemana HA-119-B. 1 ...