In his seminal paper on probabilistic Turing machines, Gill [13] asked whether the class PP is closed under intersection and union. We give a positive answer to this question. We also show that PP is closed under a variety of polynomial-time truth-table reductions. Consequences in complexity theory include the definite collapse and (assuming P<F NaN> 6= PP) separation of certain query hierarchies over PP. Similar techniques allow us to combine several threshold gates into a single threshold gate. Consequences in the study of circuits include the simulation of circuits with a small number of threshold gates by circuits having only a single threshold gate at the root (perceptrons), and a lower bound on the number of threshold gates needed to compute the parity function. 1. Introduction The class PP was defined in 1972 by John Gill [13, 14] and independently by Janos Simon [26] in 1974. PP is the class of languages accepted by a polynomial-time bounded nondeterministic Turing machine t...

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...

Much structural work on NP-complete sets has exploited SAT's d-self-reducibility. In this paper we exploit the additional fact that SAT is a d-cylinder to show that NP-complete sets are p-superterse unless P = NP. In fact, every set that is NP-hard under polynomial-time n o(1) -tt reductions is p-superterse unless P = NP. In particular no p-selective set is NP-hard under polynomialtime n o(1) -tt reductions unless P = NP. In addition, no easily countable set is NP-hard under Turing reductions unless P = NP. Self-reducibility does not seem to suffice for our main result: in a relativized world, we construct a d-selfreducible set in NP \Gamma P that is polynomial-time 2-tt reducible to a p-selective set. 1 Introduction Assume we are given a set A ` f0; 1g . Even if A is intractable there might be a way to compute some partial information about A efficiently, i.e., in polynomial time. We are interested in information of the following kind: Given a list of k strings x 1 ; : : : ;...

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 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; t...

Chang and Kadin have shown that if the difference hierarchy over NP collapses to level k, then the polynomial hierarchy (PH) is equal the kth level of the difference hierarchy over \Sigma p 2 . We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP collapses to level k, then PH collapses to i P NP (k\Gamma1)-tt j NP , the class of sets recognized in polynomial time with k \Gamma 1 nonadaptive queries to a set in NP NP and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any class C that has p m -complete sets and is closed under p conj - and NP m -reductions (alternatively, closed under p disj - and co-NP m -reductions), if the difference hierarchy over C collapses to level k, then PH C = i P NP (k\Gamma1)-tt j C . Then we show that the exact counting class C=P is closed under p disj - and co-NP m - reductions. Consequently, if the difference hiera...

Extending a line of research initiated by Lipton, we study the complexity of computing the permanent of random n by n matrices with integer values between 0 and p - 1, for any suitably large prime p. Previous

Let A be a subset of the natural numbers, and let F A n (x 1 ; : : : ; xn ) = hØA (x 1 ); : : : ; ØA (xn )i; where ØA is the characteristic function of A. An oracle Turing machine with oracle A could certainly compute F A n with n queries to A. There are some sets A (e.g., the halting set) for which F A n can be computed with substantially fewer than n queries. One key reason for this is that the questions asked to the oracle can depend on previous answers, i.e., the questions are adaptive. We examine when it is possible to save queries. A set A is terse if the computation of F A n from A requires n queries. A set A is superterse if the computation of F A n from any set requires n queries. A set A is verbose if F A 2 n \Gamma1 can be computed with n queries to A. The range of possible query savings is limited by the following theorem: F A n cannot be computed with only blog nc queries to a set X unless A is recursive. In addition we produce the following: (1) a verbose ...

This paper studies a notion called polynomial-time membership comparable sets. For a function g, a set A is polynomial-time g-membership comparable if there is a polynomialtime computable function f such that for any x 1 ; \Delta \Delta \Delta ; xm with m g(maxfjx 1 j; \Delta \Delta \Delta ; jx m jg), outputs b 2 f0; 1g m such that (A(x 1 ); \Delta \Delta \Delta ; A(xm )) 6= b. The following is a list of major results proven in the paper. 1. Polynomial-time membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomial-time membership comparable sets have polynomial-size circuits. 3. For any function f and for any constant c ? 0, if a set is p f(n)-tt -reducible to a Pselective set, then the set is polynomial-time (1 + c) log f(n)-membership comparable. 4. For any C chosen from fPSPACE;UP;FewP;NP;C=P;PP;MOD 2 P; MOD 3 P; \Delta \Delta \Deltag, if C ` P-mc(c log n) for some c ! 1, then C = P. As a corollary of the last tw...

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...

this paper was coauthored by K. Wagner)