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59
PP is Closed Under Intersection
 Journal of Computer and System Sciences
, 1991
"... 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 polynomialtime truthtable reductions. Consequences in complexity theory ..."
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Cited by 89 (9 self)
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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 polynomialtime truthtable 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 polynomialtime bounded nondeterministic Turing machine t...
Bounded Queries to SAT and the Boolean Hierarchy
 Theoretical Computer Science
, 1991
"... We study the complexity of decision problems that can be solved by a polynomialtime 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 ..."
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Cited by 64 (12 self)
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We study the complexity of decision problems that can be solved by a polynomialtime 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 similarlydefined hierarchies of functions that can be computed by a polynomialtime 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...
Approximable Sets
 Information and Computation
, 1994
"... Much structural work on NPcomplete sets has exploited SAT's dselfreducibility. In this paper we exploit the additional fact that SAT is a dcylinder to show that NPcomplete sets are psuperterse unless P = NP. In fact, every set that is NPhard under polynomialtime n o(1) tt reductions is p ..."
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Cited by 54 (10 self)
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Much structural work on NPcomplete sets has exploited SAT's dselfreducibility. In this paper we exploit the additional fact that SAT is a dcylinder to show that NPcomplete sets are psuperterse unless P = NP. In fact, every set that is NPhard under polynomialtime n o(1) tt reductions is psuperterse unless P = NP. In particular no pselective set is NPhard under polynomialtime n o(1) tt reductions unless P = NP. In addition, no easily countable set is NPhard under Turing reductions unless P = NP. Selfreducibility does not seem to suffice for our main result: in a relativized world, we construct a dselfreducible set in NP \Gamma P that is polynomialtime 2tt reducible to a pselective 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 ; : : : ;...
QueryLimited Reducibilities
, 1995
"... 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 truthtable reduction) and queries made in serial (each question ..."
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Cited by 41 (14 self)
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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 truthtable 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 informationtheoretic 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...
A Relationship between Difference Hierarchies and Relativized Polynomial Hierarchies
, 1993
"... 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 c ..."
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Cited by 39 (8 self)
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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 coNP 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 coNP m  reductions. Consequently, if the difference hiera...
On The Hardness Of Computing The Permanent Of Random Matrices
 COMPUTATIONAL COMPLEXITY
, 1992
"... 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 ..."
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Cited by 36 (1 self)
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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
PolynomialTime Membership Comparable Sets
, 1994
"... This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership 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 j ..."
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Cited by 31 (4 self)
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This paper studies a notion called polynomialtime membership comparable sets. For a function g, a set A is polynomialtime gmembership 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. Polynomialtime membership comparable sets construct a proper hierarchy according to the bound on the number of arguments. 2. Polynomialtime membership comparable sets have polynomialsize 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 polynomialtime (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 ` Pmc(c log n) for some c ! 1, then C = P. As a corollary of the last tw...
Terse, Superterse, and Verbose Sets
"... 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 w ..."
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Cited by 29 (20 self)
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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 ...
NPhard Sets are PSuperterse Unless R = NP
, 1992
"... A set A is pterse (psuperterse) 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 qtt be the class of functions reducible to A via a polynomialtime truthtable reduction of norm q, and let PF A qT be the class of ..."
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Cited by 27 (5 self)
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A set A is pterse (psuperterse) 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 qtt be the class of functions reducible to A via a polynomialtime truthtable reduction of norm q, and let PF A qT be the class of functions reducible to A via a polynomialtime Turing reduction that makes at most q queries. A set A is pterse if PF A qtt 6` PF A (q\Gamma1)T for all constants q. A is psuperterse if PF A qtt 6` PF X qT for all constants q and sets X . We show that all NPhard sets (under p tt reductions) are psuperterse, unless it is possible to distinguish uniquely satisfiable formulas from satisfiable formulas in polynomial time. Consequently, all NPcomplete sets are psuperterse unless P = UP (oneway functions fail to exist), R = NP (there exist randomized polynomialtime algorithms for all problems in NP), and the polynomialtime hierarchy collapses. This mostly solves the main open...
When Do Extra Majority Gates Help? Polylog(N) Majority Gates Are Equivalent To One
"... . Suppose that a Boolean function f is computed by a constant depth circuit with 2 m AND, OR, and NOTgatesand m majoritygates. We prove that f is computed by a constant depth circuit with 2 m O(1) AND, OR, and NOTgatesand a single majoritygate, which is at the root. One consequence ..."
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Cited by 24 (5 self)
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. Suppose that a Boolean function f is computed by a constant depth circuit with 2 m AND, OR, and NOTgatesand m majoritygates. We prove that f is computed by a constant depth circuit with 2 m O(1) AND, OR, and NOTgatesand a single majoritygate, which is at the root. One consequence is that if a Boolean function f is computed by an AC 0 circuit plus polylog majoritygates, then f is computed by a probabilistic perceptron having polylog order. Another consequence is that if f agrees with the parity function on threefourths of all inputs, then f cannot be computed by a constant depth circuit with 2 n o(1) AND, OR, and NOTgates, and n o(1) majoritygates. Key words. Majoritygate; thresholdgate; symmetric gate; circuit; parity. Subject classifications. 68Q15. 1. Introduction One of the goals of complexity theory is to find ways to reduce the use of one resource. Typically this entails a modest increase in some other resources. Recently, quasipolynomial size...