Results 1  10
of
47
GapDefinable Counting Classes
, 1991
"... The function class #P lacks an important closure property: it is not closed under subtraction. To remedy this problem, we introduce the function class GapP as a natural alternative to #P. GapP is the closure of #P under subtraction, and has all the other useful closure properties of #P as well. We s ..."
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Cited by 122 (13 self)
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The function class #P lacks an important closure property: it is not closed under subtraction. To remedy this problem, we introduce the function class GapP as a natural alternative to #P. GapP is the closure of #P under subtraction, and has all the other useful closure properties of #P as well. We show that most previously studied counting classes, including PP, C=P, and Mod k P, are "gapdefinable," i.e., definable using the values of GapP functions alone. We show that there is a smallest gapdefinable class, SPP, which is still large enough to contain Few. We also show that SPP consists of exactly those languages low for GapP, and thus SPP languages are low for any gapdefinable class. These results unify and improve earlier disparate results of Cai & Hemachandra [7] and Kobler, Schoning, Toda, & Tor'an [15]. We show further that any countable collection of languages is contained in a unique minimum gapdefinable class, which implies that the gapdefinable classes form a lattice un...
Graph Nonisomorphism Has Subexponential Size Proofs Unless The PolynomialTime Hierarchy Collapses
 SIAM Journal on Computing
, 1998
"... We establish hardness versus randomness tradeoffs for a broad class of randomized procedures. In particular, we create efficient nondeterministic simulations of bounded round ArthurMerlin games using a language in exponential time that cannot be decided by polynomial size oracle circuits with acce ..."
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Cited by 108 (6 self)
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We establish hardness versus randomness tradeoffs for a broad class of randomized procedures. In particular, we create efficient nondeterministic simulations of bounded round ArthurMerlin games using a language in exponential time that cannot be decided by polynomial size oracle circuits with access to satisfiability. We show that every language with a bounded round ArthurMerlin game has subexponential size membership proofs for infinitely many input lengths unless exponential time coincides with the third level of the polynomialtime hierarchy (and hence the polynomialtime hierarchy collapses). This provides the first strong evidence that graph nonisomorphism has subexponential size proofs. We set up a general framework for derandomization which encompasses more than the traditional model of randomized computation. For a randomized procedure to fit within this framework, we only require that for any fixed input the complexity of checking whether the procedure succeeds on a given ...
The Polynomial Method in Circuit Complexity
 In Proceedings of the 8th IEEE Structure in Complexity Theory Conference
, 1993
"... The representation of functions as lowdegree polynomials over various rings has provided many insights in the theory of smalldepth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polyno ..."
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Cited by 70 (4 self)
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The representation of functions as lowdegree polynomials over various rings has provided many insights in the theory of smalldepth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polynomials in order to prove complexity bounds. Minsky and Papert [39] used polynomials to prove early lower bounds on the order of perceptrons. Razborov [46] and Smolensky [49] used them to prove lower bounds on the size of ANDOR circuits. Other lower bounds via polynomials are due to [50, 4, 10, 51, 9, 55]. Paturi and Saks [44] discovered that rational functions could be used for lower bounds on the size of threshold circuits. Toda [53] used polynomials to prove upper bounds on the power of the polynomial hierarchy. This led to a series of upper bounds on the power of the polynomial hierarchy [54, 52], AC 0 [2, 3, 52, 19], and ACC [58, 20, 30, 37], and related classes [21, 42]. Beigel and Gi...
New Collapse Consequences Of NP Having Small Circuits
, 1995
"... . We show that if a selfreducible set has polynomialsize circuits, then it is low for the probabilistic class ZPP(NP). As a consequence we get a deeper collapse of the polynomialtime hierarchy PH to ZPP(NP) under the assumption that NP has polynomialsize circuits. This improves on the wellknown ..."
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Cited by 57 (8 self)
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. We show that if a selfreducible set has polynomialsize circuits, then it is low for the probabilistic class ZPP(NP). As a consequence we get a deeper collapse of the polynomialtime hierarchy PH to ZPP(NP) under the assumption that NP has polynomialsize circuits. This improves on the wellknown result of Karp, Lipton, and Sipser (1980) stating a collapse of PH to its second level \Sigma P 2 under the same assumption. As a further consequence, we derive new collapse consequences under the assumption that complexity classes like UP, FewP, and C=P have polynomialsize circuits. Finally, we investigate the circuitsize complexity of several language classes. In particular, we show that for every fixed polynomial s, there is a set in ZPP(NP) which does not have O(s(n))size circuits. Key words. polynomialsize circuits, advice classes, lowness, randomized computation AMS subject classifications. 03D10, 03D15, 68Q10, 68Q15 1. Introduction. The question of whether intractable sets ca...
Representing Boolean Functions As Polynomials Modulo Composite Numbers
 Computational Complexity
, 1994
"... . Define the MODm degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 01 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm degree of the OR of N variables is O( r p N ), wher ..."
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Cited by 56 (6 self)
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. Define the MODm degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 01 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm degree of the OR of N variables is O( r p N ), where r is the number of distinct prime factors of m. This is optimal in the case of representation by symmetric polynomials. The MOD n function is 0 if the number of input ones is a multiple of n and is one otherwise. We show that the MODm degree of both the MOD n and :MOD n functions is N\Omega\Gamma1/ exactly when there is a prime dividing n but not m. The MODm degree of the MODm function is 1; we show that the MODm degree of :MODm is N\Omega\Gamma30 if m is not a power of a prime, O(1) otherwise. A corollary is that there exists an oracle relative to which the MODmP classes (such as \PhiP) have this structure: MODmP is closed under complementation and union iff m is a prime power, and...
A Complexity Theory for Feasible Closure Properties
, 1991
"... The study of the complexity of sets encompasses two complementary aims: (1) establishing  usually via explicit construction of algorithms  that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as ..."
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Cited by 47 (3 self)
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The study of the complexity of sets encompasses two complementary aims: (1) establishing  usually via explicit construction of algorithms  that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as the NPcomplete sets and the PSPACEcomplete sets). For the study of the complexity of closure properties, a recent urry of results [21, 33, 49, 6, 7, 16] has established an analog of (1); these papers explicitly demonstrate many closure properties possessed by PP and C=P (and the proofs implicitly give closure properties of the function class #P). The present paper presents and develops, for function classes such as #P, SpanP, OptP, and MidP, an analog of (2): a general theory of the complexity of closure properties. In particular, we show that subtraction is hard for the closure properties of each of these classes: each is closed under subtraction if and only if it is closed under every polynom...
Computing Solutions Uniquely Collapses the Polynomial Hierarchy
 SIAM Journal on Computing
, 1993
"... Is there a singlevalued NP function that, when given a satisfiable formula as input, outputs a satisfying assignment? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterm ..."
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Cited by 40 (23 self)
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Is there a singlevalued NP function that, when given a satisfiable formula as input, outputs a satisfying assignment? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterministic function, then the polynomial hierarchy collapses to its second level. As the existence of such a function is known to be equivalent to the statement "every multivalued NP function has a singlevalued NP refinement," our result provides the strongest evidence yet that multivalued NP functions cannot be refined. We prove our result via theorems of independent interest. We say that a set A is NPSVselective (NPMVselective) if there is a 2ary partial function in NPSV (NPMV, respectively) that decides which of its inputs (if any) is "more likely" to belong to A; this is a nondeterministic analog of the recursiontheoretic notion of the semirecursive sets and the extant complexitythe...
Threshold Computation and Cryptographic Security
 SIAM JOURNAL ON COMPUTING
, 1995
"... Threshold machines are Turing machines whose acceptance is determined by what portion of the machine's computation paths are accepting paths. Probabilistic machines are Turing machines whose acceptance is determined by the probability weight of the machine's accepting computation paths. In 1975, ..."
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Cited by 37 (7 self)
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Threshold machines are Turing machines whose acceptance is determined by what portion of the machine's computation paths are accepting paths. Probabilistic machines are Turing machines whose acceptance is determined by the probability weight of the machine's accepting computation paths. In 1975, Simon proved that for unboundederror polynomialtime machines these two notions yield the same class, PP. Perhaps because Simon's result seemed to collapse the threshold and probabilistic modes of computation, the relationship between threshold and probabilistic computing for the case of bounded error has remained unexplored. In this paper, we compare the boundederror probabilistic class BPP with the analogous threshold class, BPP path , and, more generally, we study the structural properties of BPP path . We prove that BPP path contains both NP BPP and P NP[log] , and that BPP path is contained in P \Sigma p 2 [log] , BPP NP , and PP. We conclude that, unless the polynomial hierarchy collapses, boundederror threshold computation is strictly more powerful than boundederror probabilistic computation. We also consider the natural notion of secure access to a database: an adversary who watches the queries should gain no information about the input other than perhaps its length. We show, for both BPP and BPP path , that if there is any database for which this formalization of security differs from the security given by oblivious database access, then P 6= PSPACE. It follows that if any set lacking small circuits can be securely accepted, then P 6= PSPACE.
Determining Acceptance Possibility for a Quantum Computation is Hard for PH
, 1997
"... It is shown that determining whether a quantum computation has a nonzero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness ..."
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Cited by 34 (3 self)
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It is shown that determining whether a quantum computation has a nonzero probability of accepting is at least as hard as the polynomial time hierarchy. This hardness
Two queries
 In CCC
, 1999
"... We consider the question whether two queries to SAT are as powerful as one query. We show that if P NP�℄� P NP�℄then Locally either NP�coNP or NP has polynomialsize circuits. ..."
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Cited by 32 (7 self)
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We consider the question whether two queries to SAT are as powerful as one query. We show that if P NP�℄� P NP�℄then Locally either NP�coNP or NP has polynomialsize circuits.