Approximation by DNF: Examples and Counterexamples

O’Donnell, Ryan; Wimmer, Karl

doi:10.1007/978-3-540-73420-8_19

Cited by 22 publications

(10 citation statements)

References 24 publications

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“…Formally, any AC 0 [⊕] circuit of depth d computing any Approximate Majority must have size exp(Ω(n 1/2(d−1) )). On the upper bound side, it is known from the work of O'Donnell and Wimmer [12] and Amano [1] that there exist Approximate Majorities that can be computed by monotone AC 0 formulas of depth d and size exp(O(dn 1/2(d−1) )). (Note that the double exponent 1/(2(d − 1)) is now the same in the upper and lower bounds.…”

Section: Proof Outlinementioning

confidence: 99%

“…As far as we know, the study of this class of functions was initiated by O'Donnell and Wimmer [12]. See also [1,4].…”

Section: Preliminariesmentioning

confidence: 99%

“…We start with an informal description of our construction, which closely follows the constructions of O'Donnell and Wimmer [12] and Amano [1] (see also [19,7]).…”

Section: High-level Ideamentioning

confidence: 99%

“…The first basic observation, from [12], is that to compute an (ε, n)-Approximate majority, it suffices to compute the Majority function on most (say at least a 1 − (ε/4) fraction of) inputs of relative Hamming weight bounded away from 1/2; say those of weight either at most 1/2 − (ε/4 √ n) or at least 1/2 + (ε/4 √ n). So we assume that the inputs come from one of these sets.…”

Section: High-level Ideamentioning

confidence: 99%

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Rossman¹,

Srinivasan²

2019

Theory of Comput.

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We give the first separation between the power of formulas and circuits in the AC 0 [⊕] basis (unbounded fan-in AND, OR, NOT and MOD 2 gates). We show that there exist poly(n)-size depth-d circuits that are not equivalent to any depth-d formula of size n o(d) for all d ≤ O(log(n)/log log(n)). This result is obtained by a combination of new lower and upper bounds for Approximate Majorities, the class of Boolean functions {0, 1} n → {0, 1} that agree with the Majority function on a 3/4 fraction of the inputs. AC 0 [⊕ ⊕ ⊕] formula lower bound. We show that every depth-d AC 0 [⊕] formula of size s has a 1/4-error polynomial approximation over F 2 of degree O((1/d) log s) d−1. This strengthens a classic O(log s) d−1 degree approximation for circuits due to Razborov (1987). Since any polynomial that approximates the Majority function has degree Ω(√ n), this result implies an exp(Ω(dn 1/2(d−1))) lower bound on the depth-d AC 0 [⊕] formula size of all Approximate Majority functions for all d ≤ O(log n). Monotone AC 0 circuit upper bound. For all d ≤ O(log(n)/log log(n)), we give a randomized construction of depth-d monotone AC 0 circuits (without NOT or MOD 2 gates) of size exp(O(n 1/2(d−1))) that compute an Approximate Majority function. This strengthens a construction of formulas of size exp(O(dn 1/2(d−1))) due to Amano (2009).

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Section: Proof Outlinementioning

confidence: 99%

“…As far as we know, the study of this class of functions was initiated by O'Donnell and Wimmer [12]. See also [1,4].…”

Section: Preliminariesmentioning

confidence: 99%

“…We start with an informal description of our construction, which closely follows the constructions of O'Donnell and Wimmer [12] and Amano [1] (see also [19,7]).…”

Section: High-level Ideamentioning

confidence: 99%

Section: High-level Ideamentioning

confidence: 99%

See 2 more Smart Citations

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Rossman¹,

Srinivasan²

2019

Theory of Comput.

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“…Very recently, Braverman, Rao, Raz, and Yehudayoff [8] and Brody and Verbin [9] studied the power of restricted-width, read-once branching programs for this problem. The distinguishing problem is also closely related to the approximate majority problem, in which given an n-bit string x, we want to decide whether x has Hamming weight less than (1/2 − ε) n or more than (1/2 + ε) n. A large body of research has addressed the ability of constant-depth circuits to solve the approximate majority problem and its variants [1,3,4,17,20,21].…”

Section: The Distinguishing Problemmentioning

confidence: 99%

Advice Coins for Classical and Quantum Computation

Aaronson

Drucker

2011

Automata, Languages and Programming

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We study the power of classical and quantum algorithms equipped with nonuniform advice, in the form of a coin whose bias encodes useful information. This question takes on particular importance in the quantum case, due to a surprising result that we prove: a quantum finite automaton with just two states can be sensitive to arbitrarily small changes in a coin's bias. This contrasts with classical probabilistic finite automata, whose sensitivity to changes in a coin's bias is bounded by a classic 1970 result of Hellman and Cover.Despite this finding, we are able to bound the power of advice coins for space-bounded classical and quantum computation. We define the classes BPPSPACE/coin and BQPSPACE/coin, of languages decidable by classical and quantum polynomial-space machines with advice coins. Our main theorem is that both classes coincide with PSPACE/poly. Proving this result turns out to require substantial machinery. We use an algorithm due to Neff for finding roots of polynomials in NC; a result from algebraic geometry that lower-bounds the separation of a polynomial's roots; and a result on fixed-points of superoperators due to Aaronson and Watrous, originally proved in the context of quantum computing with closed timelike curves.

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Variable Influences in Conjunctive Normal Forms

Traxler

2009

Lecture Notes in Computer Science

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Approximation by DNF: Examples and Counterexamples

Abstract: Say that f : {0, 1} n → {0, 1} -approximates g : {0, 1} n → {0, 1} if the functions disagree on at most an fraction of points. This paper contains two results about approximation by DNF and other small-depth circuits:(1) For every constant 0 < < 1/2 there is a DNF of size 2 O(

Cited by 22 publications

References 24 publications

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Advice Coins for Classical and Quantum Computation

Variable Influences in Conjunctive Normal Forms

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