Probabilistic polynomials, AC0 functions and the polynomial-time hierarchy

Tarui, Jun

doi:10.1016/0304-3975(93)90214-e

Cited by 54 publications

(70 citation statements)

References 26 publications

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“…In our construction we combine two types of approximations of AC 0 circuits by low degree polynomials over R. The first one is combinatorial in the spirit of [Raz87], [Smo87], [BRS91], [Tar93] (for a comprehensive survey on polynomials in circuit complexity see e.g.…”

Section: Techniques and Proof Outlinementioning

confidence: 99%

“…Thus for such a polynomial f , P[f = F ] is very close to 1. While essentially using the same construction as [BRS91], [Tar93], utilizing tools from [VV85], we repeat the construction from scratch in Lemma 8, since we want to reason about details of the construction. We believe that any construction in this spirit would fit in our proof.…”

Section: Techniques and Proof Outlinementioning

confidence: 99%

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Polylogarithmic independence fools AC ⁰ circuits

Braverman

2008

J. ACM

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Abstract-We prove that poly-sized AC 0 circuits cannot distinguish a poly-logarithmically independent distribution from the uniform one. This settles the 1990 conjecture by Linial and Nisan [LN90]. The only prior progress on the problem was by Bazzi [Baz07], who showed that O(log 2 n)-independent distributions fool poly-size DNF formulas. Razborov [Raz08] has later given a much simpler proof for Bazzi's theorem.

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Section: Techniques and Proof Outlinementioning

confidence: 99%

Section: Techniques and Proof Outlinementioning

confidence: 99%

Polylogarithmic independence fools AC ⁰ circuits

Braverman

2008

J. ACM

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“…The proof is analogous to the one of part ii) of Corollary 4.5 using the fact that Mod m P has complete word-decreasing self-reducible languages [OL93], and that PH ⊆ BPP(Mod m P) [TO92,Ta93].…”

Section: Corollary 44 Letmentioning

confidence: 89%

New collapse consequences of NP having small circuits

Köbler

Watanabe

1995

Automata, Languages and Programming

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We show that if a self-reducible set has polynomial-size circuits, then it is low for the probabilistic class ZPP(NP). As a consequence we get a deeper collapse of the polynomial-time hierarchy PH to ZPP(NP) under the assumption that NP has polynomial-size circuits. This improves on the well-known result of Karp, Lipton, and Sipser [KL80] stating a collapse of PH to its second level Σ 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 circuit-size 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.

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“…Based on a theorem by Valiant and Vazirani [20], Beigel et al [5] and Tarui [19] gave probabilistic polynomials over the integers computing the OR function, thereby generalizing Theorem 1, albeit at the expense of a slightly larger degree. As with Theorem 1 it also gives probabilistic polynomials computing the AND function.…”

Section: Probabilistic Polynomialsmentioning

confidence: 99%

Depth Reduction for Circuits with a Single Layer of Modular Counting Gates

Hansen

2009

Computer Science - Theory and Applications

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We consider the class of constant depth AND/OR circuits augmented with a layer of modular counting gates at the bottom layer, i.e AC 0 •MOD m circuits. We show that the following holds for several types of gates G: by adding a gate of type G at the output, it is possible to obtain an equivalent probabilistic depth 2 circuit of quasipolynomial size consisting of a gate of type G at the output and a layer of modular counting gates, i.e G • MOD m circuits. The types of gates G we consider are modular counting gates and threshold-style gates. For all of these, strong lower bounds are known for (deterministic) G • MOD m circuits.

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Probabilistic polynomials, AC0 functions and the polynomial-time hierarchy

Cited by 54 publications

References 26 publications

Polylogarithmic independence fools AC ⁰ circuits

Polylogarithmic independence fools AC ⁰ circuits

New collapse consequences of NP having small circuits

Depth Reduction for Circuits with a Single Layer of Modular Counting Gates

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Probabilistic polynomials, AC0 functions and the polynomial-time hierarchy

Cited by 54 publications

References 26 publications

Polylogarithmic independence fools AC 0 circuits

Polylogarithmic independence fools AC 0 circuits

New collapse consequences of NP having small circuits

Depth Reduction for Circuits with a Single Layer of Modular Counting Gates

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Product

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Polylogarithmic independence fools AC ⁰ circuits

Polylogarithmic independence fools AC ⁰ circuits