Counting Classes are at Least as Hard as the Polynomial-Time Hierarchy

Toda, Seinosuke; Ogiwara, Mitsunori

doi:10.1137/0221023

Cited by 115 publications

(47 citation statements)

References 19 publications

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“…Hence, R is indeed an ε-error probabilistic polynomial for f . Theorem 2 immediately follows from the above and standard probabilistic polynomials for AC 0 from [3,23,24]. However, for our applications to PRGs for AC 0 , we need a slightly stronger statement, which we prove below.…”

Section: Proof Of Theoremmentioning

confidence: 88%

“…It is well-known [3,23,24] that any circuit C ∈ AC 0 (s, d) has an ε-error probabilistic polynomial P of degree (log(s/ε)) O(d) and satisfying �P� ∞ < exp (log s/ε) O(d) . This can be used to prove, for example [21], (a slightly weaker version of) Håstad's theorem [8] that says that Parity does not have subexponential-sized AC 0 circuits.…”

Section: Polynomial Approximations To Acmentioning

confidence: 99%

“…The above notion was introduced in Braverman [4] who proved the following lemma, building on earlier works of [3,23,24]. • E ∈ AC 0 (poly(s log(1/ε)), d + 3).…”

Section: Proof Of Theoremmentioning

confidence: 99%

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On polynomial approximations to AC

Harsha

Srinivasan

2018

Random Struct Algorithms

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Classical AC 0 approximation results show that any AC 0 circuit of size s and depth d has an ε-error probabilistic polynomial over the reals of degree (log(s/ε)) O(d) . We improve this upper bound to (log s) O(d) · log(1/ε), which is much better for small values of ε. We then use this result to show that (log s) O(d) · log(1/ε)-wise independence fools AC 0 circuits of size s and depth d up to error at most ε, improving on Tal's strengthening of Braverman's result that (log(s/ε)) O(d)wise independence suffices. To our knowledge, this is the first PRG construction for AC 0 that achieves optimal dependence on the error ε. We also prove lower bounds on the best polynomial approximations to AC 0 . We show that any polynomial approximating the OR function on n bits to a small constant error must have degree at least Ω( log n). This result improves exponentially on a result of Meka, Nguyen, and Vu (Theory Comput. 2016). K E Y W O R D Santi-concentration for polynomials, bounded independence, constant-depth circuits, probabilistic polynomials MOTIVATION AND RESULTSIn this paper, we study AC 0 circuits, the family of circuits of constant depth and polynomial size (in the input length) with unbounded-fanin AND and OR gates. We use AC 0 (s, d) to denote the family of AC 0 circuits of size s and depth d.

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Section: Proof Of Theoremmentioning

confidence: 88%

Section: Polynomial Approximations To Acmentioning

confidence: 99%

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On polynomial approximations to AC

Harsha

Srinivasan

2018

Random Struct Algorithms

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“…Our main result that there exists some A such that C =P A contains a BPP ⊕P A -immune set is optimal in the sense that for all oracles B, C =P B clearly is contained in PP B and thus in PP ⊕P B . However, it is also known that BPP ⊕P ⊆ Almost[⊕P] [TO92,RR95], where for any relativized class C, Almost[C] denotes the class of languages L such that for almost all oracle sets X, L is in C X [NW94]. It is an open problem (see [RR95]) whether BPP ⊕P = Almost[⊕P], so it is possible that Almost[⊕P] is a strictly larger class than BPP ⊕P .…”

Section: Conclusion and Open Problemsmentioning

confidence: 99%

Immunity and Simplicity for Exact Counting and Other Counting Classes

Rothe

1999

RAIRO-Theor. Inf. Appl.

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Ko [Ko90] and Bruschi [Bru92] independently showed that, in some relativized world, PSPACE (in fact, ⊕P) contains a set that is immune to the polynomial hierarchy (PH). In this paper, we study and settle the question of (relativized) separations with immunity for PH and the counting classes PP, C =P, and ⊕P in all possible pairwise combinations. Our main result is that there is an oracle A relative to which C =P contains a set that is immune to BPP ⊕P . In particular, this C =P A set is immune to PH A and to ⊕P A . Strengthening results of Torán [Tor91] and Green [Gre91], we also show that, in suitable relativizations, NP contains a C =P-immune set, and ⊕P contains a PP PH -immune set. This implies the existence of a C =P B -simple set for some oracle B, which extends results of Balcázar et al. [Bal85,BR88], and provides the first example of a simple set in a class not known to be contained in PH. Our proof technique requires a circuit lower bound for "exact counting" that is derived from Razborov's [Raz87] circuit lower bound for majority.

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“…The class C = P is at least as hard as the polynomial-time hierarchy, since PH ⊆ BP · C = P [33] and even PH ⊆ ZP · C = P [32]. It is at most as hard as "threshold counting," since C = P ⊆ PP, and it is not substantially easier, since PP ⊆ NP C = P .…”

Section: Introductionmentioning

confidence: 99%

Untitled

Watson¹

2015

Theory of Comput.

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Counting Classes are at Least as Hard as the Polynomial-Time Hierarchy

Cited by 115 publications

References 19 publications

On polynomial approximations to AC

On polynomial approximations to AC

Immunity and Simplicity for Exact Counting and Other Counting Classes

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