Depth-Reduction for Composites

Chen, Shiteng; Papakonstantinou, Periklis A.

doi:10.1109/focs.2016.20

Cited by 6 publications

(7 citation statements)

References 28 publications

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“…First, we note that AND circuits of small fan-in have efficient depthtwo MOD m circuits. A version was first used in [BIS90] in the context of MOD circuits, and more recently a strengthening was used to reduce the size-depth tradeoff for simulating ACC 0 circuits with SYM • AND circuits [CP19]. (Chen and Papakonstantinou [CP19] call this "linearization".)…”

Section: Proof For Any Positive Integer T M | T If and Only Ifmentioning

confidence: 99%

“…A well-known result in circuit complexity is that every ACC 0 circuit of size s can be simulated by a SYM • AND circuit of size s poly(log s) [BT94,CP19]. Therefore, the SYM • AND Hypothesis is a strengthening of the longstanding hypothesis that TC 0 ⊂ ACC 0 : the SYM • AND Hypothesis implies exponential lower bounds for simulating TC 0 circuits with ACC 0 circuits.…”

Section: Introductionmentioning

confidence: 99%

“…Since their identification over 30 years ago [Bar86,BT87], scant progress has been made on lower bounds against CC 0 circuits, and their close cousin ACC 0 which includes AND and OR in the gate basis. Some exceptions include work focusing on special cases of the problem (e.g., [BBR94, GT00, CGPT06, CW09]), uniform lower bounds [AG94], and work proving strong lower bounds but only for functions whose complexity is in QuasiNP or higher (e.g., [Wil11,CP19,MW20,CLW20]). If there has ever been a "circuit complexity winter", CC 0 circuits are at least partly to blame.…”

Section: Introductionmentioning

confidence: 99%

“…Suppose we substitute each MAJORITY gate of C with a copy of the assumed ACC 0 [m] circuit. We obtain a ACC 0 [m] circuit C ′ of depth at most c • d and of size exp( Õ(n α rd )) such that C ′ is equivalent to C. Chen and Papakonstantinou [CP19] prove that for every depth-d ′ size-s circuit D over AND, OR, and MOD m gates, where m is the product of r distinct primes, D is equivalent to a SYM • AND circuit D ′ of size at most…”

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confidence: 99%

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Smaller ACC0 Circuits for Symmetric Functions

Chapman,

Williams

2021

Preprint

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What is the power of constant-depth circuits with MOD m gates, that can count modulo m? Can they efficiently compute MAJORITY and other symmetric functions? When m is a constant prime power, the answer is well understood. In this regime, Razborov and Smolensky proved in the 1980s that MAJORITY and MOD m require superpolynomial-size MOD q circuits, where q is any prime power not dividing m. However, relatively little is known about the power of MOD m gates when m is not a prime power. For example, it is still open whether every problem decidable in exponential time can be computed by depth-3 circuits of polynomial-size and only MOD 6 gates.In this paper, we shed some light on the difficulty of proving lower bounds for MOD m circuits, by giving new upper bounds. We show how to construct MOD m circuits computing symmetric functions with non-prime power m, with size-depth tradeoffs that beat the longstanding lower bounds for AC 0 [m] circuits when m is a prime power. Furthermore, we observe that our size-depth tradeoff circuits have essentially optimal dependence on m and d in the exponent, under a natural circuit complexity hypothesis.For example, we show that for every ε > 0, every symmetric function can be computed using MOD m circuits of depth 3 and 2 n ε size, for a constant m depending only on ε > 0. In other words, depth-3 CC 0 circuits can compute any symmetric function in subexponential size. This demonstrates a significant difference in the power of depth-3 CC 0 circuits, compared to other models: for certain symmetric functions, depth-3 AC 0 circuits require 2 Ω( √ n) size [Håstad 1986], and depth-3 AC 0 [p k ] circuits (for fixed prime power p k ) require 2 Ω(n 1/6 ) size [Smolensky 1987]. Even for depth-two MOD p • MOD m circuits, 2 Ω(n) lower bounds were known [Barrington Straubing Thérien 1990].

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Section: Proof For Any Positive Integer T M | T If and Only Ifmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Smaller ACC0 Circuits for Symmetric Functions

Chapman,

Williams

2021

Preprint

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“…A considerable amount of research has gone into devising improved algorithms for checking the satisfiability of various circuit models up to depth o(log n/ log log n) (see, for example, [26,35,30,29,36,5,21,27,18,40,19,6,20,9,25]), although the question of improved algorithms for checking the satisfiability of linear size bounded depth threshold circuits is still unresolved. The improvements range from superpolynomial factors to exponential factors depending on the circuit model, size and depth.…”

Section: Introductionmentioning

confidence: 99%

Beating Brute Force for (Quantified) Satisfiability of Circuits of Bounded Treewidth

Lokshtanov

Mikhailin

Paturi

et al. 2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We investigate the algorithmic properties of circuits of bounded treewidth. Here the treewidth of a circuit C is defined as the treewidth of the underlying undirected graph of C, after the vertices corresponding to input gates have been removed. Thus, boolean formulae correspond to circuits of treewidth 1.• Our first main result is an algorithm for counting the number of satisfying assignments of circuits with n input gates, treewidth ω, and at most s · n gates. The running time of our algorithm is 2 n(1−

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Uniform proofs of ACC representations

Buss

2017

Arch. Math. Logic

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Depth-Reduction for Composites

Cited by 6 publications

References 28 publications

Smaller ACC0 Circuits for Symmetric Functions

Smaller ACC0 Circuits for Symmetric Functions

Beating Brute Force for (Quantified) Satisfiability of Circuits of Bounded Treewidth

Uniform proofs of ACC representations

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