Top-down lower bounds for depth-three circuits

Håstad, Johan; Jukna, Stasys; Pudlák, Pavel

doi:10.1007/bf01268140

Cited by 30 publications

(27 citation statements)

References 11 publications

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“…This, together with the lower bound for Majority from [3] implies that the complexity of depth-3 circuits computing the Majority function is 2 ( √ n) .…”

Section: Depth-3 Circuitsmentioning

confidence: 89%

“…We also show that this upper bound is essentially best possible for k = o( √ n): we show that for any k there exists an explicit symmetric Boolean function which cannot be computed by depth-3 circuits with bottom fanin k and less than (1 + 1/k) n /(n + 1) 2 n/k+o(n) gates. As a byproduct of the generalization of the construction of depth-3 circuits for symmetric Boolean functions we also answer an open question of Håstad et al [3], regarding the complexity of depth-3 circuits that compute the Majority function: we improve the upper bound of Klawe et al [6] for the size of depth-3 circuits for Ma-…”

Section: Depth-3 Circuitsmentioning

confidence: 97%

“…Håstad et al [3] have showed that the Majority Boolean function could not be computed by depth-3 circuits with less than 2 0.849 √ n gates and have asked to determine the asymptotics of the number of gates required for computing the Majority function by depth-3 circuits. Klawe et al [6] have showed that the Majority function can be realized by depth-3 circuits of size 2 O( √ n log n) .…”

Section: The Complexity Of σ 3 Circuits Computing Majoritymentioning

confidence: 99%

“…What Valiant has observed is that the graph representation of a linear sized Boolean circuit over B 2 has a relatively small set of edges E, whose removal separates the graph representation into components of small depth; thus, by trying all possible truth assignments to the edges in E, while keeping valid the constraints implied by the circuit, Valiant have constructed a depth-3 circuit with relatively small bottom fanin which computes the same Boolean function as the original circuit. Valiant's reduction gives a good motivation for a large line of work done on the complexity of depth-3 circuits [6,2,3,12,9,10,4,11]. For an explicit Boolean function, the largest lower bound for the number of gates in the depth-3 circuit model is due to Paturi et al [11], who showed that sufficiently dense codes cannot be accepted by depth-3 circuits of size less than 2 π √ n/6 > 2 1.281 √ n ; this lower bound was the first to give a lower bound for depth-3 circuits of the kind 2 c √ n , with c > 1.…”

Section: Depth-3 Circuitsmentioning

confidence: 99%

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The complexity of depth-3 circuits computing symmetric Boolean functions

Wolfovitz¹

2006

Information Processing Letters

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“…This, together with the lower bound for Majority from [3] implies that the complexity of depth-3 circuits computing the Majority function is 2 ( √ n) .…”

Section: Depth-3 Circuitsmentioning

confidence: 89%

Section: Depth-3 Circuitsmentioning

confidence: 97%

Section: The Complexity Of σ 3 Circuits Computing Majoritymentioning

confidence: 99%

Section: Depth-3 Circuitsmentioning

confidence: 99%

See 2 more Smart Citations

The complexity of depth-3 circuits computing symmetric Boolean functions

Wolfovitz¹

2006

Information Processing Letters

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“…One explanation is the style of bottom-up depth-reduction arguments that the previous technique employ. A second, complementary explanation is the lack of successful topdown lower bounds-in particular, via Karchmer-Wigderson games [11]-in the non-monotone boolean setting (with the exception of the depth-3 lower bound of Jukna, Pudlák and Håstad [8]). Our technique gets around the limitations of previous techniques by a novel combination of bottom-up and top-down arguments.…”

Section: Bounded-depth Formulas Vs Circuitsmentioning

confidence: 98%

Formulas vs. circuits for small distance connectivity

Rossman

2014

Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing

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We give the first super-polynomial separation in the power of bounded-depth boolean formulas vs. circuits. Specifically, we consider the problem Distance k(n) Connectivity, which asks whether two specified nodes in a graph of size n are connected by a path of length at most k(n). This problem is solvable (by the recursive doubling technique) on circuits of depth O(log k) and size O(kn 3 ). In contrast, we show that solving this problem on formulas of depth log n/(log log n) O(1) requires size n Ω(log k) for all k(n) ≤ log log n. As corollaries:(i) It follows that polynomial-size circuits for Distance k(n) Connectivity require depth Ω(log k) for all k(n) ≤ log log n. This matches the upper bound from recursive doubling and improves a previous Ω(log log k) lower bound of Beame, Impagliazzo and Pitassi [BIP98].(ii) We get a tight lower bound of s Ω(d) on the size required to simulate size-s depth-d circuits by depth-d formulas for all s(n) = n O(1) and d(n) ≤ log log log n. No lower bound better than s Ω(1) was previously known for any d(n) O(1).Our proof technique is centered on a new notion of pathset complexity, which roughly speaking measures the minimum cost of constructing a set of (partial) paths in a universe of size n via the operations of union and relational join, subject to certain density constraints. Half of our proof shows that bounded-depth formulas solving Distance k(n) Connectivity imply upper bounds on pathset complexity. The other half is a combinatorial lower bound on pathset complexity.

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On the Size of Depth-Three Boolean Circuits for Computing Multilinear Functions

Goldreich

Wigderson

2020

Lecture Notes in Computer Science

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We propose that multi-linear functions of relatively low degree over GF(2) may be good candidates for obtaining exponential 1 lower bounds on the size of constant-depth Boolean circuits (computing explicit functions). Specifically, we propose to move gradually from linear functions to multilinear ones, and conjecture that, for any t ≥ 2, some explicit t-linear functions F : ({0, 1} n ) t → {0, 1} require depth-three circuits of size exp(Ω(tn t/(t+1) )).Towards studying this conjecture, we suggest to study two frameworks for the design of depththree Boolean circuits computing multilinear functions, yielding restricted models for which lower bounds may be easier to prove. Both correspond to constructing a circuit by expressing the target polynomial as a composition of simpler polynomials. The first framework corresponds to a direct composition, whereas the second (and stronger) framework corresponds to nested composition and yields depth-three Boolean circuits via a "guess-and-verify" paradigm. The corresponding restricted models of circuits are called D-canonical and ND-canonical, respectively.Our main results are (1) a generic upper bound on the size of depth-three D-canonical circuits for computing any t-linear function, and (2) a lower bound on the size of any depththree ND-canonical circuits for computing some (in fact, almost all) t-linear functions. These bounds match the foregoing conjecture (i.e., they have the form of exp(tn t/(t+1) )). Another important result is a separation of the two models: We prove that ND-canonical circuits can be super-polynomially smaller than their D-canonical counterparts. We also reduce proving lower bounds for the ND-model to Valiant's matrix rigidity problem (for parameters that were not the focus of previous works).The study of the foregoing (Boolean) models calls for an understanding of new types of arithmetic circuits, which we define in this paper and may be of independent interest. These circuits compute multilinear polynomials by using arbitrary multilinear gates of some limited arity. It turns out that a GF(2)-polynomial is computable by such circuits with at most s gates of arity at most s if and only if it can be computed by ND-canonical circuits of size exp(s). A similar characterization holds for D-canonical circuits if we further restrict the arithmetic circuits to have depth two. We note that the new arithmetic model makes sense over any field, and indeed all our results carry through to all fields. Moreover, it raises natural arithmetic complexity problems which are independent of our original motivation.

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Top-down lower bounds for depth-three circuits

Cited by 30 publications

References 11 publications

The complexity of depth-3 circuits computing symmetric Boolean functions

The complexity of depth-3 circuits computing symmetric Boolean functions

Formulas vs. circuits for small distance connectivity

On the Size of Depth-Three Boolean Circuits for Computing Multilinear Functions

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