Threshold circuits of bounded depth

Hajnal, A.; Maass, Wolfgang; Pudlák, Pavel; Szegedy, Márió; Turán, György

doi:10.1109/sfcs.1987.59

Cited by 146 publications

(80 citation statements)

References 21 publications

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“…A related measure on a pair of boolean functions g and f , called correlation and denoted by Corr(g, f ), was defined by [15]. This measure can be defined w.r.t any distribution over the cube, but we will be solely interested in the uniform distribution for discussing correlation in this paper.…”

Section: Basic Notionsmentioning

confidence: 99%

“…sets A, B if Corr A,B (g, f ) ≥ . The usefulness of this measure in proving circuit lower bounds comes from the following connection made by [15]: …”

Section: Basic Notionsmentioning

confidence: 99%

“…By considering a simple base function that was used by [24], we show that our k-wise masked function F k+1 has (n Ω(1) ) k-party complexity whenever k is a constant. On the other hand, it is simple to verify that F k+1 is in AC [15] implies that obtaining an exponentially small upper bound on the correlation between a function f and and any boolean function that is represented by a polynomial of poly-logarithmic degree over Z m , is enough to prove that f cannot be computed in sub-exponential size by such depth three circuits. It is commonly believed that the simple function MOD q has small correlation with such low degree polynomials over Z m , if m and q are co-prime.…”

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confidence: 99%

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Discrepancy and the Power of Bottom Fan-in in Depth-three Circuits

Bogdanov

Safra

2007

48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)

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We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty 'Number on the Forehead' model. Our method is based on the notion of voting polynomial degree of functions and extends the Degree-Discrepancy Lemma in the recent work of Sherstov [24]. Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fan-in at the base layer i.e. circuits of type MAJ • SYMM • ANY O(1) cannot simulate the circuit class AC 0 in sub-exponential size. Further, even if the fan-in of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC 0 in quasi-polynomial size. This is in contrast to the classical result of Yao and Beigel-Tarui that shows that such circuits, having only MAJORITY gates, can simulate the class ACC 0 in quasi-polynomial size when the bottom fan-in is increased to poly-logarithmic size.In the second part, we simplify the arguments in the breakthrough work of Bourgain [7] for obtaining exponentially small upper bounds on the correlation between the boolean function MOD q and functions represented by polynomials of small degree over Z m , when m, q ≥ 2 are co-prime integers. Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting. This results in a slight improvement of the estimates of [7,14]. It is known that such estimates imply that circuits of type MAJ • MOD m • AND log n cannot compute the MOD q function in sub-exponential size. It remains a major open question to determine if such circuits can simulate ACC 0 in polynomial size when the bottom fan-in is increased to poly-logarithmic size.

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Section: Basic Notionsmentioning

confidence: 99%

“…sets A, B if Corr A,B (g, f ) ≥ . The usefulness of this measure in proving circuit lower bounds comes from the following connection made by [15]: …”

Section: Basic Notionsmentioning

confidence: 99%

mentioning

confidence: 99%

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Discrepancy and the Power of Bottom Fan-in in Depth-three Circuits

Bogdanov

Safra

2007

48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)

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“…Current understanding of the limitations of bounded depth threshold circuits is, however, inadequate. Exponential lower bounds for such circuits are only showed for the limited class of depth two and bounded weight [8]. For larger depth circuits, it is barely super linear lower bounds that we obtained on the number of wires [12].…”

Section: Introductionmentioning

confidence: 66%

A Satisfiability Algorithm for Some Class of Dense Depth Two Threshold Circuits

Amano

Saito

2015

IEICE Trans. Inf. ^|^ Syst.

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SUMMARYRecently, Impagliazzo et al. constructed a nontrivial algorithm for the satisfiability problem for sparse threshold circuits of depth two which is a class of circuits with cn wires. We construct a nontrivial algorithm for a larger class of circuits. Two gates in the bottom level of depth two threshold circuits are dependent, if the output of the one is always greater than or equal to the output of the other one. We give a nontrivial circuit satisfiability algorithm for a class of circuits which may not be sparse in gates with dependency. One of our motivations is to consider the relationship between the various circuit classes and the complexity of the corresponding circuit satisfiability problem of these classes. Another background is proving strong lower bounds for TC 0 circuits, exploiting the connection which is initiated by Ryan Williams between circuit satisfiability algorithms and lower bounds.

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“…This finding provided a first hint that the summation of thousands of input pulses on the membrane of a biological neuron is a really powerful computation step. In fact, it has already turned out to be quite difficult to prove for a concrete function that it cannot be computed by polynomial size-threshold circuits of depth 2 with binary weights (9), and it is currently still conceivable that depth 3 circuits of this type can even solve problems that are complete for nondeterministic polynomial time (i.e., NP-complete). The results of Esser et al (1) provide new evidence that the computational power of threshold circuits is in fact very large.…”

mentioning

confidence: 99%