On the number of threshold functions

Irmatov, Anwar

doi:10.1515/dma-1993-0407

Cited by 9 publications

(7 citation statements)

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“…The proof repeats the proof of the main theorem in [2]. The essential argument in the proof of the Odlyzko theorem is that the corresponding probability tends to zero as n ->· °°.…”

Section: Theorem 5 For Sufficiently Large Nmentioning

confidence: 56%

“…l-(D ί=ο V l / In the present paper, we generalize the estimate ,[7(/2-I)ln2/ln(n-1)]), obtained in [2], using the method suggested in the same paper. Simultaneously, we generalize the main result of the paper [3], that is caused by our goal, but seems to be of independent interest.…”

Section: -A) I Bimentioning

confidence: 78%

“…(1) the submatrix M[ lj2j 3] obtained by deleting the fourth row and the submatrix Af [1,2,4] obtained by deleting the third row are singular and each of them contains no more than L' -2 distinct columns;…”

Section: Himentioning

confidence: 99%

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On the asymptotics of the logarithm of the number of threshold functions in K-valued logic

Irmatov¹,

Kovijanić²

1998

Discrete Mathematics and Applications

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For the number P(K,n) of threshold n-ary functions of A"-valued logic, we obtain the lower boundThe key argument in our investigation is the generalization of a result obtained by Odlyzko on the subspaces spanned by ρ randomly chosen (±1)-vectors. Namely, we prove that, as η ->· oo, for any Ρ < η -(3 + Iog 2 36)n/log 2 n if Κ -2(2, respectively, for any Ρ < η -(3 + log 20 +! 36)n/log 2 + 1 n if K = 2g + 1, the probability that the linear span of p randomly chosen vectors vi, v 2 ,... ,ν,, € (E' K ) n = {±1,±3,... ,±(2β-1)} η , respectively, from E% = {0, ±1 , . . . , ±β} η » contains at least one vector from respectively, from ι=1 equals, for even K = 2g, Q ^ 1, and for β = 1, and, respectively, for odd K = IQ + 1 , β φ 1 , (ϊ -4(2 W)") '

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“…The proof repeats the proof of the main theorem in [2]. The essential argument in the proof of the Odlyzko theorem is that the corresponding probability tends to zero as n ->· °°.…”

Section: Theorem 5 For Sufficiently Large Nmentioning

confidence: 56%

Section: -A) I Bimentioning

confidence: 78%

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On the asymptotics of the logarithm of the number of threshold functions in K-valued logic

Irmatov¹,

Kovijanić²

1998

Discrete Mathematics and Applications

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“…In 1993 inequality (2) was improved in the following way (see [9]): P(2, η) > 2" 2(1 -7/ln ' 2) P (2, [7(n -1) In 21 ln(n -1)])…”

Section: Introductionmentioning

confidence: 99%

Bounds for the number of threshold functions

Irmatov¹

1996

Discrete Mathematics and Applications

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We obtain a lower bound and refine Schl fli's upper bound for the number of threshold functions. As a consequence it is shown that the assertion that the number of threshold functions is asymptotically equal to is equivalent to the assertion that the portion of the collections consisting of η -1 different (1, -l)-vectors vj , . . ., v,,_ ι of length η such that span(vj , . . ., v"_ i) n { 1, -1 }" coincides with the set of all vectors of the form (±νι, ...,±ν,,_ι) tends to 1 as η -» °°.The work was supported by the Russian Foundation for Basic Research, grant 95-01-00369.

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“…In the paper [6] a combination of an original geometric construction with the result from the paper [12] made it possible to improve the inequality (5) upto:…”

mentioning

confidence: 99%

A Lower Bound of the Number of Threshold Functions in Terms of Combinatorial Flags on the Boolean Cube

Irmatov

2018

Preprint

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and q W l := |span(w i n−l+1 , . . . , w in ) ∩ E|. Then for any weights p = (p 1 , . . . , p 2 n ), p i ∈ R, 2 n i=1 p i = 1 we have for the number of threshold functions P (2, n) the following lower boundand the right side of the inequality doesn't depend on the choice of p. Here the indices used in the numerator correspond to vectors from span(w i 1 , . . . , w in ) ∩ E = w i 1 , . . . , w in , . . . w i q W n .

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On the number of threshold functions

Cited by 9 publications

References 0 publications

On the asymptotics of the logarithm of the number of threshold functions in K-valued logic

On the asymptotics of the logarithm of the number of threshold functions in K-valued logic

Bounds for the number of threshold functions

A Lower Bound of the Number of Threshold Functions in Terms of Combinatorial Flags on the Boolean Cube

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