On the Number of Two-Dimensional Threshold Functions
نویسندگان
چکیده
منابع مشابه
On number of two-dimensional threshold functions
2-D threshold function f is a function defined on the integer interior of a rectangular area [0, m] × [0, n] on a plane and takes values from the set {0, 1} such that there is a straight line separating the pre-images f −1 (0) and f −1 (1). For the number of such functions only asymptotic bounds were known. We give an exact formula for the number of 2-D threshold functions and derive an asympto...
متن کاملNote on the Number of Two-Dimensional Threshold Functions
A two-dimensional threshold function of k-valued logic can be viewed as coloring of the points of a k × k square lattice into two colors such that there exists a straight line separating points of different colors. For the number of such functions only asymptotic bounds are known. We give an exact formula for the number of two-dimensional threshold functions and derive more accurate asymptotics.
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Let m,n ≥ 2, m ≤ n. It is well-known that the number of (twodimensional) threshold functions on an m× n rectangular grid is t(m,n) = 6 π (mn) +O(mn logn) +O(mn log logn) = 6 π (mn) +O(mn logm). We improve the error term by showing that t(m,n) = 6 π (mn) +O(mn).
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It is known that the minimal teaching set of any threshold function defined on a two-dimensional rectangular grid consists of 3 or 4 points. Exact formulae for the numbers of functions corresponding to these values are proposed. Also, we prove that the average cardinality of the minimal teaching set of a threshold function is asymptotically 72 . Corollaries of these results concerning special a...
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ژورنال
عنوان ژورنال: SIAM Journal on Discrete Mathematics
سال: 2010
ISSN: 0895-4801,1095-7146
DOI: 10.1137/090750184