Truth functions realizable by single threshold organs

A threshold function is a mapping from {0, 1}n into {0, 1} satisfying: there exists integers w1, w2,..., wn, t such that f(x1, x2,..., xn) = 1 iff Sigmai=1n wi xi \geq t. Two chains of conditions necessary for a function to be a threshold function are discussed. Early parts of the two chains are equ...

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Published in2nd Annual Symposium on Switching Circuit Theory and Logical Design (SWCT 1961) pp. 225 - 245
Main Author Elgot, Calvin C.
Format Conference Proceeding
LanguageEnglish
Published IEEE 01.10.1961
Subjects
Sufficient conditions
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DOI10.1109/FOCS.1961.39

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Abstract A threshold function is a mapping from {0, 1}n into {0, 1} satisfying: there exists integers w1, w2,..., wn, t such that f(x1, x2,..., xn) = 1 iff Sigmai=1n wi xi \geq t. Two chains of conditions necessary for a function to be a threshold function are discussed. Early parts of the two chains are equivalent. One chain constitutes a sufficient condition while it is not known whether the other more intrinsic condition is sufficient or not. It is shown that any set of threshold functions of n variables realizable by a common set of weights is included in a maximal chain of threshold functions of length 1 + 2n realizable by a common set of weights. If f depends on at most 5 variables or if f or its dual has at most 4-prime implicants, then f is a threshold function iff it is 2-monotonic.
AbstractList A threshold function is a mapping from {0, 1}n into {0, 1} satisfying: there exists integers w1, w2,..., wn, t such that f(x1, x2,..., xn) = 1 iff Sigmai=1n wi xi \geq t. Two chains of conditions necessary for a function to be a threshold function are discussed. Early parts of the two chains are equivalent. One chain constitutes a sufficient condition while it is not known whether the other more intrinsic condition is sufficient or not. It is shown that any set of threshold functions of n variables realizable by a common set of weights is included in a maximal chain of threshold functions of length 1 + 2n realizable by a common set of weights. If f depends on at most 5 variables or if f or its dual has at most 4-prime implicants, then f is a threshold function iff it is 2-monotonic.
Author Elgot, Calvin C.
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Snippet A threshold function is a mapping from {0, 1}n into {0, 1} satisfying: there exists integers w1, w2,..., wn, t such that f(x1, x2,..., xn) = 1 iff Sigmai=1n wi...
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StartPage 225
SubjectTerms Sufficient conditions
Title Truth functions realizable by single threshold organs
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