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Monotone Boolean functions as combinatorially piecewise linear maps

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Beynon, Meurig (1987) Monotone Boolean functions as combinatorially piecewise linear maps. University of Warwick. Department of Computer Science. (Department of Computer Science research report). (Unpublished)

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Abstract

The class of combinatorially piecewise linear (cpl) maps was first introduced to solve a geometric problem concerning the representability of piecewise linear functions as pointwise maxima of minima of linear functions. Such maps correspond in a canonical fashion to monotone boolean functions. This paper describes how a monotone boolean function in n variables whose prime implicants and prime clauses are non-rivial defines a partition of the symmetric group on n symbols into a set of "singular cycles" representing relations between transpositions of adjacent symbols. Several possible approaches to the classification of such cycles are described, and some characteristic properties of singular cycles are identified. The potential computational significance of singular cycles is indicated with reference to new combinatorial models for monotone boolean formulae and circuits that arise directly form the
appropriate theory of computational equivalence and replaceability. The prospects for application to monotone boolean complexity are briefly examined. A catalogue of known relations is included as an Appendix.

Item Type: Report
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science > Computer Science
Library of Congress Subject Headings (LCSH): Linear operators, Functions
Series Name: Department of Computer Science research report
Publisher: University of Warwick. Department of Computer Science
Official Date: November 1987
Dates:
DateEvent
November 1987Completion
Number: Number 109
DOI: CS-RR-109
Institution: University of Warwick
Theses Department: Department of Computer Science
Status: Not Peer Reviewed
Publication Status: Unpublished
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