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1987
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-trivial 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 from the appropriate theory of computational equivalence and replaceability. The prospects for application to monotone boolean function complexity are briefly examined, A catalogue of known relations is included as an Appendix.
Information and Computation
Complexity of DNF minimization and isomorphism testing for monotone formulas2008 •
1999 •
We introduce the concept of multilinear partition of a point set V⊂Rn and the concept of multilinear separability of a function f:V→K={0, ..., k-1}. Based on well known relationships between linear partitions and minimal pairs, we derive formulae for the number of multilinear partitions of a point set in general position and of the set K2. The (n, k, s)-perceptrons partition the input space V into s+1 regions with s parallel hyperplanes. We obtain results on the capacity of a single (n, k, s)-perceptron, respectively for V⊂Rn in general position and for V=K2. Finally, we describe a fast polynomial-time algorithm for counting the multilinear partitions of K2
Proceedings 1999 29th IEEE International Symposium on Multiple-Valued Logic (Cat. No.99CB36329)
The number of cascade functions1999 •
SIAM Journal on Computing
Boltzmann Samplers, Pólya Theory, and Cycle Pointing2011 •
The IMA Volumes in Mathematics and Its Applications
Generalized Rook Polynomials and Orthogonal Polynomials1989 •
2010 •
2003 •
2007 •
Journal of Computer and System Sciences
Complexity of reachability problems for finite discrete dynamical systems2006 •
Discrete Event Dynamic Systems
A Control Theory for Boolean Monomial Dynamical Systems2010 •
2001 •
2003 IEEE Aerospace Conference Proceedings (Cat. No.03TH8652)
A novel model-based diagnosis engine: theory and applications2003 •
Systems and synthetic biology
Monotone and near-monotone biochemical networks2007 •