Chamber geometry and specification numbers of Boolean threshold functions
The specification number $σ_n(f)$ of a Boolean threshold function $f$ on $n$ variables is the least number of points whose $f$-values determine $f$ uniquely among all threshold functions. Its essential points form the unique minimum such set. We develop Zuev's geometric interpretation: the threshold functions are the chambers of a central hyperplane arrangement in the $(n+1)$-dimensional space of weights and thresholds, and the essential points of a function correspond exactly to the facets of its chamber, so the specification number is the chamber's facet number. The lower bound $σ_n(f)\ge n+1$ becomes the fact that a pointed full-dimensional cone has at least $n+1$ facets, with equality...
| Authors | Martin Anthony |
|---|---|
| Primary category | cs.DM |
| Published | 28 June 2026 |
| Last updated at source | 28 June 2026 |
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