🤖 AI Summary
This study investigates the specification number of Boolean threshold functions—the minimum number of input points required to uniquely determine a function—and its geometric underpinnings. By identifying Boolean threshold functions with chambers in central hyperplane arrangements, the work precisely characterizes the specification number as the number of facets of the corresponding chamber. It establishes deep combinatorial and geometric connections via threshold polytopes and Chow vectors. The main contributions include proving that the average specification number is at most \(2n\), thereby resolving an open question posed by Gutekunst et al. regarding its asymptotic order being \(\Theta(n)\); demonstrating the geometric invariance of minimally specified functions under variable expansion, which answers the fourth operation-related question raised by Lozin et al.; and extending these results to polynomial threshold functions.
📝 Abstract
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 for simplicial chambers. The average specification number $\overlineσ_n$ becomes an average facet count. We evaluate this average exactly via the resonance arrangement and bound it through a theorem of Fukuda, Tamura, and Tokuyama, obtaining $\overlineσ_n\le 2n$; hence $\overlineσ_n=Θ(n)$. This settles a question of Gutekunst, Mészáros, and Petersen. The method also extends to polynomial threshold functions.
The same geometry links threshold functions with a threshold zonotope, whose vertices are modified Chow vectors. Its one-skeleton is the one-inclusion graph, and a vertex's degree is the specification number of that function.
Finally, we treat the operations of Lozin et al. on functions of minimum specification number. Adding a variable and extending on a variable both take the product of a chamber closure with a half-line, preserving simpliciality. For the symmetric-variables extension we give an exact thresholdness criterion and show that minimum specification number is preserved whenever the extension is a threshold function. We also resolve a question they pose concerning a fourth operation.