A geometric connection to threshold logic via cubical lattices

A cut-complex is a cubical complex whose vertices are strictly separable from the rest of the vertices of the n-cube by a hyperplane of R ⁿ . These objects render geometric presentations for threshold Boolean functions, the core objects of study in threshold logic. By applying cubical lattices and geometry of rotating hyperplanes, we prove necessary and sufficient conditions to recognize the cut-complexes with 2 or 3 maximal faces. This result confirms a positive answer to an old conjecture of Metropolis-Rota concerning cubical lattices.