Canonicalization of Threshold Logic Representation and Its Applications

Threshold logic functions gain revived attention due to their connection to neural networks employed in deep learning. Despite prior endeavors in the characterization of threshold logic functions, to the best of our knowledge, the quest for a canonical representation of threshold logic functions in the form of their realizing linear inequalities remains open. In this paper we devise a procedure to canonicalize a threshold logic function such that two threshold logic functions are equivalent if and only if their canonicalized linear inequalities are the same. We further strengthen the canonicity to ensure that symmetric variables of a threshold logic function receive the same weight in the canonicalized linear inequality. The canonicalization procedure invokes $O(m)$ queries to a linear programming (resp. an integer linear programming) solver when a linear inequality solution with fractional (resp. integral) weight and threshold values is to be found, where $m$ is the number of symmetry groups of the given threshold logic function. The guaranteed canonicity allows direct application to the classification of NP (input negation, input permutation) and NPN (input negation, input permutation, output negation) equivalence of threshold logic functions. It may thus enable applications such as equivalence checking, Boolean matching, and library construction for threshold circuit synthesis.