Implementations of two Algorithms for the Threshold Synthesis Problem
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A linear pseudo-Boolean constraint (LPB) is an expression of the form a_1 ·ℓ_1 + … + a_m ·ℓ_m ≥ d, where each ℓ_i is a literal (it assumes the value 1 or 0 depending on whether a propositional variable x_i is true or false) and a_1, …, a_m, d are natural numbers. An LPB represents a Boolean function, and those Boolean functions that can be represented by exactly one LPB are called threshold functions. The problem of finding an LPB representation of a Boolean function if possible is called threshold recognition problem or threshold synthesis problem. The problem has an O(m^7 t^5) algorithm using linear programming, where m is the dimension and t the number of terms in the DNF input. It has been an open question whether one can recognise threshold functions through an entirely combinatorial procedure. Smaus has developed such a procedure for doing this, which works by decomposing the DNF and "counting" the variable occurrences in it. We have implemented both algorithms as a thesis project. We report here on this experience. The most important insight was that the algorithm by Smaus is, unfortunately, incomplete.
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