A Pretty Complete Combinatorial Algorithm for the Threshold Synthesis Problem
A linear pseudo-Boolean constraint (LPB) [1,4,5] is an expression of the form a 1ℓ1 + … + a m ℓ m ≥ d. Here each ℓ i is a literal of the form x i or 1 –x i . An LPB can be used to represent a Boolean function; e.g. 2x 1 + x 2 + x 3 ≥ 2 represents the same function as the propositional formula x1 ∨ (x2 ∧ x3).
Functions that can be represented by a single LPB are called threshold functions. The problem of finding the LPB for a threshold function given as disjunctive normal form (DNF) is called threshold synthesis problem. The reference on Boolean functions  formulates the research challenge of recognising threshold functions through an entirely combinatorial procedure. In fact, such a procedure had been proposed in [3,2] and was later reinvented by us . In this paper, we report on an implementation of this procedure for which we have run experiments for up to m = 22. It can solve the biggest problems in a couple of seconds.
Unable to display preview. Download preview PDF.
- 1.Chai, D., Kuehlmann, A.: A fast pseudo-Boolean constraint solver. In: Proceedings of the 40th Design Automation Conference, pp. 830–835. ACM (2003)Google Scholar
- 2.Coates, C.L., Kirchner, R.B., Lewis II, P.M.: A simplified procedure for the realization of linearly-separable switching functions. IRE Transactions on Electronic Computers (1962)Google Scholar
- 3.Coates, C.L., Lewis II, P.M.: Linearly-separable switching functions. Journal of Franklin Institute 272, 366–410 (1961); Also in an expanded version, GE Research Laboratory, Schenectady, N.Y., Technical Report No.61-RL-2764EGoogle Scholar
- 4.Crama, Y., Hammer, P.L.: Boolean Functions: Theory, Algorithms, and Applications. Encyclopedia of Mathematics and its Applications. Cambridge University Press (May 2011)Google Scholar
- 6.Schilling, C.: Solving the Threshold Synthesis Problem of Boolean Functions by Translation to Linear Programming. Bachelor thesis, Universität Freiburg (2011)Google Scholar
- 8.Wenzelmann, F.: Solving the Threshold Synthesis Problem of Boolean Functions by a Combinatorial Algorithm. Bachelor thesis, Universität Freiburg (2011)Google Scholar