A Pretty Complete Combinatorial Algorithm for the Threshold Synthesis Problem

  • Christian Schilling
  • Jan-Georg Smaus
  • Fabian Wenzelmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8288)


A linear pseudo-Boolean constraint (LPB) [1,4,5] is an expression of the form a 11 + … + a m 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 [4] 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 [7]. 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.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Christian Schilling
    • 1
  • Jan-Georg Smaus
    • 2
  • Fabian Wenzelmann
    • 1
  1. 1.Institut für InformatikUniversität FreiburgGermany
  2. 2.IRITUniversité de ToulouseFrance

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