  A linear pseudo-Boolean constraint (LPB) is an expression of the form $a_1
\cdot \ell_1 + \dots + a_m \cdot \ell_m \geq d$, where each $\ell_i$ is a
literal (it assumes the value 1 or 0 depending on whether a propositional
variable $x_i$ is true or false) and $a_1, \dots, a_m, d$ 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.
