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.
翻译:线性伪Boolean 限制 (LPB) 是表单 $_ 1\ cdot\ cdot\ ell_ 1 +\ dots + a_ m\ cdot\ cdot\ ell\ mm\ geq d$, 其中每个$\ ell_ i$ 是字形的( 假设值 1 或 0, 取决于价格变量$x美元是真实的还是假的) 和 $a_ 1,\ dots, a_ m, d$ 是自然数字。 LPB 代表一个 Boolean 函数, 而那些完全由 LPB 代表的 Boolean 函数被称为起始函数 。 如果可能的话, 找到一个 Boolean 函数的 LPB 代表 代表 = a_ m\ mm\ cd\ cd\ m\ m = m lets 。 问题在于, 我们通过一个完全复调程序来识别阈值 。 Smaus 已经通过一个不完全的 commissional commissional commissional 运行了这个过程。