Suppose we have a Boolean expression $$(\neg x_{1}\wedge\neg x_{2})\vee\left(\neg x_{1}\wedge\neg x_{3}\right),$$ which we need to be true. Is there a method to convert this to a linear expression of the form $$a_1x_1+a_2x_2+a_3x_3<1,$$ $$x_1,x_2,x_3\in\{0,1\}$$where $a_1,a_2,a_3$ are real constants. Such that the Boolean expression is satisfied iff the integer linear expression is satisfied. The question is how to find $a_1,a_2,a_3$ uniquely? There is a method to convert later linear expression into the Boolean expression but we need other way around.
Thanks a lot.
You may take
$$ f(x_1,x_2,x_3)=(1+\varepsilon)x_1+(1-\varepsilon)(x_2+x_3) $$
where $\varepsilon \in (0,\frac{1}{2})$ is a constant.
When $(x_1,x_2,x_3)$ satisfies your condition, one has $f(x_1,x_2,x_3) \leq 1-\varepsilon$.
When $(x_1,x_2,x_3)$ does not satisfy your condition, one has $f(x_1,x_2,x_3) \geq 2(1-\varepsilon)$.
Note that there is no “uniqueness”.