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I would like to define a pseudo-boolean function $f$ such that $f(x) = 0$ for all logically valid combinations of $x\in{0,1}$ and $f(x) > 0$ for all logically invalid combinations of $x\in{0,1}$.

Suppose I have the following constraint where $x_i \in {0,1}$:

$x_1 + x_2 + x_3 \leq 2$

$x_1$ $x_2$ $x_3$ Valid
0 0 0 Yes
0 0 1 Yes
0 1 0 Yes
0 1 1 Yes
1 0 0 Yes
1 0 1 Yes
1 1 0 Yes
1 1 1 No

After binary expansion, it becomes quite trivial to see that the case $\{1,1,1\}$ is the only case that causes an invalid result. As such, $f = x_1 \land x_2 \land x_3$ is an appropriate pseudo boolean function.

Is there an efficient way to reach this conclusion without a binary expansion?

Alternatively, I can surmise that the "invalid" cases are:

$x_1 + x_2 + x_3 > 1$

Since we are dealing with binary values, this is equivalent to:

$x_1 + x_2 + x_3 \geq 2$

That looks a little bit closer to $x_1 \land x_2 \land x_3$, but I'm still not sure...

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  • $\begingroup$ You're asking for an efficient way to do something, which presumably means you're asking for an efficient algorithm. What are the inputs to the algorithm? Presumably some kind of specification of a constraint. But what kinds of constraints are allowed? Only a single linear constraint? A conjunction of multiple linear constraints? Something more general? $\endgroup$
    – D.W.
    Feb 16, 2022 at 7:11
  • $\begingroup$ I suggest you read up on en.wikipedia.org/wiki/Tseytin_transformation, en.wikipedia.org/wiki/Adder_(electronics), cs.stackexchange.com/q/82033/755 $\endgroup$
    – D.W.
    Feb 16, 2022 at 7:13

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