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I'm wondering about systems of equations that take on a finite set of values for those variables. Is there a theory behind that? In particular, solving systems where the variable takes on values of either 0 or 1. The problem is probably NP-hard, but I'm wondering where to look to learn more about it.

Like for example,

$a^2 + bc - a = 0$

$b(a + d) - b = 0$

$c(a + d) - c = 0$

$d^2 + bc - d = 0$ where a, b, c, d are from {0, 1}.

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    can't you just try all alternatives? 2^4 isn't that big number.2012-12-08
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    Sounds like you are interested in 0-1 programming. You might start with http://en.wikipedia.org/wiki/Linear_programming#Integer_unknowns2012-12-08

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