# Partially defined boolean function

Consider boolean function $$f(x_{1}, x_{2}, \dots, x_{n})$$. The value of $$f$$ is defined on some set of inputs, and some inputs are undefined (let us label undefined value with $$?$$). It is possible to set $$f$$‘s value on some inputs, no matter if they were initially defined.

The problem is to set some of the inputs such that:

• all initially undefined inputs become defined (i. e. no $$?$$ in function truth table);
• $$f$$ is monotone in the common sense.

The optimization part is to use the minimum possible assignments. How can one find an optimal way in the time $$\mathcal{O} \left( p(n) \left( 2^{n} \right)^{2} \right)$$, where $$p(n)$$ is polynomial?