Methods, apparatus, and computer software product for making a decision based on the semantics of formal logic are provided. In an exemplary embodiment, two custom computing apparatuses are used to resolve the satisfiability of a logical formula and provide an example. In this embodiment, the two custom computing apparatuses operate in concert to explore the space of possible satisfying examples. This Abstract is provided for the sole purpose of complying with the Abstract requirement rules. This Abstract is submitted with the explicit understanding that it will not be used to interpret or to limit the scope or the meaning of the claims.