Zeon and Idem-Clifford Formulations of Boolean Satisfiability

Abstract

The Boolean satisfiability problem (SAT) is the problem of determining whether the variables of a given Boolean formula can be consistently replaced by true or false in such a way that the formula evaluates to true. In fact, SAT was the first known NP-complete problem. In recent years, SAT has found numerous industrial applications, particularly in model checking tools. In the current work, three approaches to Boolean satisfiability based on Clifford subalgebras are presented. In the first approach, an “idem-Clifford” algebraic test for satisfiability is presented. This test is straightforward to implement symbolically (e.g., using Mathematica), but does not yield the specific solution sets for a given formula. In the second approach, nilpotent adjacency matrix methods are extended to Boolean formulas in order to determine not only whether or not a Boolean formula is satisfiable but to explicitly obtain all solutions. This approach requires the construction of a graph associated with a given Boolean formula. Finally, a “new” algebraic framework is developed that combines the convenience of the first approach with the power of the second, recovering explicit solutions without the need to construct graphs. The algebraic formalism presented here readily lends itself to symbolic computations and provides the theoretical basis of a Clifford-algebraic SAT solver.