Alternating Fixed Points in Boolean Equation Systems as Preferred Stable Models

We formally characterize alternating fixed points of boolean equation systems as models of (propositional) normal logic programs. To precisely capture this relationship, we introduce the notion of a preferred stable model of a logic program, and define a mapping that associates a normal logic program with a boolean equation system such that the solution to the equation system can be “read off” the preferred stable model of the logic program. We show that the preferred model cannot be calculated a-posteriori (i.e. compute stable models and choose the preferred one) but its computation rather is intertwined with the stable-model computation itself. This definition reveals a very natural relationship between the evaluation of alternating fixed points in boolean equation systems and the Gelfond-Lifschitz transformation used in stable-model computation. For alternation-free boolean equation systems, we show that the logic programs we derive are stratified, while for formulas with alternation, the corresponding programs are non-stratified. Consequently, our mapping of boolean equation systems to logic programs preserves the computational complexity of evaluating the solutions of special classes of equation systems (e.g., linear-time for the alternation-free systems, exponential for systems with alternating fixed points).