A Modular Reduction of Regular Logic to Classical Logic

In this paper we first define a reduction Æ that transforms an instance of Regular-SAT into a satisfiability equivalent instance Æ of SAT. The reduction Æ has interesting properties: (i) the size of Æ is linear in the size of , (ii) Æ transforms regular Horn formulas into Horn formulas, and (iii) Æ transforms regular 2-CNF formulas into 2-CNF formulas. Second, we describe a new satisfiability testing algorithm that determines the satisfiability of a regular 2-CNF formula in time O(j j log j j); this algorithm is inspired by the reduction Æ. Third, we introduce the concept of renamable-Horn regular CNF formula and define another reduction Æ0 that transforms a renamable-Horn instance of Regular-SAT into a renamable-Horn instance Æ0 of SAT. We use this reduction to show that both membership and satisfiability of renamable-Horn regular CNF formulas can be decided in time O(j j log j j).