Finding Tractable Formulas in NNF

Many applications in Computer Science require to represent knowledge and to reason with non normal form formulas. However, most of the advances in tractable reasoning are applied only to CNF formulas. In this paper, we extend tractability to several classes of non normal formulas which are of high practical interest. Thus, we rst deene three non normal Horn-like classes of formulas F1 ^ F2 ^ : : : ^ Fn where each Fi is constituted by a disjunction of two optional terms Fi = N N F ? i _C + i : the rst one is in Negation Normal Form (NNF) composed exclusively with negative literals and the second one is a conjunction of positive propositions. These formulas codify the same problems that the Horn formulas but with signiicantly, even exponentially, less propositional symbols. Second, we deene sound and refutational complete inference rule sets for each class. Our third contribution consists in the design of a sound, complete and strictly linear running time algorithm for each class. As a result, the time required by our linear algorithms running on the de-ned non normal Horn-like formulas can be exponentially less than that required by the existing linear Horn-SAT algorithms.