The complexity of the falsifiability problem for pure implicational formulas

Abstract

Since it is unlikely that any NP-complete problem will ever be efficiently solvable, one is interested in identifying those special cases that can be solved in polynomial time. We deal with the special case of Boolean formulas where the logical implication → is the only operator and any variable (except one) occurs at most twice. For these formulas we show that an infinite hierarchy $S_1 \subseteq S_2$... exists such that we can test any formula from S _i for falsifiability in time O(n ^Sl), where n is the number of variables in the formula. We describe an algorithm that finds a falsifying assignment, if one exists. Furthermore we show that the falsifiability problem for ⋃S_i is NP-complete by reducing the SAT-Problem. In contrast to the hierarchy described by Gallo and Scutella for Boolean formulas in CNF, where the test for membership in the k-th level of the hierarchy needs time O(n ^k), our hierarchy permits a linear time membership test. Finally we show that S ₁ is neither a sub-nor a superset of some commonly known classes of Boolean formulas, for which the SAT-Problem has linear time complexity (Horn formulas, 2-SAT, nested satisfiability).