"The Complexity of the Falsifiability Problem for Pure Implicational Formulas"
Article by Peter Heusch, available as BibTeX Source, postscript file and compressed postscript file.
Informatik, Universität zu Köln
Preprint Key:	zpr95-189
Keywords:	algorithms, data structures, logic in computer science, NP-complete
MSC codes:	03B99, 68Q15, 68T15
This article was published in 1995 by Springer in the proceedings "Proceedings of the 20th International Symposium, Mathematical Foundations of Computer Science. MFCS '95. 28 Aug - 1 Sep 1995. Prague, Czech Republic" (edited by J. Wiedermann, P. Hajek), series "Lecture Notes in Computer Science", volume 969, number 189, pages 221--226.
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 a is the only operator and any variable (except one) occurs at most twice. For these formulas we show that an infinite hierarchy S_1subseteq S_2cdots exists such that we can test any formula from S_i for falsifiability in time O(ni), 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 igcup_{i=1}^infty 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(nk), 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).