Complexity Results of Subclasses of the Pure Implicational Calculus

About 50 years ago Lukasiewicz, Tarski, see 6, 4] and others studied the implicational calculus, i. e. the set PIF (pure implicational formulas) of those propositional formulas that are constructed exclusively from Boolean variables and the propositional implication ! as the only connective. Obviously this class of formulas is not able to represent all Boolean functions. While every formula in PIF is satissable, the falsiiability problem remains NP{complete. In 3] it was shown that the falsiiability problem for PIF{formulas with every variable occurring at most twice besides one variable z occurring an arbitrary number of times remains NP{complete, it is solvable in polynomial time O(n k) when the number of occurrences of z is bounded by k. In this paper we present some further subclasses of PIF{formulas for which the falsiiabil-ity problem remains NP{complete, e. g. for those PIF{formulas F where every implicant on the path from the root of the tree that corresponds to F to the rightmost variable contains at most 3 variables. If however subformulas of type u ! (v ! w) are forbidden, then the falsiiability problem for this class is solvable in linear time. We present some further classes of PIF{formulas where falsiiability is solvable in polynomial time.