The Complexity of Read-Once Resolution

Abstract

We investigate the complexity of deciding whether a propositional formula has a read-once resolution proof. We give a new and general proof of Iwama–Miynano's theorem which states that the problem whether a formula has a read-once resolution proof is i>NP-complete. Moreover, we show for fixed i>kg2 that the additional restriction that in each resolution step one of the parent clauses is a i>k-clause preserves the i>NP-completeness. If we demand that the formulas are minimal unsatisfiable and read-once refutable then the problem remains i>NP-complete. For the subclasses i>MU(i>k) of minimal unsatisfiable formulas we present a pol-time algorithm deciding whether a i>MU(i>k)-formula has a read-once resolution proof. Furthermore, we show that the problems whether a formula contains a i>MU(i>k)-subformula or a read-once refutable i>MU(i>k)-subformula are i>NP-complete.