Generalising unit-refutation completeness and SLUR via nested input resolution

We introduce two hierarchies of clause-sets, SLURk and UCk, based on the classes SLUR (Single Lookahead Unit Refutation), introduced in 1995, and UC (Unit refutation Complete), introduced in 1994. The class SLUR, introduced in [Annexstein et al, 1995], is the class of clause-sets for which unit-clause-propagation (denoted by r1) detects unsatisfiability, or where otherwise iterative assignment, avoiding obviously false assignments by look-ahead, always yields a satisfying assignment. It is natural to consider how to form a hierarchy based on SLUR. Such investigations were started in [Cepek et al, 2012] and [Balyo et al, 2012]. We present what we consider the "limit hierarchy" SLURk, based on generalising r1 by rk, that is, using generalised unit-clause-propagation introduced in [Kullmann, 1999, 2004]. The class UC, studied in [Del Val, 1994], is the class of Unit refutation Complete clause-sets, that is, those clause-sets for which unsatisfiability is decidable by r1 under any falsifying assignment. For unsatisfiable clause-sets F, the minimum k such that rk determines unsatisfiability of F is exactly the "hardness" of F, as introduced in [Ku 99, 04]. For satisfiable F we use now an extension mentioned in [Ansotegui et al, 2008]: The hardness is the minimum k such that after application of any falsifying partial assignments, rk determines unsatisfiability. The class UCk is given by the clause-sets which have hardness <= k. We observe that UC1 is exactly UC. UCk has a proof-theoretic character, due to the relations between hardness and tree-resolution, while SLURk has an algorithmic character. The correspondence between rk and k-times nested input resolution (or tree resolution using clause-space k+1) means that rk has a dual nature: both algorithmic and proof theoretic. This corresponds to a basic result of this paper, namely SLURk = UC_k.