Generalising Unit-Refutation Completeness and SLUR via Nested Input Resolution

Abstract

The class SLUR (Single Lookahead Unit Resolution) was introduced in Schlipf, Annexstein, Franco, and Swaminathan [43] as an umbrella class for efficient SAT solving, with in fact linear time SAT decision (while the recognition problem was not considered). Čepek, Kučera, and Vlček [12], Balyo,Štefan Gurský, Kučera, and Vlček [2] extended this class in various ways to hierarchies covering all of CNF (all clause-sets). We introduce a hierarchy SLUR k which we argue is the natural "limit" of such approaches.The second source for our investigations is the class UC of unit-refutation complete clause-sets introduced in del Val [20]. Via the theory of (tree-resolution based) "hardness" of clause-sets as developed in Kullmann [36,37], Ansótegui, Bonet, Levy, and Manyà [1] we obtain a natural generalisation UC k , containing those clause-sets which are "unit-refutation complete of level k", which is the same as having hardness at most k. Utilising the strong connections to (tree-)resolution complexity and (nested) input resolution, we develop fundamental methods for the determination of hardness (the level k in UC k ).A fundamental insight now is that SLUR k = UC k holds for all k. We can thus exploit both streams of intuitions and methods for the investigations of these hierarchies. As an application we can easily show that the hierarchies from [12, 2] are strongly subsumed by SLUR k . 1Finally we consider the problem of "irredundant" clause-sets in UC k . For 2-CNF we show that strong minimisations are possible in polynomial time, while already for (very special) Horn clause-sets minimisation is NPcomplete. We conclude with an extensive discussion of open problems and future directions.

Unified Characterisations of Resolution Hardness Measures

“…The special case of extension of "hardness" was first mentioned (as one of two possibilities) by [4]. Our motivation was that the extension of clause-sets falsifiable by unit-clause propagation yields precisely the class SLUR ( [30,33]). …”

“…See [44,48,30,33] for equivalent descriptions in this setting, where especially the algorithmic approach, via generalised unit-clause propagation r k , is notable: hardness is the minimal level k of generalised unit-clause propagation needed to derive a contradiction under any instantiation. As shown in [44,Corollary 7.9], and more generally in [48, Theorem 5.14], we have…”

“…[20,7]; by definition CANON(0) = UC 0 . See Subsection 7.2 in [33] and Subsection 9.2 in [32] for further results.…”

“…Based on the notion of "k-resolution" introduced in [39], the "asymmetric width" was introduced in [44,47,48] (and further studied in [33,32,34]). 5) Different from the symmetric width, only one parent clause needs to have size at most k (while there is no restriction on the other parent clause nor on the resolvent):…”

“…The sets UC k of all clause-sets F with hd(F ) ≤ k yield the basic hierarchy, and we have SAT decision in time O(n(F ) 2 hd(F )−2 · (F )). The special case UC 1 = UC was introduced in [59] for the purpose of Knowledge Compilation (KC), and in [30,33] it is shown that UC = SLUR holds, where SLUR is the class introduced in [54] as an umbrella class for polynomial-time SAT solving. By [6,33] we get that membership decision for UC k with k ≥ 1 is coNP-complete.…”