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.

Abstract. The class SLUR (Single Lookahead Unit Resolution) was introduced in [22] as an umbrella class for efficient SAT solving. [7,2] 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 [10]. Via the theory of (tree-resolution based) "hardness" of clause-sets as developed in [19,20,1] we obtain a natural generalisation UC k , containing those clause-sets which are "unitrefutation 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 [7,2] are strongly subsumed by SLUR k . We conclude with a discussion of open problems and future directions.

Knowledge Compilation (KC) studies compilation of boolean functions f into some formalism F , which allows to answer all queries of a certain kind in polynomial time. Due to its relevance for SAT solving, we concentrate on the query type "clausal entailment" (CE), i.e., whether a clause C follows from f or not, and we consider subclasses of CNF, i.e., clause-sets F ∈ CLS with special properties (CNF itself is not suitable for CE queries unless P=NP). In this report we do not allow auxiliary variables (except of the Outlook), and thus F needs to be equivalent to f . We consider the hierarchies UC k ⊆ WC k ⊂ CLS (k ∈ N0), which were introduced in [26,27], and where each level allows CE queries. The first two levels are well-known classes for KC, namely UC0 = WC0 is the same as PI as studied in KC, that is, f is represented by the set of all prime implicates, while UC1 = WC1 is the same as UC, the class of unit-refutation complete clause-sets introduced in [20]. We show that for each k there are (sequences of) boolean functions with polysize representations in UC k+1 , but with an exponential lower bound on representations in WC k . Such a separation was previously only know for k = 0. We also consider PC ⊂ UC, the class of propagation-complete clause-sets introduced in [52,11]. Strengthening [2], we show that there are (sequences of) boolean functions with polysize representations in UC, while there is an exponential lower bound for representations in PC. These separations are steps towards a general conjecture determining the representation power of the hierarchies PC k ⊂ UC k ⊆ WC k . The strong form of this conjecture also allows auxiliary variables, as discussed in depth in the Outlook.

We study the problem of finding good CNF-representations F of systems of linear equations S over the two-element field, also known as systems of XORconstraints x1 ⊕ · · · ⊕ x k = ε, ε ∈ {0, 1}, or systems of parity-constraints. The number of equations in S is m, the number of variables is n. These representations are used as parts of SAT problems F * ⊃ F , such that F has "good" properties for SAT solving in the context of F * ; here F * may for example represent the problem of finding the key for a cryptographic cipher. The basic quality criterion is "arc consistency" (AC), that is, for every partial assignment ϕ to the variables of S, all assignments xi = ε forced by ϕ are determined by unit-clause propagation on the result ϕ * F of the application.We show there is no AC-representation of polynomial size for arbitrary S. We use the lower bound on monotone circuits for monotone span programs from [2], and we show a close relation between monotone circuits and ACrepresentations, based on the work in [9]. We then turn to constructing good representations. We analyse the basic translation F = X1(S), which translates each constraint on its own, by splitting up x1 ⊕ · · · ⊕ x k into sums x1 ⊕ x2 = y2, y2 ⊕ x3 = y3, . . . , y k−1 ⊕ x k = ε, introducing auxiliary variables yi. We show that X1(S * ), where S * is obtained from S by considering all derived equations, is an AC-representation of S. The derived equations are obtained by adding up the equations of all sub-systems S ′ ⊆ S. There are 2 m such S ′ , and computing an AC-representation is fixed-parameter tractable (fpt) in the parameter m, improving [61], which showed fpt in n.To obtain stronger representations, instead of mere AC we consider the class PC of propagation-complete clause-sets, as introduced in [12]. The stronger criterion is F ∈ PC, which requires for all partial assignments, possibly involving also the auxiliary (new) variables in F , that forced assignments can be determined by unit-clause propagation. Using "propagation hardness" phd(F ) ∈ N0 as introduced in [37], we have F ∈ PC ⇔ phd(F ) ≤ 1. We show that X1 applied to a single equation (m = 1) yields a translation in PC, i.e., phd(X1(S)) ≤ 1. Then we study m = 2. Now S * has two equations more, and X1(S * ) is an AC-representation, but the "distance" to PC is arbitrarily high, i.e., phd(X1(S * )) is unbounded (using results from [54]). We show two possibilities to remedy this (for m = 2). On the one hand, if instead of unit-clause propagation we allow (arbitrary) resolution with clauses of length at most 3 (i.e., 3-resolution), and only require refutation of inconsistencies after (arbitrary, partial) instantiations, then even just X1(S) suffices. On the other hand, with a more intelligent translation, which avoids duplication of equivalent auxiliary variables yi, we obtain a (short) representation in PC. We conjecture that also the general case can be handled this way, that is, computing a representation F ∈ PC of S is fpt in m.Recall that the two-element field Z 2 has elements 0, 1, where ad...

We present a general framework for "good CNF-representations" of boolean constraints, to be used for translating decision problems into SAT problems (i.e., deciding satisfiability for conjunctive normal forms). We apply it to the representation of systems of XOR-constraints ("exclusive-or"), also known as systems of linear equations over the two-element field, or systems of parity constraints, or as systems of equivalences (XOR is the negation of an equivalence).The general framework defines the notion of "representation", and provides several methods to measure the quality of the representation, by measuring the complexity ("hardness") needed for making implicit "knowledge" of the representation explicit (to a SAT-solving mechanism). We obtain general upper and lower bounds.Applied to systems of XOR-constraints, we show a super-polynomial lower bound on "good" representations under very general circumstances. A corresponding upper bound shows fixed-parameter tractability in the number of constraints.The measurement underlying this upper bound ignores the auxiliary variables needed for shorter representations of XOR-constraints. Improved upper bounds for special cases take them into account, and a rich picture begins to emerge, under the various hardness measurements.