Local autarkies searching for the dynamic partition ... - ResearchGate

Local autarkies searching for the dynamic partition of CNF formulaeÉric Grégoire Bertrand Mazure Lakhdar SaïsUniversité Lille-Nord de FranceCRIL - CNRS UMR 8188Artois, F-62307 Lens{gregoire,mazure,sais}@cril.frAbstractIn this paper an original dynamic partition of formulaein Conjunctive Normal Form (CNF) is presented. It isbased on the autarky concept first introduced by Monienand Speckenmeyer and further investigated by Kullmannand Van Gelder. Intuitively, an autarky is a partial assignmentsatisfying some clauses while not affecting any literalin any other clause, leading to a partition of the CNF formula.Autarkies can play a dramatic role in the efficiency ofmodern SAT solvers. The approach in this paper aims to dynamicallyextend the current partial assignment to a localautarky thanks to an inference rule based on unit propagation.More precisely, at each node of the search tree, it ischecked whether the current decision literal can be mademonotone by subsuming all the clauses where it appearsnegatively. The formal framework is detailed and its technicalfeatures discussed.1. IntroductionThe SAT problem, i.e., the problem of checking whethera set of Boolean clauses is satisfiable or not, is centralin many computer science and artificial intelligencedomains, including e.g. constraint satisfaction problems(CSP), planning, non-monotonic reasoning and VLSIcorrectness checking. Today, SAT has gained a considerableaudience with the advent of a new generation ofSAT solvers able to solve large complex instances encodingreal-world applications. These solvers also appearto provide crucial base-building blocks for variousother problem-solving technologies, like SMT solving, theoremproving, model finding and QBF solving.These solvers, called modern SAT solvers [19, 6], arebased on the standard unit propagation technique [4] combinedin a smart way with efficient data structures, togetherwith: (i) restart policies [7, 12], (ii) activity-based variableselection heuristics (VSIDS-like) [3, 19], and (iii) clauselearning [17, 1, 19]. Modern SAT solvers can be interpretedas extended versions of the well-known DPLL-like(short for Davis, Putnam, Logemann and Loveland) procedureaugmented with these various enhancements. Let usstress that the well-known resolution rule still plays a dramaticrole in the efficiency of modern SAT solvers whichcan be understood as a particular form of general resolution[2]. These lookback-based SAT solvers are particularlyefficient on SAT instances encoding practical applications.Other lookahead-based SAT solvers are designed using efficientlocal processing techniques and variable branchingheuristics (e.g. Satz [14], kcnfs [5], March-DL [8]). Thislast family of solvers are particularly efficient on randomand crafted SAT instances.The recent dramatic breakthrough in solving complexindustrial SAT instances can often be related to the presenceof hidden structures exploited in a smart way by theSAT solvers. Indeed, such instances are very suitable fordecomposition-based techniques. However, most of the decompositiontechniques are much time consuming. Lightdivide-and-conquer approaches have been proposed previously,using for example structure-based variable orderingheuristics to recursively decompose the SAT instances [9].In [15], a dynamic decomposition method based on hypergraphseparators is proposed. The authors give a nice integrationof the separator decomposition into variable orderingof a modern SAT solver that speedups the searchon real-world SAT instances. Other approaches are basedon the exploitation of autarky assignments originally proposedby Monien and Speckenmeyer [18] and further investigatedby Kullmann [13] and Van Gelder [21]. Morerecently, Liffton and Sakallah [16] proposed a novel algorithmthat searches for autarkies directly using a standardsatisability solver and explore the potential of trimming autarkiesin minimal unsatisable subsets (MUS) and minimalcorrection subsets (MCS) extraction flows. Let us stress thatthe dynamic exploitation of autarkies assignments in DPLLlikesearch procedures actually goes back to Oxusoff’s andRauzy’s work [20]. In their paper, these latter authors pro-

posed several variable branching heuristics to dynamicallypartition the CNF formula into disjoint sets of clauses. Moreprecisely, the proposed variable ordering heuristics aim topartition the formula: the next variable to assign is selectedin such a way that its assignment leads to a new sub-formulaincluded in previous ones at the same branch of the tree.The approach that will be presented in this paper followsthis idea but is heuristic-independent. More precisely, let xbe the next variable to assign, the proposed approach triesto subsume the clauses where ¬x appears and then deducethat such a literal is monotone 1 . This clearly extends the localautarky assignments, using subclause deduction. Indeed,a partial assignment that is not a local autarky can be transformedinto a local autarky.The paper is organized as follows. After some preliminarydefinitions and notations, basic and local autarkies,Oxusoff’s and Rauzy’s dynamic approach are described insection 3. Then our generalized local autarkies are formallypresented in section 4. Finally, we conclude by introducingseveral promising perspectives opened by this framework.2. Technical background2.1. Preliminary definitions and notationsA CNF formula Σ is a set (interpreted conjunctively) ofclauses, where a clause is a disjunction of literals. A literalis a positive (x) or negated (¬x) propositional variable.The two literals x and ¬x are called complementary. Wenote by ¯l the complementary literal of l. For a set of literalsL, ¯L is defined as {¯l | l ∈ L}. A unit clause is a clausecontaining only one literal (called unit literal), while a binaryclause contains exactly two literals. The empty clause,noted ⊥, is interpreted as false (unsatisfiable), whereas theempty CNF formula, noted ⊤, is interpreted as true (satisfiable).We define the size |Σ| of a CNF formula Σ as∑c∈Σ|c|, where |c| is the number of literals in c. The numberof clauses in Σ is denoted C Σ .The set of variables occurring in Σ is noted V Σ . A setof literals is complete if it contains one literal for each variablein V Σ , and fundamental if it does not contain complementaryliterals. An assignment ρ of a Boolean formula Σis a function that associates a value ρ(x) ∈ {false,true}to some of the variables x ∈ V Σ . ρ is complete if it assignsa value to every x ∈ V Σ , and partial otherwise. An assignmentis alternatively represented by a fundamental set of literals,in the obvious way. A model of a formula Σ is an assignmentρ that makes the formula true; noted ρ Σ.The following notations will also be used throughout thepaper:1 A monotone literal is a literal that occurs either positively or negativelyin the CNF formula (but not in both polarities).• Σ| x will denote the formula obtained from Σ by assigningx the truth-value true. Formally Σ| x = {c | c ∈Σ, {x, ¬x} ∩ c = ∅} ∪ {c\{¬x} | c ∈ Σ, ¬x ∈ c}(that is: the clauses containing x are removed;and those containing ¬x are simplified). This notationis extended to assignments: given an assignmentρ = {x 1 ,...,x n }, Σ| ρ is defined as(...((Σ| x1 )| x2 )...| xn ).• Σ ∗ denotes the formula Σ closed under unit propagation,defined recursively as follows: (1) Σ ∗ = Σ if Σdoes not contain any unit clause, (2) Σ ∗ =⊥ if Σ containstwo unit-clauses {x} and {¬x}, (3) otherwise,Σ ∗ = (Σ| x ) ∗ where x is the literal appearing in a unitclause of Σ. A clause c is deduced by unit propagationfrom Σ, noted Σ ∗ c, if (Σ|¯c ) ∗ =⊥.Let c 1 and c 2 be two clauses of a formula Σ. We say thatc 1 (respectively c 2 ) subsumes (respectively is subsumed by)c 2 (respectively by c 1 ) iff c 1 ⊆ c 2 . If c 1 subsumes c 2 , thenc 1 c 2 (the converse is not true). Also Σ and Σ − c 2 areequivalent with respect to satisfiability.2.2. DPLL searchDPLL [4] is a tree-based backtrack search procedure; ateach node of the search tree, the assigned literals (both thedecision literal and the propagated ones) are labeled withthe same decision level starting from 1 and increased at eachdecision (or branching). After backtracking, some variablesare unassigned, and the current decision level is decreasedaccordingly. At the ith level, the current partial assignmentρ can be represented as a sequence of decision-propagationsteps of the form 〈(x i k ),xi k 1,x i k 2,...,x i k nk〉 where the firstliteral x i k corresponds to the decision literal x k assigned atthe ith level and each x i k jfor 1 ≤ j ≤ n k represents unitpropagated literals at the ith level. Let x ∈ ρ, we note δ(x)the assignment level of x.3. Autarky assignmentsThe autarky concept has been introduced in [18].Definition 1 Let Σ be a CNF formula and ρ a partial assignment.ρ is an autarky if Σ| ρ ⊆ Σ i.e. ∀c ∈ Σ, ¯ρ ∩ c = ∅or ρ ∩ c ≠ ∅.For a CNF formula, computing an autarky when it existsis clearly (F)NP-complete. However, autarkies can be usedto improve the efficiency of SAT solvers [21, 13]. Indeed,the refutation of Σ| ρ leads to the proof of the unsatisfiabilityof Σ. Following definition 1, the property prop:inclusioncan be obtained.Property 1 If ρ is an autarky of the CNF formula Σ thenΣ Σ| ρ .

2nd inclusionbetweenΣ and Σ| {l1,¬l2,l3,l4}Σ is proved unsatisfiable1st inclusionbetweenΣ| {l1} et Σ| {l1,¬l2,l3}Σ| {l1,¬l2,l3,l4,l5}l6⊥Σ| {l1,¬l2,l3,l4}l5l2⊥Σ|l1Σ| {l1,¬l2,l3}¬l6⊥l4l6⊥l1l3/ΣCut(2nd local autarky)¬l2Σ| {l1,¬l2}Cut(1st local autarky)Cut(2nd local autarky)¬l5Σ| {l1,¬l2,l3,l4,¬l5}Figure 1. DPLL search tree and local autarkiesIn [10, 20] the authors exploit this property to prune thesearch tree of DPLL-like procedures. To this end, a local autarkyconcept is defined and exploited.Definition 2 A partial assignment ρ = {l 0 ,...,l i } of aCNF formula Σ is a local autarky if ∃ρ ′ = {l 0 ,...,l j } withj < i such that Σ| ρ ⊆ Σ| ρ ′.Property 1 also applies for local autarkies. Indeed, fromDefinition 2 and Property 1, Σ| ρ ′ Σ| ρ can be deduced.This can be used to prune the search tree of the DPLL algorithm.Indeed, if ρ is a local autarky with respect to the partialassignment ρ ′ then Σ| ρ ′ is satisfiable if and only if Σ| ρis satisfiable. All branches not explored between ρ ′ and ρcan be pruned if Σ ρ is proved unsatisfiable.The following example illustrates this property.Example 1 Let Σ = {(l 1 ∨ l 2 ∨ ¬l 4 ),(¬l 1 ∨ l 4 ),(¬l 1 ∨l 2 ∨l 3 ),(¬l 2 ),(l 5 ∨l 6 ),(l 5 ∨ ¬l 6 ),(¬l 5 ∨l 6 ),(¬l 5 ∨ ¬l 6 )}.The search tree of the DPLL algorithm represented inFigure 1 is obtained thanks to the exploitation of local autarkies.This corresponds to the application of the local autarkypruning rule as defined by Jeannicot et al. in [10]. Inthe following example, the unit and monotone literals propagatedduring search are not mentioned for clarity reasons.From the search tree, the following formulae associatedto the different nodes are obtained:• Σ| {l1} = {(l 4 ),(l 2 ∨ l 3 ),(¬l 2 ),(l 5 ∨ l 6 ),(l 5 ∨ ¬l 6 ),(¬l 5 ∨ l 6 ),(¬l 5 ∨ ¬l 6 )}• Σ| {l1,¬l 2} = {(l 4 ),(l 3 ),(l 5 ∨ l 6 ),(l 5 ∨ ¬l 6 ),(¬l 5 ∨l 6 ),(¬l 5 ∨ ¬l 6 )}• Σ| {l1,¬l 2,l 3} = {(l 4 ),(l 5 ∨ l 6 ),(l 5 ∨ ¬l 6 ),(¬l 5 ∨l 6 ),(¬l 5 ∨ ¬l 6 )}¬l6⊥//• Σ| {l1,¬l 2,l 3,l 4} = {(l 5 ∨l 6 ),(l 5 ∨¬l 6 ),(¬l 5 ∨l 6 ),(¬l 5 ∨¬l 6 )}• Σ| {l1,¬l 2,l 3,l 4,l 5} = {(l 6 ),(¬l 6 )}• Σ| {l1,¬l 2,l 3,l 4,¬l 5} = {(l 6 ),(¬l 6 )}Consequently, the following inclusion holds:Σ| {l1,¬l 2,l 3} ⊂ Σ| {l1,¬l 2}Σ| {l1,¬l 2,l 3} ⊂ Σ| {l1}Σ| {l1,¬l 2,l 3,l 4} ⊂ Σ| {l1,¬l 2,l 3}Σ| {l1,¬l 2,l 3,l 4} ⊂ Σ| {l1,¬l 2}Σ| {l1,¬l 2,l 3,l 4} ⊂ Σ| {l1}Σ| {l1,¬l 2,l 3,l 4} ⊂ Σ) First application ofpruning rules of local⎞autarkySecond application⎟⎠of pruning rules oflocal autarkyClearly, the local autarky pruning rule is a generalizationof the monotone literal pruning one. Indeed, if l is amonotone literal, then Σ| l is the formula obtained from Σwhere all clauses containing l are removed. In consequenceΣ| l ⊂ Σ and {l} is a local autarky for Σ. Using the pruningrule based on local autarky allows the simplification bymeans of the monotone literal rule. Let us note that monotoneliterals are not exploited in modern SAT solvers thatexploit lazy data structures like watched literals. In otherwords, the occurrence number of a given literal at each nodeof the search tree is not recorded or updated.At a given level i of the search tree, finding the smallestlevel j < i such that Σ| {l0,...,l i} is included inΣ| {x0,x 1,...,x j} is computationally hard. However the followingproperty allows an incremental detection that ensuresthe best pruning.Property 2 ([20]) If Σ| {l0,...,l i} ⊆ Σ| {l0,...,l j} withj < i, then Σ| {l0,...,l i} is also included in Σ| {l0,...,x j+1},Σ| {l0,...,l j+2}, ..., and Σ| {l0,...,l i−1}.To avoid useless inclusion tests, it is more convenient touse the following corollary.Corollary 1 If Σ| {l0,...,l i} ⊄ Σ| {l0,...,l j}, then Σ| {l0,...,l i}is not included in all the following formulas: Σ| {l0,...,l j−1},Σ| {l0,...,l j−2}, ..., Σ| {l0} and Σ.The efficiency of the detection of the greatest value i−j,called a partition, depends only on the efficiency of the inclusiontest between two formulas associated to two consecutivenodes in the current branch of the search tree. Thistest can be performed efficiently using adapted data structures.Finally, to ensure that all inclusions are detected, it isnecessary to remove subsumed clauses during the DPLLprocedure, as illustrated by the following example.Example 2 Let Σ = {(¬l 1 ∨ l 3 ),(l 2 ∨ l 4 ),(¬l 2 ∨ l 3 ∨ l 4 )},and Σ| {l 1} = {(l 3 ),(l 2 ∨ l 4 ),(¬l 2 ∨ l 3 ∨ l 4 )}Σ| {l1,l 2} = {(l 3 ),(l 3 ∨ l 4 )}

In this case no inclusion is detected. But if the subsumedclauses are removed from Σ| {l1,l 2}, then Σ| {l1,l 2} ={(l 3 )} ⊂ Σ| {l1}.The subsumption test can be achieved efficiently usingfor example the recent approach by Lintao Zhang [22].In the next section, a generalization of the (local) autarkyassignments is introduced.4. Generalized (local) autarkyThe generalization is obtained by substituting the syntacticalinclusion operation (⊂) by the semantic logical consequencerelationship () in Definitions 1 and 2.Proposition 1 (Generalized (local) autarky) Let Σ be aCNF formula, ρ = {l 0 ,...,l j ,...,l i } is a generalizedlocal autarky if there exists ρ ′ = {l 0 ,...,l j } such thatΣ| ρ ′ Σ| ρ . If ρ ′ = ∅ then ρ is a generalized autarky.This is clearly a generalization of the previous concept,because if Ω ⊂ Σ then Σ Ω while the converse is false,as illustrated by the following example.Example 3 Let Σ = {(a ∨ b ∨ c),(¬a ∨ b)} and Ω =Σ| ¬a = {(b ∨ c)}, clearly Σ Ω but Ω ⊄ Σ.The most important feature of local autarkies is the incrementaldetection which ensures the greatest inclusion orpartition (see Property 2). This property is preserved for thegeneralized autarky assignments.Property 3 If Σ| {l0,...,l j} Σ| {l0,...,l j,...,l i}, thenΣ| {l0,...,l j,...,l i} is a logical consequence of all the followingformulae: Σ| {l0,...,l j+1}, Σ| {l0,...,l j+2}, ... andΣ| {l0,...,l i−1}.As before, the following corollary avoids useless logicalconsequence tests (CoNP-Complete) in order to computethe greatest partition.Corollary 2 If Σ| {l0,...,l j} Σ| {l0,...,l j,...,l i}, thenΣ| {l0,...,l i} is not a logical consequence of the followingformulae: Σ| {l0,...,l j−1}, Σ| {l0,...,l j−2}, ..., Σ| {l0} andΣ.We have seen that the propagation of monotone literalsis a special case of the local autarky detection. The followingproperty shows that the classical unit propagation is alsoa special case of the generalized local autarkies.Property 4 Let Σ be a CNF formula and l a literal occurringin a unit clause of Σ, we have Σ Σ| {l} .By extension, it can be shown that:Corollary 3 Let Σ a CNF formula, Σ Σ ∗ .More generally, if Σ l with l a literal of the CNF formulaΣ, then Σ Σ| {l} . This means that the detection ofgeneralized local autarkies includes the propagation due toany implied literal.The use of generalized local autarkies within a DPLL algorithmcan be achieved in the same way than for local autarkies.Unfortunately, if the test of inclusion can be computedefficiently for local autarkies, the detection of generalizedautarkies requests a CoNP-complete test.4.1. Generalized (local) autarky modulo unit propagationIn order to pratically exploit the generalized local autarkies,it is necessary to weaken the deductive relation thatis used. Accordingly, the full consequence relationship ()is weakened down to the mere unit propagation rule ( ∗ ),which can be computed in linear-time.Definition 3 Let Σ and Ω be two sets of clauses, Ω is adeductive consequence of Σ restricted to unit propagation,noted Σ ∗ Ω, if and only if ∀c ∈ Ω, Σ ∗ c.Clearly, if Σ ∗ Ω then Σ Ω. In consequence, Proposition1 can be adapted to introduce the generalized local autarkymodulo unit propagation.Proposition 2 (Generalized (local) autarky modulounit propagation) Let Σ be a CNF formula,ρ = {l 0 ,...,l j ,...,l i } is a generalized local autarky modulounit propagation if there exists ρ ′ = {l 0 ,...,l j } suchthat Σ| ρ ′ ∗ Σ| ρ . If ρ ′ = ∅ then ρ is a generalized autarkymodulo unit propagation.Earlier in the paper, it has been shown that the propagationsof monotone, unit and more generally implied literalsare special cases of local autarky detection. Likewise,the detection of local autarky modulo unit propagation automaticallycomputes the propagations of monotone and unitliterals, too. The following property proves useful in that respect:Property 5 Let Σ and Ω be two CNF formulae. If Ω ⊆ Σ,then Σ ∗ Ω.This property is easily proved. Indeed, each clause c of Ωis present in Σ, also (Σ|¯c ) ∗ = ⊥ because the clause c of Σ isreduced to an empty clause by the propagation of ¯c. In consequence,for each clause c of Ω Σ ∗ c and by definitionΣ ∗ Ω. The converse is false, as illustrated by the followingexample.Example 4 Let Σ = {(a ∨ b ∨ c),(a ∨ ¬b),(b ∨ ¬c),(c ∨¬a)} and Ω = Σ| b = {(a),(c ∨ ¬a)}, we have Σ ∗ Ω butΩ ⊄ Σ.

So, Property 5 ensures that all classical local autarkiesare also detected by generalized local autarkies modulo unitpropagation. In consequence, the simplification of monotoneliterals is implicitly included during the detection ofgeneralized local autarkies modulo unit propagation. Similarly,the simplification by unit propagation is a specialcase of generalized local autarkies modulo unit propagations(Property 6).Property 6 For any set of clauses Σ, Σ ∗ Σ ∗ .Compared with generalized autarkies, only the simplificationby the implied literals are not included in the generalizedautarkies modulo unit propagation, as illustrated bythe following example:Example 5 Let Σ = {(a ∨ b ∨ c),(a ∨ b ∨ ¬c),(a ∨ ¬b ∨¬c),(¬a ∨ b ∨ ¬c),(¬a ∨ ¬b ∨ c),(a ∨ ¬b ∨ c)}. We haveΣ a but Σ ∗ a.However, the restriction has been introduced to ensurethat generalized (local) autarkies modulo unit propagationcan be computed in polynomial time. Its implementation ina DPLL-like procedure is discussed in the next section.4.2. Some useful properties and implementationIn this section, some practical properties about the generalized(local) autarkies modulo unit propagation are presented.More precisely, it is shown how the detection ofthese kinds of autarkies can be computed efficiently duringthe computation of a DPLL-like algorithm.The first property is about the time complexity of deductionrestricted to unit propagation.Property 7 Let Σ and Ω be two sets of clauses. CheckingΣ ∗ Ω is in O(C Ω × (|Σ| + |c|)) where c is the longestclause of Ω.Unfortunately a polynomial-time complexity does notguarantee practical efficiency. In this respect, we also reducein a significant way the number of unit propagationsthat are needed to detect local autarkies modulo unit propagation.Property 8 Let Σ a set of clauses and l a literal. Σ ∗ Σ| lif and only if ∀c ∈ Σ such that ¯l ∈ c, Σ ∗ c.The above property suggests that after the assignment ofa literal l, clauses containing ¬l need to be checked, only.From properties 7 and 8, the following corollary is easilyobtained:Corollary 4 Let Σ be a set of clauses and l a literal. CheckingΣ ∗ Σ| l is in O(Occ(¯l)×C Σ ) where Occ(¯l) is the numberof occurrences of ¯l in Σ.1.11.21.31.41.51.61.71.8Algorithm 1: DPLL-autarkinput : A CNF formula Σoutput: true if Σ is satisfiable, false otherwiseΣ ∗ ←UnitPropagation(Σ);if Σ ∗ = ⊥ then return false;else if Σ ∗ = ∅ then return true;elsel←DecisionHeuristic(Σ ∗ );if DPLL-autark(Σ ∗ ∧ (l)) then return true;else return AutarkyModuloUP(Σ ∗ ,l);endThis corollary shows that detecting a generalized localautarky modulo unit propagation between two consecutivenodes of a DPLL-like algorithm is a simple task that is verycheap w.r.t. time-complexity. However, there can exist generalizedlocal autarkies modulo unit propagation betweentwo non-consecutive nodes. The following property and itscorollary show how useless tests can be avoided and that anincremental detection is possible.Property 9 Let Σ be a set of clauses and{l 0 ,...,l j ,...,l i } a set of literals. If Σ| {l0,...,l j} ∗Σ| {l0,...,l j,...,l i}, then Σ| {l0,...,l j,l j+1} ∗ Σ| {l0,...,l j,...,l i},Σ| {l0,...,l j,l j+1,l j+2} ∗ Σ| {l0,...,l j,...,l i}, ... andΣ| {l0,...,l j,...,l i−1} ∗ Σ| {l0,...,l j,...,l i}.Corollary 5 Let Σ be a set of clauses and{l 0 ,...,l j ,...,l i } a set of literals. If Σ| {l0,...,l j} ∗Σ| {l0,...,l j,...,l i}, then Σ| {l0,...,l j−1} ∗ Σ| {l0,...,l j,...,l i},Σ| {l0,...,l j−2} ∗ Σ| {l0,...,l j,...,l i}, ... and Σ ∗Σ| {l0,...,l j,...,l i}.Detecting generalized local autarkies modulo unit propagationbetween two non-consecutive nodes of a DPLL-likealgorithm depends on such a detection between two consecutivenodes. Accordingly, we insert this detection processwithin a DPLL-like procedure. It is performed duringthe backtrack step. Compared with a forward detection, thismethod avoids the useless unit propagations when the developedbranch is proved unsatisfiable. The following algorithmsdescribe this detection process (see Algorithm 2)and its integration within a DPLL-like procedure (see Algorithm1).Algorithm 1 details the DPLL-autark functionwhich is a DPLL-like algorithm using the simplificationby the generalized (local) autarkies modulo unit propagation.In this algorithm, the UnitPropagation andDecisionHeuristic functions are the DPLL-likesolvers usual ones. All kinds of decision heuristics (MOMS[11], VSIDS [19], etc.) can be used within DPLL-autark.When a call to the AutarkyModuloUP function is made,the Σ| l branch has been already explored and proved unsatisfiable.If Σ ∗ Σ| l then the Σ|¯l branch can be pruned since

Algorithm 2: AutarkyModuloUPinput : a set of clauses Σ and a literal l such that Σ| lhas been proved unsatisfiableoutput: true if Σ|¯l is satisfiable, false otherwise(Σ|¯l) ∗ ←UnitPropagation(Σ ∧ (¯l));if (Σ|¯l) ∗ = ⊥ then return false;else if (Σ|¯l) ∗ = ∅ then return true;elseautarkyFound ←true;// Σ ∗ Σ| l iff ∀c = (¯l ∨ c ′ ) ∈ Σ, Σ ∗ c ′foreach c ∈ Σ s.t. c = (¯l ∨ l 1 ∨ l 2 ∨ · · · ∨ l n ) doif UnitPropagation((Σ|¯l) ∗ ∧ (¯l 1 ) ∧(¯l 2 ) ∧ · · · ∧ ( l ¯ n ))= ⊥ then(Σ|¯l) ∗ ←(Σ|¯l) ∗ ∧ (l 1 ∨ l 2 ∨ · · · ∨ l n );endelse autarkyFound ←false;endif autarkyFound then return true;else return DPLL-autark((Σ|¯l) ∗ );end2.12.22.32.42.52.62.72.82.92.102.112.122.132.14an autarky is detected. The AutrakyModuloUP functioncomputes this test and, depending on the result, iteither explores or prune the branch. This function is detailedin Algorithm 2.For efficiency reasons, Algorithm 2 begins with the unitpropagation of ¯l. Indeed, each clause tested during the verificationof Σ ∗ Σ l implies the unit literal ¯l. In order toavoid several propagations of ¯l, the propagation of ¯l is doneonce at the beginning of the procedure (see line 2.1).When a clause c ′ = (l 1 ∨ l 2 ∨ · · · ∨ l n ) is proved to bea logical consequence through unit propagation of (Σ|¯l) ∗ ,it is added to (Σ|¯l) ∗ (see line 2.8). These additional clausesspeed up the computation of all unit propagations done duringthe rest of the procedure and reduce the size of the subtreebuilt during the exploration of the branch Σ|¯l when noautarky has been detected. Note that these additions are performedin constant space. Indeed, the c ′ clause is obtainedfrom a c clause of Σ that contains ¯l. This c clause is satisfiedduring the unit propagation of ¯l at the beginning of theprocedure. The addition of c ′ can be done with a simple removalof the ¯l literal from c. This process can be achievedin constant time and space with an adequate data structure.Finally, even if the generalized local autarky modulo unitpropagation is not detected, this approach is able to produceseveral sub-clauses. It is also possible to stop the detectionas soon as Σ ∗ Σ| l is proved (at the line 2.10). We believethat the spent time to produce sub-clauses is offset bythe time saved to explore a smaller sub-tree. Two versionscan be derived, one that only checks for generalized localautarkies modulo unit propagation and an other one that additionallyproduces sub-clauses. In the future, we plan toimplement these two versions in order to measure and com-Σ| {a,b,c,d,e}ef⊥Σ| {a,b,c,d}d⊥¬fΣ| {a,b,c}cfCurrent branchΣ| {a,b}b//Σ| {a}aCut (unit literal)¬eΣCut (unit literal)¬cInexplored branchΣ| {a,b,c,d,¬e}¬f////Inexplored BranchCut (unit literal)Figure 2. Levels of literals in DPLLCNFLevels of literalsFormula δ(a) δ(ā) δ(b) δ(¯b) δ(c) δ(¯c) δ(d) δ( ¯d) δ(e) δ(ē) δ(f) δ( ¯f)Σ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞Σ| {a} 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞Σ| {a,b} 0 ∞ 0 ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞ ∞Σ| {a,b,c} 0 ∞ 0 ∞ 1 ∞ ∞ ∞ ∞ ∞ ∞ ∞Σ| {a,b,¯c} 0 ∞ 0 ∞ ∞ 0 ∞ ∞ ∞ ∞ ∞ ∞Σ| {a,b,c,d} 0 ∞ 0 ∞ 1 ∞ 1 ∞ ∞ ∞ ∞ ∞Σ| {a,b,c,d,e} 0 ∞ 0 ∞ 1 ∞ 1 ∞ 2 ∞ ∞ ∞Σ| {a,b,c,d,e,f} 0 ∞ 0 ∞ 1 ∞ 1 ∞ 2 ∞ 3 ∞Σ| {a,b,c,d,e, ¯f} 0 ∞ 0 ∞ 1 ∞ 1 ∞ 2 ∞ ∞ 2Σ| {a,b,c,d,ē} 0 ∞ 0 ∞ 1 ∞ 1 ∞ ∞ 1 ∞ ∞Σ| {a,b,c,d,ē,f} 0 ∞ 0 ∞ 1 ∞ 1 ∞ ∞ 1 2 ∞Σ| {a,b,c,d,ē, ¯f} 0 ∞ 0 ∞ 1 ∞ 1 ∞ ∞ 1 ∞ 1Table 1. Levels of literals of the search treedepicted in Fig. 2.pare their practical strengths.4.3. Local autarkies and implication levelsIn the previous algorithm, only the current branchof DPLL can be cut when an autarky is found. It couldbe interesting to cut several branches to process a nonchronologicalbacktracking like it is done in modern implementationsof DPLL. In this section, it is shown that thisgoal can be reached by computing the largest jump with respectto an order of unit propagations. The standardconcepts of level for a literal (section 2.2) and implicationgraph (Definition 5) are used in that respect.We recall that δ(l) the level of a given literal l is given bythe number of branches not explored before its assignment.All assigned literals between two calls of the decisionheuristics have the same level, as illustrated in the next example.An unassigned literal has an infinite level. In Table1, the level of all literals at each node of the search tree depictedin Figure 2 is detailed.

Definition 4 Let Σ be a CNF formula, {l 1 ,l 2 ,...,l i } apartial assignment of Σ and c a clause of Σ| {l1,l 2,...,l i}. Thelevel of c, noted δ(c), is defined in the following way:• If c ∈ Σ, δ(c) = 0 ;• Else δ(c) = max{δ(l) s.t. l ∈ c et ¯l ∈ {l 1 ,l 2 ,...,l i }}The clause level matches the level of the last literal of theclause assigned to false.The fundamental process to detect a generalized autarkymodulo unit propagation consists in checking whether aclause is implied by unit propagation. The sequence ofgenerated implications can be depicted by an acyclic orientedgraph, called implication graph. Usually such a graphis used in all CDCL (conflict-driven clauses learning) approachesto learn clauses. It was first introduced in [17].Definition 5 An implication graph, noted G = (V, A) is agraph such that:1. V is the set of vertices where each vertex, noted s(l), isan assigned literal during unit propagation;2. A is the set of edges. For a vertex s(l i ) theset of predecessors is defined by the vertices{s(l 1 ),...,s(l i−1 ),s(l i+1 ),...,s(l n )} if there existsa clause c = (¯l 1 ,..., l i−1¯ ,l i , l i+1¯ ,..., l ¯ n )and a current partial assignment ρ such that{l 1 ,...,l i−1 ,l i+1 ,...,l n } ⊆ ρ. Each arc〈s(l j ),s(l i )〉 is labeled by c.The roots of the implication graph representing the implicationof a given clause by unit propagation are the verticeslabelled by the negation of each literal of the clause. Byconstruction, it is straightforward that this graph is acyclic.Example 6 Let Σ be the CNF formula built on the following14 clauses:c 1 : (¬a ∨ c) c 6 : (¬b ∨ e) c 11 : (¬i ∨ l ∨ ¬k)c 2 : (¬a ∨ d) c 7 : (¬e ∨ i) c 12 : (¬i ∨ ¬l)c 3 : (¬b ∨ d) c 8 : (¬e ∨ h) c 13 : (¬k ∨ ¬l)c 4 : (¬d ∨ f) c 9 : (¬g ∨ h ∨ ¬f) c 14 : (c ∨ e)c 5 : (¬d ∨ g) c 10 : (¬h ∨ k)The implication graph corresponding to the implication byunit propagation of the clause (ā ∨ ¯b) is depicted in Fig. 3.The goal is to compute the minimum levels of implicationof the clauses that are used during the detection of thegeneralized local autarkies modulo unit propagation. So avalue, noted η(l), is associated to each vertex of the graphin the following way:• Roots vertices have a null value;• Let a vertex s(l) and {s(l 1,1 ),...,s(l 1,m ),s(l 2,1 ),...,s(l 2,n ),...,s(l k,1 ),...,s(l k,p )} its set of predecessorssuch that 〈s(l 1,1 ),s(l)〉,...,〈s(l 1,m ),s(l)〉 are labelledby c 1 , 〈s(l 2,1 ),s(l)〉,...,〈s(l 2,n ),s(l)〉 are labelledby c 2 , ..., and 〈s(l k,1 ),s(l)〉,...,〈s(l k,p ),s(l)〉are labelled by c k , thencacbc1c2c3c6ccdcec4c5c7cfcgc8ciFigure 3. Implication graph representing animplication of a clause by unit propagationη(s(l))=min[max(δ(c 1 ),η(s(l 1,1 )),...,η(s(l 1,m ))),...,max(δ(c k ),η(s(l k,1 )),...,η(s(l k,p )))].Example 7 Let us consider the same CNF formula as in example6 together with the following clause levels:δ(c 1 ) = 8 δ(c 5 ) = 6 δ(c 9 ) = 12 δ(c 13 ) = 7δ(c 2 ) = 3 δ(c 6 ) = 0 δ(c 10 ) = 2 δ(c 14 ) = 5δ(c 3 ) = 7 δ(c 7 ) = 3 δ(c 11 ) = 1δ(c 4 ) = 9 δ(c 8 ) = 3 δ(c 12 ) = 4For each vertex of the implication graph depicted in Fig. 3,the following values are obtained:η(s(a)) = 0 η(s(b)) = 0 η(s(c)) = 8 η(s(d)) = 3η(s(e)) = 0 η(s(f)) = 9 η(s(g)) = 6 η(s(i)) = 3η(s(h)) = 3 η(s(¯l)) = 4 η(s(k)) = 3 η(s(¯k)) = 4So, the implication level of the clause (ā,¯b) is equal tomax(η(s(k)),η(s(¯k))) = 4.Thus, it is possible to compute an implication level forall the clauses used in the detection of generalized local autarkiesmodulo unit propagation. A level can be assigned tothe detected autarky. It is defined as the greatest implicationlevel computed for each implied clause. In consequence, ifthe level of the autarky is j, it is possible to backjump safelyup to this decision level. Indeed, for a sub-formula Σ i obtainedfrom Σ and a partial assignment {l 0 ,...,l j ,...,l i },if all the clauses of Σ i are implied modulo unit propagationat level k ≤ j, then Σ j ∗ Σ i . All branches between jand the current level i are pruned.Furthermore, even when no autarky is found, we haveseen that the clauses that are proved to be implied during theautarky detection, are added to the formula. These clausesare added only until the current decision level is achieved.Since its minimal implication level has been computed, itis possible to keep the implied clauses until backtrackingto their corresponding levels. This will reduce the size ofsearch tree even more.5. ConclusionThis paper presents a formal framework for the detectionand use of autarky assignments, with the goal to im-c9c9c12chc¯lc11c10c11ckc¯k

prove the efficiency of SAT solvers. Several contributionsabout the generalization of classical autarkies have beenprovided. The first one introduces the concept of autarkymodulo unit propagation, which extends the usual definitionby substituting the syntactical inclusion operation by asemantic-one based on unit propagation. It has also beenproved that propagating unit literals are special cases ofthis generalization. A second extension introduces the conceptof implication level, aiming at detecting larger partitions.Interestingly enough, when no generalized local autarkycan be found, the approach is able to produce severalsub-clauses from the formula. Each implied clause canbe exploited in order to reduce the search space until backtrackingto its corresponding implication level. Finally, it isshown how these different approaches could be integratedin DPLL-like SAT solvers.As future works, we plan to investigate how to integrateand implement the approach in modern SAT solvers. Thismight prove very succesful for efficiently solving industrialrelatedSAT instances. It is well-known that many of theseinstances can be partitioned into several disconnected componentsafter assigning a small set of variables. Anotherinteresting path of research would be to design powerfulclauses ordering heuristics for searching for local autarkiesmodulo unit propagation.References[1] Roberto J. Bayardo, Jr. and Robert C. Schrag. Using CSPlook-back techniques to solve real-world SAT instances. InProceedings of the Fourteenth National Conference on ArtificialIntelligence (AAAI’97), pages 203–208, 1997.[2] Paul Beame, Henry A. Kautz, and Ashish Sabharwal. Understandingthe power of clause learning. In Proceedings of theEighteenth International Joint Conference on Artificial Intelligence(IJCAI’03), pages 1194–1201, 2003.[3] Laure Brisoux, Éric Grégoire, and Lakhdar Saïs. Improvingbacktrack search for SAT by means of redundancy. In InternationalSyposium on Methodologies for Intelligent Systems,pages 301–309, 1999.[4] Martin Davis, George Logemann, and Donald W. Loveland.A machine program for theorem-proving. Communicationsof the ACM, 5(7):394–397, 1962.[5] Olivier Dubois and Gilles Dequen. A backbone-searchheuristic for efficient solving of hard 3–SAT formulae. InProceedings of the Seventeenth International Joint Conferenceon Artificial Intelligence (IJCAI’01), 2001.[6] Niklas Eén and Niklas Sörensson. An extensible sat-solver.In Proceedings of the Sixth International Conference on Theoryand Applications of Satisfiability Testing (SAT’03), pages502–518, 2002.[7] Carla P. Gomes, Bart Selman, and Henry Kautz. Boostingcombinatorial search through randomization. In Proceedingsof the Fifteenth National Conference on Artificial Intelligence(AAAI’97), pages 431–437, 1998.[8] Marijn J.H. Heule and Hans van Maaren. March dl: Addingadaptive heuristics and a new branching strategy. Journal onSatisfiability, Boolean Modeling and Computation, 2:47–59,2006.[9] Jinbo Huang and Adnan Darwiche. A structure-based variableordering heuristic for SAT. In Proceedings of the 18thInternational Joint Conference on Artificial Intelligence (IJ-CAI), pages 1167–1172, 2003.[10] Serge Jeannicot, Laurent Oxusoff, and Antoine Rauzy.évaluation sémantique en calcul propositionnel : une propriétéde coupure pour rendre plus efficace la procédure dedavis et putnam. Revue d’Intelligence Artificielle, 2(1):41–60, 1988. (in French).[11] Robert G. Jeroslow and Jinchang Wang. Solving propositionalsatisfiability problems. Annals of Mathematics andArtificial Intelligence, 1:167–187, 1990.[12] Henri Kautz, Eric Horvitz, Yongshao Ruan, Carla Gomes,and Bart Selman. Dynamic restart policies. In Proceedingsof the Eighteenth National Conference on Artificial Intelligence(AAAI’02), pages 674–682, 2002.[13] Oliver Kullmann. Investigations on autark assignments. DiscreteApplied Mathematics, 107(1-3):99–137, 2000.[14] Chu Min Li and Anbulagan. Heuristics based on unit propagationfor satisfiability problems. In Proceedings of theFifteenth International Joint Conference on Artificial Intelligence(IJCAI’97), pages 366–371, 1997.[15] Wei Li and Peter van Beek. Guiding real-world SAT solvingwith dynamic hypergraph separator decomposition. In16th IEEE International Conference on Tools with ArtificialIntelligence (ICTAI 2004), 15-17 November 2004, Boca Raton,FL, USA, pages 542–548, 2004.[16] Mark H. Liffiton and Karem A. Sakallah. Searching for autarkiesto trim unsatisfiable clause sets. In Proceedings of the11th International Conference on Theory and Applicationsof Satisfiability Testing (SAT’2008), volume 4996 of LNCS,pages 182–195. Springer Verlag, 2008.[17] Joao P. Marques-Silva and Karem A. Sakallah. GRASP -A New Search Algorithm for Satisfiability. In Proceedingsof IEEE/ACM International Conference on Computer-AidedDesign, pages 220–227, 1996.[18] Burkhard Monien and Ewald Speckenmeyer. Solving satisfiabilityin less than 2 n steps. Discrete Applied Mathematics,10:287–295, 1985.[19] Matthew W. Moskewicz, Conor F. Madigan, Ying Zhao, LintaoZhang, and Sharad Malik. Chaff: Engineering an efficientSAT solver. In Proceedings of the 38th Design AutomationConference (DAC’01), pages 530–535, 2001.[20] Laurent Oxusoff and Antoine Rauzy. L’évaluationsémantique en calcul propositionnel. Thèse de doctorat, Universitéde Provence, Marseille (France), January 1989.[21] Allen Van Gelder. Autarky pruning in propositional modelelimination reduces failure redundancy. Journal of AutomatedReasoning, 23(2):137–193, 1999.[22] Lintao Zhang. On subsumption removal and on-the-fly CNFsimplification. In Proceedings of the Eighth InternationalConference on Theory and Applications of Satisfiability Testing(SAT’05), pages 482–489, 2005.