Solution Learning and Solution Directed ... - Semantic Scholar

Solution Learning and Solution Directed Backjumping, Revisited Ian P Gent and Andrew G D Rowley 1 Abstract. We make two significant contributions. First, we introduce a new technique for solution based learning in Quantified Boolean Formulae (QBF’s.) This takes advantage of the structure of QBF’s to improve on previous methods. More information is extracted from each solution learnt, so fewer states need to be visited later in search. Unfortunately, our empirical results suggest that our learning technique does not do well on non-random benchmarks. Our second contribution is an important negative result which helps to explain this poor performance. We show empirically that solutiondirected backjumping (SBJ) provides little or no reduction in search on non-random instances from QBFLib, while it does cause an overhead. All solution learning methods exploit the power of SBJ, so neither our new method nor existing ones can be effective unless SBJ is. As well as explaining the lack of success of learning methods, this suggests that even SBJ may cause unnecessary overheads unless the existing set of benchmarks is inadequate. Finally, for random instances for which SBJ is effective, we show that our new learning technique can improve median universal backtracks by an order of magnitude and improve runtime by a factor of eight.

1

INTRODUCTION

Quantified Boolean Formulae (QBFs) can be seen as an extension to Satisfiability (SAT) formulae with the addition of a prefix in which every variable is universally or existentially quantified. This increase in expressivity comes at a price; the decision problem of whether a QBF is true or false is PSPACE-complete. This allows QBF to be used to solve many other PSPACE problems such as games, conditional planning problems and hardware verification. There have been many advances in QBF solver technology. Many of the more recent algorithms are based on an extension to the DPLL[2] procedure for SAT. The first such example is presented in [1]. Many other SAT techniques have been extended and added to this procedure, starting with conflict and solution directed backjumping [5], in which pruning of the search tree is achieved through look-back techniques. This provided a great improvement in both run-time and branching rate. Learning was then introduced[6, 7, 9]. This had much less impact than occurred in SAT. While improvements were seen on some problems, the algorithms actually performed worse on others. In this paper, we will examine the structure of the QBF to introduce a new technique for solution learning which reduces the number of previously visited solutions with significantly less overhead than would be possible with previous methods. We will show by empirical evaluation that this method does not perform well on non-random benchmark problems. This poor performance will then be explained 1

Department of Computer Science, University of St Andrews, St. Andrews, Fife, Scotland, UK; e-mail: {ipg, agdr}@dcs.st-and.ac.uk

by showing how solution-directed backjumping (SBJ) also does not perform well on non-random benchmarks. Since all learning techniques are based on the power of the backjumping techniques, learning cannot be successful if backjumping is not. We will finally show that on random benchmarks where SBJ does perform well, our new learning technique significantly improves performance with up to an order of magnitude improvement in median universal backtracks.

2

BACKGROUND

We provide some basic details of QBF, but refer the reader elsewhere for a more detailed introduction [1]. A QBF is of the form QC where Q is a sequence of quantifiers Q1 v1 ...Qn vn , where each Qi quantifies (∃ or ∀) a variable vi and each variable occurs exactly once in the sequence Q. C is a Boolean formula in conjunctive normal form (CNF), a conjunction of clauses where each clause is a disjunction of literals. Each literal is a variable and a sign. The literal is said to be negative if negated and positive otherwise. Universals and existentials are those variables quantified by a universal or existential quantifier respectively. A QBF of the form ∃v1 Q2 v2 ...Qn vn C is true if either Q2 v2 ...Qn vn C[v1 := T ] or Q2 v2 ...Qn vn C[v1 := F ] is true, where T represents true and F represents false. Similarly, a QBF of the form ∀v1 Q2 v2 ...Qn vn C is true if both Q2 v2 ...Qn vn C[v1 := T ] and Q2 v2 ...Qn vn C[v1 := F ] are true. A QBF is trivially false if it contains an all-universal clause or an empty clause. An all-universal clause is a clause in which all the literals are universal. Such a clause is false since we must be able to satisfy the formula for all combinations of universals, including the combination which makes all literals false in the all universal clause. This is true so long as the clause is not a tautology, which we remove with preprocessing. A QBF is trivially true if the variables are assigned such that all the clauses are satisfied by some literal. We can remove a variable from a clause that does not satisfy the clause, thus when no variables satisfy a clause it is empty. Similarly we can remove a clause that has been satisfied and so a trivially true QBF will have an empty set of clauses. In [1], a procedure is presented to decide the truth of a QBF. This is similar to the DPLL procedure for SAT[2]. The first difference is that the order of variables is now determined by the order in which the variables occur in the prefix. Secondly, when the formula is found true, the procedure backtracks to the most recently assigned universal that has not had both true and false values explored. Further propagation is possible using unit or single existential clauses. A unit or single-existential clause is defined to be a clause in which only one existential variable is left to be assigned and all universal variables are quantified after the existential in the quantifier prefix. In this clause, the existential must be assigned true to avoid an all-universal clause.

Conflict Directed Backjumping[8] is not covered in this paper. For a detailed description of Conflict Directed Backjumping for QBF see [5]. Solution Directed Backjumping can be performed during QBF search when assignments result in an empty set of clauses. The first stage is to find a small set of universal assignments from the current set of assignments which leaves the formula satisfied. This is known as the reason for the solution or the solution set. To find the solution set, each clause is examined in turn. Where a clause is satisfied by at least one existential, nothing is added to the solution set. Where a clause is only satisfied by a universal assignment, we choose one of the universals in the clause to be added to the set. We only backtrack on a universal variable when it occurs in the solution set. When both true and false assignments to a universal result in the QBF formula being true, we can combine the solution sets from both these assignments and pass the new solution set back to be used for universals assigned earlier. For more details see [5]. Three versions of solution learning have been proposed to date [6, 7, 9]. In solution learning we take an extended solution set and add it to the formula of the QBF. The extended solution set is the solution set with the addition of some existential variables, and the values assigned to the variables. The existentials are taken from the clauses which are satisfied by at least one existential variable such that there is at least one variable to satisfy each clause in the extended solution set. A DNF formula, D, is added to the QBF such that C ∨ D ⇐⇒ C i.e. the satisfiability of the original formula is not affected. The extended solution set is added as a cube (the DNF equivalent of a clause) of this DNF formula, with the value assigned to the variables being the sign of the respective literals in the cube. When the variables in the set are assigned to the same values, this cube will indicate that the formula is true, and so no further search will be done. Furthermore, when the cube is single-universal, i.e. it only contains one universal literal and all existential literals are quantified after the universal in the prefix, the universal variable can be assigned so as not to satisfy the cube, and thus direct search away from previously visited states.

3

SOLUTION LEARNING REVISITED

When performing SBJ, when a clause is satisfied by more than one universal assignment, we have to make a heuristic choice as to which universal to choose. The set which we learn in solution learning is therefore influenced by this decision. Ideally, we would like to learn the combination of all these solution sets so that we do not visit these solutions again. To take advantage of current learning techniques, we would also prefer to learn these solutions using DNF cubes. As we shall see, this leads to an exponential increase in the size of the formula. We can also gain additional information from the clauses of the QBF, which solution sets never capture. For example, take the clause (u1 ∨ u2 ∨ e1 ), where the variables are quantified with the prefix ∀u1 ∀u2 . . . ∀un ∃e1 . . . ∃em (we will deal with formulae not of the form ∀∃ later), and say that e1 is false and all other clauses in the formula are satisfied by at least one existential. Call this set of existential assignments E1 . For the formula to be satisfied at least one of u1 and u2 must be true. For any combination of the assignments to u1 and u2 that satisfies this clause we can satisfy the formula using the same set of existential assignments E1 . Standard solution learning does not capture this information; unless both universals satisfy the clause, the solution set can only contain one of u1 and u2 . When both universals satisfy the clause, we only choose one of them. So long as the formula u1 ∨ u2 is true, the formula in the example

remains satisfied. This, in itself, is not difficult to learn. It is unlikely though that this will be the only clause not satisfied by an existential. Suppose that (u1 ∨ u2 ∨ e1 ) ∧ (u3 ∨ u4 ∨ e2 ) are the only clauses not satisfied by a set of existential assignments E2 in which e1 = e2 = F . Then the QBF is true so long as the (u1 ∨ u2 ) ∧ (u3 ∨ u4 ) is true, since E2 satisfies all other clauses. Call this formula L1 . Say we now find another solution in which the only clauses not satisfied by an existential are (u5 ∨ u6 ∨ e3 ) ∧ (u7 ∨ u8 ∨ e5 ). We must now learn (u5 ∨ u6 ) ∧ (u7 ∨ u8 ). Call this formula L2 . So now we have that the QBF formula is true if L1 ∨ L2 is true. But L1 and L2 are CNF formulae so this is not a DNF. The conversion of this to DNF, using naive application of Boolean rules, would increase the formula size exponentially. We avoid this exponential increase, by using new variables. Notice that the disjunctive formulae are of the form I1 ∧ I2 ∧ . . . ∧ In , where Ii is a CNF clause. This is in the form of a DNF cube if we simply introduce variables Ii to stand for the clauses. The I is used as an indicator variable, to indicate the truth value of this clause. We can do this by adding additional formulae to the QBF during preprocessing. Notice that each I formula is the set of universals that are contained within a clause of the formula. So we can say that In is true iff clause n of the CNF formula is satisfied by a universal. This can easily be encoded into a set of DNF cubes: e.g. for the clause (u1 ∨ u2 ∨ e3 ) we can add the cubes (u1 ∧ u2 ∧ Ii ) ∨ (u1 ∧ I i ) ∨ (u2 ∧ I i ). So when u1 or u2 or both satisfy clause i, the variable Ii is set true by DNF unit propagation[9]. Similarly, Ii is set false when both u1 and u2 are false. We call this set of equivalences the indicator formula so as to distinguish it from the learned formula. So now we can replace L1 ∨ L2 in the above by (Ii ∧ Ij ) ∨ (Ik ∧ Il ) where Ii is defined as above and Ij , Ik and Il are defined by appropriate equivalences in a similar way to Ii . Adding the indicator formula as a set of DNF cubes does not affect the satisfiability of the original QBF so long as we quantify the indicator variables existentially after the universals that they represent. This means that they must be set by unit propagation and cannot be set at a branching node. This makes the DNF cubes trivially false, so C ∨ D ⇐⇒ C is equal to C ∨ F ⇐⇒ C. We can therefore add the indicator formula at the preprocessing stage, as the same variable Ii can be used no matter how many times the clause ci is involved in learning. This allows the indicator variables to be readily available when they are required in the learned cubes. This will also allow the learned cubes to guide the search once a solution has been found. With the above set of clauses, if we set u1 = T , this will cause the indicator assignment Ii = T . Similarly, setting u3 = T will set Ij = T . If we now use the existentials to satisfy the remaining clauses, we will learn the cube (Ii ∧ Ij ). If now at some later stage we set u1 = F , u2 = T , this will again set Ii = T . This cube will now be a unit cube and so Ij must be set false. Through the equivalences, this will force u3 = F and u4 = F , thus guiding search away from the previous solution. This type of guidance is performed in standard solution learning. Standard learning would not, however, avoid the later state where u1 = F since only the state where u1 = T will be recorded. This is the crux of our method, and is entirely new for learning methods for QBF. When backtracking on universals, we will never set universals to values for which a previous existential assignment works. We adapt SBJ slightly. Given a set of clauses not satisfied by existentials, we construct an indicator set from the indicator variable from those clauses. We use this for backjumping and for learning. On backtracking, an indicator variable is replaced by the universal that was used to set the indicator. This universal is then backtracked

• During preprocessing, add the indicator formula such that In,m is true iff clause n is satisfied by at least one universal quantified in the mth quantifier. • When a solution is found to the QBF, take the set of clauses not satisfied by an existential and extract the set of indicators that we need to set true to satisfy these clauses. • Add the indicator set as a cube of the DNF formula of the QBF. • Use the indicator set for backjumping in the QBF search. • When joining two indicator sets during SBJ, add the resultant set as a DNF cube.

1e+5 No. Universal Backtracks performed by GRL

upon. So if the clause i = (u1 ∨ u2 ∨ e3 ) is in a solution with u1 = T, u2 = T, e3 = F , we have the indicator set {Ii }. When we backtrack to Ii , we notice that it was set by (DNF) unit propagation from the assignment of u1 . We therefore jump back to u1 and set u1 = F . When both values have been assigned to the universal, the indicator sets of the two assignments can be combined, and the universal removed. Since this combined set contains only indicators, this can be added as a learned cube. This cube represents all the information learned from the two assignments to the universal and so if we ever get to a place where this cube is true, we know that we can solve the remaining QBF. Adding this cube pushes us away from such states. Take the clauses i = (u1 ∨u2 ∨e4 ), j = (u3 ∨e5 ) and k = (u3 ∨e6 ). If we find a solution with u1 = T, u2 = T, u3 = T we will learn the cube (Ii ∨ Ij ). Upon returning to the assignment of Ij we will remove Ij from the solution set, replacing it with u3 . If we then find a solution with u1 = T, u2 = T, u3 = F then we will learn the cube (Ii ∨ Ik ). Upon returning to the assignment of Ik we will remove Ik from the solution set. When we return to u3 having set it both true and false, we can union the solution sets and remove u3 . This results in the solution set Ii which can be added as a cube. So far we have only covered the situation where there are no existentials quantified outside the universals. If such a set of existentials exists and we change one of them after learning some cubes, the cubes that we have learned will become invalid. This is the same as only learning the solution set instead of the extended solution set in solution learning. To avoid this situation we add a condition to our learned cubes. If we change an existential we remove all cubes that contain universals that are quantified inside the existential. Note that we still never need to remove the indicator formula. For example take the QBF with quantifiers ∃e1 ∀u2 , u3 ∃e4 . Say that we find a solution in which clause i = (u2 ∨ u3 ∨ e4 ) is the only clause not satisfied by an existential. So we can now learn the cube (Ii ). If we then find that we need to change the assignment to e4 , we can keep this learned cube. If, however, we need to change the assignment to e1 we must throw this learned cube away. The situation is further complicated when we consider different levels of universal quantification. The indicator formula must be extended. We now say that In,m is true iff clause n is satisfied by a universal from quantifier level m. When a solution is found, we add cubes as before using these indicators, one per clause. When a clause is satisfied only by universals and the universals come from two different levels, we must choose between the levels heuristically. We assume that the level chosen is the outermost level. For example, take the QBF with quantifiers ∀u1 , u2 ∃e3 , e4 ∀u5 , u6 ∃e7 , e8 . Say that we find a solution in which the only clause not satisfied by an existential is clause i = (u5 ∨ u6 ∨ e7 ). Here we will learn the cube (Ii,3 ). Note that the subscript now states that we have a solution so long as clause i is satisfied by a universal from the third quantifier. In summary, our new learning technique performs the following operations:

1e+4 1e+3 1e+2

aya casToilet Connect k letTree rin rinR3cnf s schBec narRobots

1e+1 1e+0 1e+0

Figure 1.

1e+1 1e+2 1e+3 1e+4 No. Universal Backtracks performed by CSBJ

1e+5

CSBJ vs GRL on all QBFLib benchmarks.

• When we change an existential, remove any learned DNF cube that contains a universal quantified after the existential. The method presented has overheads related to the addition of the indicator variables; we now need to solve the indicator formula in addition to the original QBF. The indicator formula can be solved entirely by unit propagation once the appropriate universals have been assigned, and so it introduces only minimal overheads. The overhead is significantly less than if the indicator formula was not introduced. If we attempt to use standard solution learning to perform the same operations as here, we will introduce a potentially exponential increase in the size of the formula.

4

EXPERIMENTAL ANALYSIS

To determine the effectiveness of the new learning procedure, we compared an algorithm that performed solution and conflict directed backjumping with one that performed the new solution learning method and conflict directed backjumping. The algorithms presented so far for learning[6, 7, 9] perform either just conflict directed learning, or both conflict and solution learning. There is no algorithm that just performs solution learning and so a comparison between our learning method and previous learning is difficult. Hypothesis 1 The new learning technique presented in this paper performs fewer universal backtracks than SBJ on non-random benchmark problems from QBFLib. In order to test the hypothesis we compared two implementations of a complete QBF search based algorithm, CSBJ and GRL. CSBJ is a local implementation of conflict and solution directed backjumping. GRL is the same as CSBJ except that the SBJ procedure is replaced by the new solution learning technique as described in this paper. Throughout this paper, we eliminate inessential differences between algorithms by implementing all using the same local QBF solving library, which implements a two-literal watching scheme and a clause watching scheme [3]. The heuristic used on both algorithms is to choose the literal of a variable from the outermost quantifier set that occurs in the most clauses if existential, or the least clauses if universal. Both algorithms were run on a cluster of computers using Intel Pentium II processors running at 450 Mhz, each with 386MB RAM. Each algorithm had a timeout value of 1200 seconds per problem. The set of problems used were all the structured instances available from QBFLib (www.qbflib.org) as of 28th November 2003. The graph presented here is in the following format. Each point represents the number of universal backtracks performed on one of the QBF problems. The number of universal backtracks performed

1e+8


1e+4 1e+3 1e+2

No. Universal Backtracks performed by CSBJ

No. Universal Backtracks performed by SBJ

1e+5

1e+1 1e+0

1e+6 1e+5 1e+4 1e+3


1e+2 1e+1 1e+0

1e+0

Figure 2.

1e+1 1e+2 1e+3 1e+4 No. Universal Backtracks performed by BT

1e+5

BT vs SBJ on all QBFLib benchmarks.

by the CSBJ algorithm is plotted against the number of universal backtracks performed by GRL. A line is drawn to show where the two values are equal. If a point lies above this line, then on the problem represented GRL performs more universal backtracks than CSBJ, and vice versa for a point below the line. The runs where one of CSBJ and GRL timed out are not shown on any of the graphs. Fig. 1 shows the number of universal backtracks performed by the two algorithms. Most of the points lie on the line. Some lie very slightly below the line, suggesting that GRL performed slightly fewer backtracks on these instances. No points lie above the line. We see that most of the time, the new learning technique does not help in solving the benchmark problems. The fact that GRL never performs worse than CSBJ and sometimes performs better shows that the hypothesis is correct. It is less obvious whether it is worth performing the new solution learning technique given these results. Not shown here is the run time of the algorithms. GRL can run two orders of magnitude slower than CSBJ where the number of universal backtracks are the same. The implementation of GRL is not likely to be the most efficient; for example we may be able to deal with the indicator variables implicitly rather than explicitly. It is unlikely however that GRL would ever perform the same as CSBJ where the number of universal backtracks are the same since GRL will needlessly add cubes on which search is still performed.

5

1e+7

SOLUTION DIRECTED BACKJUMPING REVISITED

In the evaluation of solution and conflict directed backjumping for QBF[5], the experiments were only ever performed on the combination of both conflict and solution directed backjumping. These experiments show that the combination of both backjumping methods outperform an algorithm where neither are used. It is therefore hard to see how much SBJ actually contributes on its own. With both solution learning and the new solution learning techniques failing to produce good results, it would seem that a better analysis of SBJ is required. Both these techniques rely on the power of SBJ; if SBJ does not help, it is unlikely the solution learning methods will either. Hypothesis 2 Solution Directed Backjumping does not cause a significant decrease in the number of universal backtracks performed by a QBF algorithm on structured QBF instances. In order to test the hypothesis, we performed two comparisons on implementations of a tree-search based QBF algorithm, BT and SBJ, and CBJ and CSBJ. BT contains no backjumping. SBJ is the same as BT, except that it implements solution directed backjumping. CBJ is the same as BT, except that it implements conflict directed

1e+0 1e+1 1e+2 1e+3 1e+4 1e+5 1e+6 1e+7 1e+8 No. Universal Backtracks performed by CBJ

Figure 3.

CBJ vs CSBJ on all QBFLib benchmarks.

backjumping. CSBJ implements both conflict and solution directed backjumping. As a further measure, we also ran each algorithm with two different heuristics. These were: (a) choose the literal of a variable from the outermost quantifier that occurs in the most clauses if existential, or the least clauses if universal; and (b) choose the literal of the variable in the outermost quantifier with the smallest variable index. For an explanation of the graph presented here see section 4. For each problem there are two points, one for each heuristic. The heuristic used in each case is not shown. Fig. 2 shows the number of universal backtracks performed by the algorithms. The points for most of the sets lie on the line where SBJ and BT perform the same number of backtracks. A few sets break this pattern. These are the Narrizzano Robot problems, the Rintanen R3CNF problems and some of the Scholl/Becker problems. On the latter two, only a small decrease in the number of universal backtracks is seen when using SBJ, whereas for the former, there are quite a few problems for which SBJ performs less universal backtracks than BT. Fig. 3 shows a similar graph for CBJ vs CSBJ. The results here are the same. From the results we see that most of the time, SBJ performs the same number of universal backtracks as BT. This suggests that SBJ does not help on these instances. On a few instances, this is not the case. This is most obvious for the robot problems. This suggests that the hypothesis is incorrect, and that SBJ does help on some structured instances. This is also confirmed in the comparison of CBJ and CSBJ, where on the same sets of instances the same results are seen. Further investigation into these problems reveals that these particular problems have a certain element of randomness built into them. The R3CNF problems appear to be entirely random. The Scholl/Becker benchmarks are partial circuit implementation problems into which random errors have been introduced. The robot problems have a basic structure to them onto which many random permutations are then made. These findings lead to the conclusion that SBJ only works when a certain structural property is missing (or present) in the problems. This property appears to be removed with randomisation of the problems, and thus allows SBJ to work on these problems. These results also help to explain why solution learning techniques, including the one presented in this paper, have not shown good results on non-random benchmark problems. This indicates that we need to take a serious look at our current solution directed techniques before continuing on research in this direction, unless the current set of benchmarks is inadequate.

0.9 0.8 0.7 0.6

1e+3

0.5 0.4

1e+2 1e+1

0.3 0.2 0.1

1e+0

1e+6

0 50 100 150 200 250 300 350 400 450 Number of Clauses in Each Problem

6

EXPERIMENTS ON RANDOM INSTANCES

From the above experiments, we see that SBJ works best on random problems. This means that our new solution learning technique is also likely to show improvements on random problems. Here we will show that this is the case. Hypothesis 3 The new solution learning technique performs less universal backtracks than SBJ on random problems. As with the preceding experiments, we used CSBJ and GRL to test the hypothesis with the same heuristic, test machinery and timeout as was used in section 4. As a problem set, we used randomly generated problems using Gent and Walsh model A[4], with 20 variables per quantifier, 5 variables per clause and a varying number of clauses from 25 to 450 in steps of 25. All problems had an existential innermost quantifier set and there were a varying number of quantifier sets from 2 to 11, where even numbered sets have an outermost universal quantifier, and odd numbered sets have an outermost existential quantifier. The graphs presented are in the following form. Each point represents the median number of universal backtracks performed on 100 problems (y axis) against the number of clauses in the problems (xaxis). Also plotted is the percentage of problems that were satisfiable. The graphs show the familiar phase transition as the problems go from 100% satisfiable to 100% unsatisfiable [4]. Figs. 4 and 5 show the typical phase transition curves with the outermost quantifier universal with 4 quantifiers and the outermost quantifier existential with 5 quantifiers respectively. The phase transitions for other numbers of quantifiers showed similar trends. We see in these figures that with the outermost quantifier existential, GRL takes about half as many universal backtracks as CSBJ on average. With the outermost quantifier universal, GRL takes about one order of magnitude less universal backtracks than CSBJ. From the results we see that GRL performs much better than CSBJ on average on random problems. It seems that with the outermost quantifier existential, this improvement is not as marked as when the outermost quantifier is universal. These results show that the hypothesis is correct, and that we can gain an order of magnitude improvement in universal backtracks by using our new solution learning technique. Run time is not shown; on the problems with universal outermost, GRL runs about 8 times faster than CSBJ at the phase transition. On those with existential outermost, GRL runs about 2 times slower slow as CSBJ at the phase transition. These values are not really indicative of the techniques effectiveness since GRL is not optimised compared to CSBJ.

1e+5

Figure 5.

7

1 0.9 0.8 0.7

1e+4

0.6 1e+3

0.5 0.4

1e+2

0.3 0.2

1e+1

0.1 1e+0

Figure 4. Typical phase transition curves for GRL and CSBJ with outermost quantifier universal.

Fraction Satisfiable CSBJ-EAEAE GRL-EAEAE

Fraction of Problems Satisfiable

1e+4

1 Median Universal Backtracks Performed on 100 Problems

1e+5

Fraction Satisfiable CSBJ-AEAE GRL-AEAE

Fraction of Problems Satisfiable

Median Universal Backtracks Performed on 100 Problems

1e+6

0 50 100 150 200 250 300 350 400 450 Number of Clauses in Each Problem

Typical phase transition curves for GRL and CSBJ with outermost quantifier existential.

CONCLUSIONS AND FURTHER WORK

In this paper, we have presented a new technique for solution learning for QBF. This technique will revisit less solutions than would be done with previous solution learning techniques, and requires less overhead than would be added if previous techniques were trivially extended. We have evaluated the new technique with respect to solution directed backjumping and shown that it does not appear to perform well, providing very little reduction in the number of universal backtracks performed. We then performed an experimental analysis of solution directed backjumping to show that this technique also does not perform as well as was previously thought. We used this as an explanation for the poor performance of solution learning techniques including our own, and then show that on random problems where SBJ works well, our new technique works better, with an order of magnitude improvement in universal backtracks on some problems. As future work, it would be interesting to compare the new learning technique to solution learning. We would also like to see if there is any applicability of this learning technique to conflict directed learning, since conflict learning appears to work well on QBF. We would like to extend the learning technique so that cubes do not necessarily need to be thrown away when existential variables are changed. This could result in improvements on problems where similar solutions are found in vastly different parts of the search tree. Finally, the results shown here suggest that it would be important to find the property of the structured problems that seems to stop SBJ from working well. If problems could be reformulated to remove this property, further improvements in QBF solving may be seen.

ACKNOWLEDGEMENTS The first author is supported by a Royal Society of Edinburgh SEELLD Support Fellowship and the second by EPSRC grant GR/55382. The authors are members of the APES and CPPOD research groups and thank other members. We also thank Massimo Narizzano.

REFERENCES [1] M. Cadoli, M. Schaerf, A. Giovanardi, and M. Giovanardi, ‘An algorithm to evaluate quantified boolean formulae and its experimental evaluation’, Journal of Automated Reasoning, 28(2), 101–142, (2002). [2] M. Davis, G. Logemann, and D. Loveland, ‘A machine program for theorem-proving’, Commun. ACM, 5(7), 394–397, (1962). [3] I. Gent, E. Giunchiglia, M. Narizzano, A. Rowley, and A. Tacchella, ‘Watched data structures for QBF solvers’, in Theory and Applications of Satisfiability Testing. Springer, (2003). [4] I. Gent and T. Walsh, ‘Beyond NP: The QSAT phase transition’, in Proc. AAAI 99. AAAI / MIT Press, (1999).

[5] E. Giunchiglia, M. Narizzano, and A. Tacchella, ‘Backjumping for quantified boolean logic satisfiability’, in IJCAI, pp. 275–281, (2001). [6] E. Giunchiglia, M. Narizzano, and A. Tacchella, ‘Learning for quantified boolean logic satisfiability’, in Proc. AAAI 2002, pp. 649–654, (2002). [7] R. Letz, ‘Lemma and model caching in decision procedures for quantified boolean formulas’, in Automated Reasoning with Analytic Tableaux and Related Methods, pp. 160–175. Springer, (2002). [8] P. Prosser, ‘Hybrid algorithms for the constraint satisfaction problem’, Computational Intelligence, 9(3), 268–299, (1993). [9] L. Zhang and S. Malik, ‘Towards a symmetric treatment of satisfaction and conflicts in quantified boolean formula evaluation’, in Proc. CP2002, pp. 200–215. Springer, (2002).