Efficient Conflict Driven Learning in a Boolean Satisfiability Solver

In this section, we will take a look at different learningschemes employed in the existing SAT solvers, and proposesome new schemes based on different cuts of the implicationgraph.Rel_sat [7] is one of the first SAT solvers to incorporatelearning and non-chronological backtracking. The rel_satconflict analysis engine generates the conflict clause byrecursively resolving the conflicting clause with its antecedent,until the resolved clause includes only decision variables ofthe current decision level and variables assigned at decisionlevels smaller than the current level.In the implication graph representation, the rel_sat engine willput all variables assigned at the current decision level, exceptfor the decision variable, on the conflict side; and put all thevariables assigned prior to the current level and the currentdecision variable on the reason side. In our example, theconflict clause added would be:(V 11 ’+ V 6 + V 13 + V 4 ’+ V 8 ’ + V 17 + V 19 ’)We will call this learning scheme rel_sat scheme.ReasonSide-V 6 (1)V 11 (5)-V 13 (2)-V 12 (5)V 16 (5)Cut 3: cut does notinvolve conflictV 4 (3)V 8 (2)V 1 (5)V 3 (5)-V 2 (5) -V 10 (5)ConflictSide-V 5 (5)Figure 4. Different cuts-V 17 (1)V 18 (5)-V 18 (5)First UIP CutV 19 (3)Last UIP CutBy adding a conflict clause, the reason for the conflict isstated. The maximum decision level of the variables (exceptthe current decision level variable) in this conflict clause is thedecision level to backtrack. After backtracking, the conflictclause will become a unit clause, and since the currentdecision variable is the unit literal, it is forced to flip. Such aclause that causes a flip of the variable is called an assertingclause. It is always desirable for a conflict clause to be anasserting clause. The unit variable in the asserting clause willbe forced to assume a value and take the search to a new spaceto resolve the current conflict. To make a conflict clause anasserting clause, the partition needs to have one UIP of thecurrent decision level on the reason side, and all verticesassigned after this UIP on the conflict side. Thus, afterbacktracking, the UIP vertex will become a unit literal, andmake the clause an asserting clause.Another learning scheme is implemented in GRASP [6].GRASP’s learning scheme is different from rel_sat’s in thesense that it tries to learn as much as possible from a conflict.There are two cases when a conflict occurs in GRASP’slearning engine. If the current decision variable is a realdecision variable (explained later), the GRASP learningengine will add each reconvergence between UIPs in currentdecision level as a learned clause. In our example, if V 11 is areal decision (i.e. it has no antecedent), when the conflictoccurs, the GRASP engine will add one clause into thedatabase corresponding to the UIP reconvergences shown inFigure 4. This corresponds to the clause (V 2 ’ + V 6 + V 11 ’ +V 13 ).Moreover, GRASP will also include a conflict clause thatcorresponds to the partition where all the variables assigned atthe current decision level after the first UIP will be put on theconflict side. The rest of the vertices will be put on the reasonside. This corresponds to the FirstUIP cut as shown in figure3. The clause added will be (V 10 + V 8 ’ + V 17 + V 19 ’).After backtracking, the conflict clause will be an assertingclause, which forces V 10 to flip. Note that V 10 was not adecision variable before. In GRASP, such a decision is a fakedecision. The decision level of V 10 remains unchanged. Thisessentially means that we are not done at the current decisionlevel yet. We will call this mode of the analysis engine flipmode.If the deduction found that flipping the decision variable stillleads to a conflict, the GRASP conflict analysis engine willenter backtracking mode. Besides the clauses that have to beadded in the flip mode, it also adds another clause called theback clause. The cut for the back clause will put all thevertices at the current decision level (including the fakedecision variable) on the conflict side, and all other vertices onthe reason side. For our example, suppose the decisionvariable V 11 is actually a fake decision variable, withantecedent clause (V 21 + V 20 + V 11 ’). Then, besides the twoclauses added before, the GRASP engine will add anotherclause(V 21 + V 20 + V 6 + V 13 + V 4 ’ + V 8 ’ + V 17 + V 19 ’)This clause is a conflicting clause, and it only involvesvariables assigned before the current decision level. Theconflict analysis engine will continue to resolve this conflict,and bring the solver to an earlier decision level. We will callthis learning scheme the GRASP scheme.Besides these two learning schemes, many more options exist.One obvious learning scheme is to add only the decisionvariables involved in the conflict to the conflict clause. In ourimplication graph representation, this corresponds to makingthe cut such that only the decision variables are in the reasonside, and all the other variables are in the conflict side. Wewill call this scheme Decision scheme. Note that it is no goodto include all the decision variables of the current search treein the conflict clause, because such a combination of decisions