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Conflict Driven Clause Learning

Now a days Boolean Satisfiability (SAT) solvers are much effective in practical applications include AI planning, Bioinformatics, Software test pattern generation, Software package dependencies, Hardware and Software model checking, and cryptography. The Boolean Satisfiability problem (SAT) is one of the most studied NP-Complete problems because of its significance in both theoretical research and practical applications. Given a Boolean formula, the SAT problem asks for an assignment of variables so that the entire formula evaluates to true. Modern SAT solvers are really fast, one of the main reason is that state-of-the-art SAT solvers uses Conflict-Driven Clause Learning (CDCL) inside of it. CDCL SAT solvers have been applied in many cases with remarkable success.

The inside organization of CDCL SAT solvers is primarily inspired by DPLL solvers (See DPLL algorithm for more details). Conflict Driven Clause Learning was proposed by:

• J. P. Marques-Silva and Karem A. Sakallah, in their paper “GRASP: A Search Algorithm for Propositional Satisfiability”. and
• R. J. Bayardo Jr. and R. C. Schrag, in their paper “Using CSP look-back techniques to solve real world SAT instances.”

Algorithm

To have a clear idea about the CDCL algorithm we must have background knowledge about:

# SAT (Satisfiability).
# Unit clause rule (unit propagation). 
# Boolean constraint propagation (BCP).
# Resolution
# DP Algorithm.
# DPLL algorithm

Background

SAT (Satisfiability):

Satisfiability problem - Given a CNF formula, find satisfying assignment to a propositional logic (CNF) formula.

The following formulas is in CNF:

$CNF=(\neg A\lor \neg C)\land (B\lor C)$

where A,C,B = variables or literals. Both $(\neg A\lor \neg C)$ and $(B\lor C)$ are clauses, $\neg A$ = negation of A

$CNF=(A'+C')(B+C)$

where A,C,B = variables or literals. Both $(A'+C')$ and $(B+C)$ are clauses. $A'$ = negation of A

If we provide this CNF formula to any SAT solvers than one solution for this formula could be:

$A=Flase,B=Flase,C=True$

If we just apply brute force search to find out satisfiabilty assignment, as there are 3 (A,B,C) variables and there are 2 possible values (0 or 1) for each variables then we have $2^{4}=16$ possibilities. Surely we can apply brute force algorithm. But if we have million of variables and clauses then we can't apply brute force search. The responsibility of a SAT solver is to find a satisfying assignment efficiently and quickly applying different heuristics from complex CNF formula.

Unit clause rule (unit propagation):

If an unsatisfied clause has all but one of its literals or variables evaluate to 0 (False), then free literals must be implied to be 1 (True). For example, if the below unsatisfied clause is evaluated with

$A=False\land B=False$ then must be $C=True$

$clause=(A\lor B\lor C)$

Boolean constraint propagation (BCP):

The iterated application of the unit clause rule is referred to as unit propagation or Boolean constraint propagation (BCP).

Resolution:

Consider two clause $clause1=(A\lor B\lor C)$ and $clause2=(A\lor B\lor \neg C)$ and applying resolution refutation we get $clause=(A\lor B)$,
by cancelling out $\neg C$ and $C$.

DP Algorithm:

Davis, Putnam (1960) had developed this algorithm. Property of this algorithms are:

• Iteratively select variable for resolution till no more variable are left.
• Can discard all original clauses after each iteration.

See more details here DP Algorithm

Resolution of caluses

DPLL Algorithm:

Davis, Putnam, Logemann, Loveland (1962) had developed this algorithm. Property of this algorithms are:

• It is based on search.
• Most successful and the basis for all most all modern SAT solvers.
• It uses learning and chronological back tracking (1996).

See more details here DPLL algorithm. An example with visualization of DPLL algorithm having chronological back tracking:

• 
  All clauses makes a CNF formula
• 
  Pick a variable
• 
  Make a decision, variable a = Flase (0), thus green clauses becomes True
• 
  After several decision making, we find an implication graph that lead to a conflict.
• 
  Now Backtrack to immediate level and by force assign opposite value to that variable
• 
  But force decision still lead to another conflict
• 
  Backtrack to previous level and make a force decision
• 
  Make a new decision, but it lead to a conflict
• 
  Make a forced decision, but again it lead to a conflict
• 
  Backtrack to previous level
• 
  Continue in this way and the final implication graph

CDCL (Conflict Driven Clause Learning)

The main difference of CDCL and DPLL is that CDCL's back jumping is non - chronological.

Conflict Driven Clause Learning works as follows.

1. Select a variable and assign 0 or 1. This is called decision state. Remember the assignment.
2. Apply Boolean constraint propagation (Unit propagation).
3. Build the implication graph.
4. If there is any conflict then analysis the conflict and non - chronologically backtrack to appropriate decision level.
5. Otherwise continue from step 1 until all variable values are assigned.

Example

A visual example of CDCL algorithm:

• 
  At first pick a branching variable. Yellow circles means make a decision.
• 
  Now apply unit propagation and it yields X4 must be True. Gray circles means force variable assignment during unit propagation. This graph is called implication graph.
• 
  Pick another branching variable. Yellow circles means make a decision.
• 
  Apply unit propagation and find another new implication graph.
• 
  Here two variable X8 and X12 forced to be 0 and 1.
• 
  Pick another branching variable.
• 
  Find implication graph.
• 
  Pick another branching variable.
• 
  Find implication graph. Finally found a CONFLICT !!
• 
  Find the cut that lead to this conflict and from the cut find the conflict clause.
• 
  Find the clause by its negation.
• 
  Add the conflict clause to the rest of the clauses.
• 
  Non- chronological back jump to appropriate decision level.
• 
  Back jump and set variable values accordingly.

Pseudocode

algorithm CDCL (cnfFormula,variables):
    if (UnitPropagation(cnfFormula,variables) == CONFLICT):
         return UNSAT
    else:
         decisionLevel = 0 #Decision level
         while (not AllVariablesAssigned(cnfFormula,variables)):
              (variable, val) = PickBranchingVariable(cnfFormula,variables) #Decide stage
              decisionLevel += 1 #Increment decision level due to new decision
              insert (variable, val) in variables
              if (UnitPropagation(cnfFormula,variables) == CONFLICT):
                 backtrackLevel = ConflictAnalysis(cnfFormula,variables)
                 if( backtrackLevel < 0):
                    return UNSAT
                 else:
                    Backtrack(cnfFormula,variables,backtrackLevel)
                    decisionLevel  = backtrackLevel  #Decrement decision level due to backtracking
         return SAT

In addition to the main CDCL function, the following auxiliary functions are used:

• UnitPropagation: consists of the iterated application of the unit clause rule. If an unsatisfied clause is identified, then a conflict indication is returned.
• PickBranchingVariable: consists of selecting a variable to assign and the respective value.
• ConflictAnalysis consists of analyzing the most recent conflict and learning a new clause from the conflict. The organization of this procedure is described in section 4.4.
• Backtrack backtracks: to the decision level computed by Conflict- Analysis.
• AllVariablesAssigned: tests whether all variables have been assigned, in which case the algorithm terminates indicating that the CNF formula is satisfiable. An alternative criterion to stop execution of the algorithm is to check whether all clauses are satisfied. However, in modern SAT solvers that use lazy data structures, clause state cannot be maintained accurately, and so the termination criterion must be whether all variables are assigned.

Completeness

DPLL is a sound and complete algorithm for SAT. CDCL SAT solvers implement DPLL, but can learn new clauses and backtrack non chronologically. Clause learning with conflict analysis does not affect soundness or completeness. Conflict analysis identifies new clauses using the resolution operation. Therefore each learnt clause can be inferred from the original clauses and other learnt clauses by a sequence of resolution steps. If cN is the new learnt clause, then ϕ is satisfiable if and only if ϕ ∪ {cN} is also satisfiable. Moreover, the modified backtracking step also does not affect soundness or completeness, since backtracking information is obtained from each new learnt clause.

Applications

The main application of CDCL algorithm is in different SAT solver including

• MiniSAT
• Zchaff SAT
• Z3
• ManySAT etc.

CDCL algorithm has made SAT solvers so powerful that these solvers are being used in several real world application area like - AI planning, Bioinformatics, Software test pattern generation, Software package dependencies, Hardware and Software model checking, and cryptography.

Related algorithms

Related algorithm to CDCL are the DP and DPLL algorithm described before. DP algorithm uses resolution refutation and it has potential memory access problem. Where as DPLL algorithm is OK for randomly generated instances but bad for instances generated in practical application. We have better approach CDCL to solve such problems. Applying CDCL provides less state space search in compare to DPLL.

• 
  DPLL: no learning and chronological backtracking.
• 
  CDCL: conflict driven clause learning and non - chronological backtracking.

References

1. M. Davis, H. Putnam. A Computing Procedure for Quantification Theory. J. ACM 7(3): 201-215 (1960)
2. M. Davis, G. Logemann, D.W. Loveland. A machine program for theorem-proving. Commun. ACM 5(7): 394-397 (1962)
3. M.W. Moskewicz, C. F. Madigan, Y. Zhao, L. Zhang, S. Malik. Chaff: Engineering an Efficient SAT Solver. DAC 2001: 530-535
4. J. P. Marques-Silva and Karem A. Sakallah, “GRASP: A Search Algorithm for Propositional Satisfiability”, IEEE Trans. Computers, C-48, 5:506-521,1999.
5. R. J. Bayardo Jr. and R. C. Schrag “Using CSP look-back techniques to solve real world SAT instances.” Proc. AAAI, pp. 203-208, 1997.
6. L. Zhang, C. F. Madigan, M.W. Moskewicz, S. Malik. Efficient Conflict Driven Learning in Boolean Satisfiability Solver. ICCAD 2001: 279-285
7. Handbook of Satisfiability Armin Biere, Marijn Heule, Hans van Maaren and Toby Walsch IOS Press, 2008.
8. Presentation - "SAT-Solving: From Davis-Putnam to Zchaff and Beyond" by Lintao Zhang. (Several pictures are taken from his presentation)

Other material

Beside CDCL there are other approaches which are used to speed up SAT solvers. Some of them are -

• Restart
• Watched literal
• Failed Literal
• Stochastic search