Conflict Driven Clause Learning

Background

Background knowledge about the following issues is needed to have a clear idea about the CDCL algorithm.

Boolean satisfiability problem

The satisfiability problem consists in finding a satisfying assignment for a given formula in conjunctive normal form (CNF).

An example of such a formula is:

or, using a common notation:

where A,B,C are Boolean variables, A ′ , C ′ , B , and C are literals, and A ′ + C ′ and B + C are clauses.

A satisfying assignment for this formula is e.g.:

since it makes the first clause true (since A ′ is true) as well as the second one (since C is true).

This examples uses three variables (A, B, C), and there are two possible assignments (True and False) for each of them. So one has 2 3 = 8 possibilities. In this small example, one can use brute-force search to try all possible assignments and check if they satisfy the formula. But in realistic appilcations with millions of variables and clauses brute force search is impractical. The responsibility of a SAT solver is to find a satisfying assignment efficiently and quickly by applying different heuristics for complex CNF formulas.

Unit clause rule (unit propagation)

If an unsatisfied clause has all but one of its literals or variables evaluated at False, then the free literal must be True in order for the clause to be True. For example, if the below unsatisfied clause is evaluated with A = F a l s e and B = F a l s e we must have C = T r u e in order for the clause ( A ∨ B ∨ C ) to be true.

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 clauses ( A ∨ B ∨ C ) and ( ¬ C ∨ D ∨ ¬ E ) . The clause ( A ∨ B ∨ D ∨ ¬ E ) , obtained by merging the two clauses and removing both ¬ C and C , is called the resolvent of the two clauses.

DP algorithm

The DP algorithm has been introduced by Davis and Putnam in 1960. Informally, the algorithm iteratively performs the following steps until no more variables are left in the formula:

See more details here DP Algorithm

DPLL algorithm

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

See more details at DPLL algorithm. An example with visualization of DPLL algorithm having chronological backtracking:

CDCL (Conflict-Driven Clause Learning)

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

Conflict-Driven Clause Learning works as follows.

Example

A visual example of CDCL algorithm:

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 solvers including:

The CDCL algorithm has made SAT solvers so powerful that they are being used effectively in several real world application areas like AI planning, bioinformatics, software test pattern generation, software package dependencies, hardware and software model checking, and cryptography.

Related algorithms

Related algorithms to CDCL are the DP and DPLL algorithm described before. The DP algorithm uses resolution refutation and it has potential memory access problem. Whereas the DPLL algorithm is OK for randomly generated instances, it is bad for instances generated in practical applications. CDCL is a more powerful approach to solve such problems in that applying CDCL provides less state space search in comparison to DPLL.