In computer science, conflict-driven clause learning (CDCL) is an algorithm for solving the Boolean satisfiability problem (SAT). Given a Boolean formula, the SAT problem asks for an assignment of variables so that the entire formula evaluates to true. The internal workings of CDCL SAT solvers were inspired by DPLL solvers. The main difference between CDCL and DPLL is that CDCL's backjumping is non-chronological.
Conflict-driven clause learning was proposed by Marques-Silva and Karem A. Sakallah (1996, 1999) [1] [2] and Bayardo and Schrag (1997). [3]
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: [4]
where A,B,C are Boolean variables, , , , and are literals, and and are clauses.
A satisfying assignment for this formula is e.g.:
since it makes the first clause true (since is true) as well as the second one (since 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 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 applications 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.
If a 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 and we must have in order for the clause to be true.
The iterated application of the unit clause rule is referred to as unit propagation or Boolean constraint propagation (BCP).
Consider two clauses and . The clause , obtained by merging the two clauses and removing both and , is called the resolvent of the two clauses.
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Conflict-driven clause learning works as follows.
A visual example of CDCL algorithm: [4]
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 affects neither soundness nor 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. [5]
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 to CDCL are the Davis–Putnam algorithm and DPLL algorithm. The DP algorithm uses resolution refutation and it has potential memory access problem.[ citation needed ] 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.
In logic and computer science, the Boolean satisfiability problem (sometimes called propositional satisfiability problem and abbreviated SATISFIABILITY, SAT or B-SAT) is the problem of determining if there exists an interpretation that satisfies a given Boolean formula. In other words, it asks whether the variables of a given Boolean formula can be consistently replaced by the values TRUE or FALSE in such a way that the formula evaluates to TRUE. If this is the case, the formula is called satisfiable. On the other hand, if no such assignment exists, the function expressed by the formula is FALSE for all possible variable assignments and the formula is unsatisfiable. For example, the formula "a AND NOT b" is satisfiable because one can find the values a = TRUE and b = FALSE, which make (a AND NOT b) = TRUE. In contrast, "a AND NOT a" is unsatisfiable.
In boolean logic, a disjunctive normal form (DNF) is a canonical normal form of a logical formula consisting of a disjunction of conjunctions; it can also be described as an OR of ANDs, a sum of products, or — in philosophical logic — a cluster concept. As a normal form, it is useful in automated theorem proving.
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