GRASP: A Search Algorithm for Propositional Satisfiability
IEEE Transactions on Computers
A Computing Procedure for Quantification Theory
Journal of the ACM (JACM)
Unit Refutations and Horn Sets
Journal of the ACM (JACM)
A machine program for theorem-proving
Communications of the ACM
Chaff: engineering an efficient SAT solver
Proceedings of the 38th annual Design Automation Conference
Symbolic Logic and Mechanical Theorem Proving
Symbolic Logic and Mechanical Theorem Proving
Propositional Logic: Deduction and Algorithms
Propositional Logic: Deduction and Algorithms
Efficient conflict driven learning in a boolean satisfiability solver
Proceedings of the 2001 IEEE/ACM international conference on Computer-aided design
Automated theorem proving: A logical basis (Fundamental studies in computer science)
Automated theorem proving: A logical basis (Fundamental studies in computer science)
Improved Conflict-Clause Minimization Leads to Improved Propositional Proof Traces
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
SAT '09 Proceedings of the 12th International Conference on Theory and Applications of Satisfiability Testing
A new clause learning scheme for efficient unsatisfiability proofs
AAAI'08 Proceedings of the 23rd national conference on Artificial intelligence - Volume 3
Towards understanding and harnessing the potential of clause learning
Journal of Artificial Intelligence Research
A generalized framework for conflict analysis
SAT'08 Proceedings of the 11th international conference on Theory and applications of satisfiability testing
Contributions to the theory of practical quantified boolean formula solving
CP'12 Proceedings of the 18th international conference on Principles and Practice of Constraint Programming
Efficient clause learning for quantified boolean formulas via QBF pseudo unit propagation
SAT'13 Proceedings of the 16th international conference on Theory and Applications of Satisfiability Testing
Concurrent clause strengthening
SAT'13 Proceedings of the 16th international conference on Theory and Applications of Satisfiability Testing
SAT'13 Proceedings of the 16th international conference on Theory and Applications of Satisfiability Testing
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The dominant propositional satisfiability solvers of the past decade use a technique often called conflict-driven clause learning (cdcl), although nomenclature varies. The first half of the decade concentrated on deriving the best clause from the conflict graph that the technique constructs, also with much emphasis on speed. In the second half of the decade efforts have emerged to exploit other information that is derived by the technique as a by-product of generating the conflict graph and learning a conflict clause. The main thrust has been to strengthen the conflict clause by eliminating some of its literals, a process often called conflict-clause minimization, but more accurately described as conflictclause width reduction, or strengthening. This paper first introduces implication sequences as a general framework to represent all the information derived by the CDCL technique, some of which is not represented in the conflict graph. Then the paper analyzes the structure of this information. The first main result is that any conflict clause that is a logical consequence of an implication sequence may be derived by a particularly simple form of resolution, known as linear input regular. A key observation needed for this result is that the set of clauses in any implication sequence is Horn-renamable. The second main result is that, given an implication sequence, and a clause C derived (learned) from it, it is NP-hard to find a minimum-cardinality subset of C that is also derivable. This is in sharp contrast to the known fact that such a minimum subset can be found quickly if the derivation is restricted to using only clauses in the conflict graph.