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Many formal descriptions of DPLL-based SAT algorithms either do not include all essential proof techniques applied by modern SAT solvers or are bound to particular heuristics or data structures. This makes it difficult to analyze proof-theoretic properties or the search complexity of these algorithms. In this paper we try to improve this situation by developing a nondeterministic proof calculus that models the functioning of SAT algorithms based on the DPLL calculus with clause learning. This calculus is independent of implementation details yet precise enough to enable a formal analysis of realistic DPLL-based SAT algorithms.

This paper describes the proof calculus LD for clausal propositional logic, which is a linearized form of the well-known DPLL calculus extended by clause learning. It is motivated by the demand to model how current SAT solvers built on clause learning are working, while abstracting from decision heuristics and implementation details. The calculus is proved sound and terminating. Further, it is shown that both the original DPLL calculus and the conflict-directed backtracking calculus with clause learning, as it is implemented in many current SAT solvers, are complete and proof-confluent instances of the LD calculus.