Conflict-Driven Clause Learning SAT Solvers

One of the main reasons for the widespread use of SAT in many applications is that Conflict-Driven Clause Learning (CDCL) Boolean Satisfiability (SAT) solvers are so effective in practice. Since their inception in the mid-90s, CDCL SAT solvers have been applied, in many cases with remarkable success, to a number of practical applications. Examples of applications include hardware and software model checking, planning, equivalence checking, bioinformatics, hardware and software test pattern generation, software package dependencies, and cryptography. This chapter surveys the organization of CDCL solvers, from the original solvers that inspired modern CDCL SAT solvers, to the most recent and proven techniques. The organization of CDCL SAT solvers is primarily inspired by DPLL solvers. As a result, and even though the chapter is self-contained, a reasonable knowledge of the organization of DPLL is assumed. In order to offer a detailed account of CDCL SAT solvers, a number of concepts have to be introduced, which serve to formalize the operations implemented by any DPLL SAT solver. DPLL corresponds to backtrack search, where at each step a variable and a propositional value are selected for branching purposes. With each branching step, two values can be assigned to a variable, either 0 or 1. Branching corresponds to assigning the chosen value to the chosen variable. Afterwards, the logical consequences of each branching step are evaluated. Each time an unsat-isfied clause (i.e. a conflict) is identified, backtracking is executed. Backtracking corresponds to undoing branching steps until an unflipped branch is reached. When both values have been assigned to the selected variable at a branching step, backtracking will undo this branching step. If for the first branching step both values have been considered, and backtracking undoes this first branching step, then the CNF formula can be declared unsatisfiable. This kind of back-tracking is called chronological backtracking. An alternative backtracking scheme is non-chronological backtracking, which is described later in this chapter. A more detailed description of the DPLL algorithm is given in Part 1, Chapter 3.