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Decision procedures : an algorithmic point of view by Daniel Kroening, Ofer Strichman

By Daniel Kroening, Ofer Strichman

A choice strategy is an set of rules that, given a call challenge, terminates with an accurate yes/no solution. right here, the authors specialise in theories which are expressive sufficient to version genuine difficulties, yet are nonetheless decidable. in particular, the publication concentrates on determination tactics for first-order theories which are ordinary in computerized verification and reasoning, theorem-proving, compiler optimization and operations examine. The strategies defined within the booklet draw from fields reminiscent of graph conception and common sense, and are sometimes utilized in undefined. The authors introduce the elemental terminology of satisfiability modulo theories after which, in separate chapters, research selection approaches for every of the next theories: propositional common sense; equalities and uninterpreted capabilities; linear mathematics; bit vectors; arrays; pointer common sense; and quantified formulation.

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An assignment to all the variables. The second category is based on a stochastic search: the solver guesses a full assignment, and then, if the formula is evaluated to false under this assignment, starts to flip values of variables according to some (greedy) heuristic. Typically it counts the number of unsatisfied clauses and chooses the flip that minimizes this number. There are various strategies that help such solvers avoid local minima and avoid repeating previous bad moves. DPLL solvers, however, are considered better in most cases, at least at the time of writing this chapter (2007), according to annual competitions that measure their performance with numerous CNF instances.

Comments This repeated process is called Boolean constraint propagation (BCP). BCP is applied in line 2 because unary clauses at this stage are unit clauses. Name Analyze-Conflict() Output Minus 1 if a conflict at decision level 0 is detected (which implies that the formula is unsatisfiable). Otherwise, a decision level which the solver should backtrack to. Description A detailed description of this function is delayed to Sect. 4. Briefly, it is responsible for computing the backtracking level, detecting global unsatisfiability, and adding new constraints on the search in the form of new clauses.

Let l, m, n be the number of AND, OR, and NOT gates, respectively, in ϕ. Derive a formula parameterized by l, m and n that expresses the ratio of the number of CNF clauses in Tseitin’s encoding to that in the one-sided encoding suggested here. 8 Glossary The following symbols were used in this chapter: Symbol Refers to . . α |= ϕ |= ϕ T First used on page . . 1 Propositional Logic We assume that the reader is familiar with propositional logic. The syntax of formulas in propositional logic is defined by the following grammar: formula : formula ∧ formula | ¬formula | (formula) | atom atom : Boolean-identifier | true | false Other Boolean operators such as OR (∨) can be constructed using AND (∧) and NOT (¬).

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