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

By Daniel Kroening, Ofer Strichman

A selection method is an set of rules that, given a call challenge, terminates with an accurate yes/no solution. right here, the authors specialize in theories which are expressive adequate to version genuine difficulties, yet are nonetheless decidable. in particular, the e-book concentrates on selection tactics for first-order theories which are frequent in automatic verification and reasoning, theorem-proving, compiler optimization and operations learn. The recommendations defined within the e-book draw from fields equivalent to graph idea and good judgment, and are often utilized in undefined. The authors introduce the elemental terminology of satisfiability modulo theories after which, in separate chapters, learn selection tactics for every of the next theories: propositional good judgment; equalities and uninterpreted services; linear mathematics; bit vectors; arrays; pointer common sense; and quantified formulation.

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**Additional resources for Decision procedures : an algorithmic point of view**

**Sample text**

Every formula in the theory deﬁnes a language, which is the set of “words” (the assignments, in the case of quantiﬁer-free formulas) that satisfy it. We now deﬁne what it means that one theory is more expressive than another. 24 (expressiveness). Theory A is more expressive than theory B if every language that can be deﬁned by a B-formula can also be deﬁned by an A-formula, and there exists at least one language deﬁnable by an A-formula ✄ that cannot be deﬁned by a B-formula. We denote the fact that theory B is B ≺ A ✂ ✁ less expressive than theory A by B ≺ A.

While a formulation of this problem in propositional logic is rather trivial with n · (n − 1) variables, currently no SAT solver (which, recall, implicitly perform resolution) can solve this problem in a reasonable amount of time for n larger than several tens, although the size of the CNF itself is relatively small. As an experiment, we tried to solve this problem for n = 20 with three leading SAT solvers: Siege4 [171], zChaﬀ-04 [133] and HaifaSat [82]. On a Pentium 4 with 1 GB of main memory, none of the three could solve this problem within three hours.

This new implication re-starts the BCP process at level 3. Clause c9 is a special kind of a conﬂict clause, called an asserting clause: it forces an immediate implication after backtracking. Analyze-Conflict can be designed to generate asserting clauses only, as indeed most competitive solvers do. 3 In the case of learning a unary clause, the solver backtracks to the ground level. 34 2 Decision Procedures for Propositional Logic Aside: Multiple Conﬂict Clauses More than one conﬂict clause can be derived from a conﬂict graph.