By P. Braffort

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In FroCos 96: Frontiers of Combining Systems, Applied Logic, pages 103–120. Kluwer Academic Publishers, 1996. N. Tillmann and W. Schulte. Parameterized unit tests. In FSE 05: Foundations of Software Engineering, pages 253–262. ACM Press, 2005. G. Yorsh and M. Musuvathi. A combination method for generating interpolants. In CADE 05: Conference on Automated Deduction, LNCS 3632, pages 353–368. Springer-Verlag, 2005. G. Yorsh, T. Reps, and M. Sagiv. Symbolically computing most-precise abstract operations for shape analysis.

Ln is a conﬂicting clause. Then, as before, if there is no decision literal in M then Fail applies, and otherwise T -Backjump applies. Even if always immediately after backjumping the theory lemma is forgotten, the termination, soundness and completeness results hold. Modeling the previous reﬁnements and theory propagation. This requires the following additional rule: Deﬁnition 15. The Theory Propagate rule is: ⎧ ⎨ M |=T l l or ¬l occurs in F M || F =⇒ M l || F if ⎩ l is undeﬁned in M The purpose of this rule is to prune the search by assigning a truth value to literals that are T -entailed by M .

Then, for each one of these consequences, Theory Propagate is immediately applied by DPLL(X). Note that hence M never becomes T -inconsistent. • If Theory Propagate is not applicable, then UnitPropagate is eagerly applied by DPLL(X) (this is implemented using the two-watched-literals scheme). • DPLL(X) applies Fail or T -Backjump if a conﬂicting clause is detected. T -Backjump works as explained in Example 6, but there is a diﬀerence: a literal l at a node in the graph can now also be due to an application of Theory Propagate.