Abstract
Combination of theories underlies the design of satisfiability modulo theories (SMT) solvers. The Nelson-Oppen framework can be used to build a decision procedure for the combination of two disjoint decidable stably infinite theories.
We here study combinations involving an arbitrary first-order theory. Decidability is lost, but refutational completeness is preserved. We consider two cases and provide complete (semi-)algorithms for them. First, we show that it is possible under minor technical conditions to combine a decidable (not necessarily stably infinite) theory and a disjoint finitely axiomatized theory, obtaining a refutationally complete procedure. Second, we provide a refutationally complete procedure for the union of two disjoint finitely axiomatized theories, that uses the assumed procedures for the underlying theories without modifying them.
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Fontaine, P., Merz, S., Weidenbach, C. (2012). Combination of Disjoint Theories: Beyond Decidability. In: Gramlich, B., Miller, D., Sattler, U. (eds) Automated Reasoning. IJCAR 2012. Lecture Notes in Computer Science(), vol 7364. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31365-3_21
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DOI: https://doi.org/10.1007/978-3-642-31365-3_21
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