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Conflict-Driven Satisfiability for Theory Combination: Transition System and Completeness

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Abstract

Many applications depend on solving the satisfiability of formulæ involving propositional logic and first-order theories, a problem known as Satisfiability Modulo Theory. This article presents a new method for satisfiability modulo a combination of theories, named CDSAT, for Conflict-Driven SATisfiability. CDSAT also solves Satisfiability Modulo Assignment problems that may include assignments to first-order terms. A conflict-driven procedure assigns values to variables to build a model, and performs inferences on demand in order to solve conflicts between assignments and formulæ. CDSAT extends this paradigm to generic combinations of disjoint theories, each characterized by a collection of inference rules called theory module. CDSAT coordinates the theory modules in such a way that the conflict-driven reasoning happens in the union of the theories, not only in propositional logic. As there is no fixed hierarchy with propositional logic at the center and the other theories as satellites, CDSAT offers a flexible framework for model search. We prove the soundness, completeness, and termination of CDSAT, identifying sufficient requirements on theories and modules that ensure these properties.

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Notes

  1. In fact, it is an inference system in the sense of [12].

  2. \({\textsf {UndoClear}}\) is a renaming of Undo [5].

  3. \({\textsf {UndoDecide}}\) replaces T-backjump-decide [9] or Semantic split [15].

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Acknowledgements

We thank Dejan Jovanović for fruitful discussions. Part of this work was done when the first author was visiting the Computer Science Lab of SRI International, whose support is greatly appreciated. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of NSF, DARPA, or the U.S. Government.

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Authors and Affiliations

  1. Università degli Studi di Verona, Strada Le Grazie 15, 37134, Verona, EU, Italy

    Maria Paola Bonacina

  2. SRI International, 333 Ravenswood Avenue, Menlo Park, CA, 94025, USA

    Stéphane Graham-Lengrand & Natarajan Shankar

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  1. Maria Paola Bonacina
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  2. Stéphane Graham-Lengrand
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  3. Natarajan Shankar
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Correspondence to Maria Paola Bonacina.

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This research was funded in part by NSF Grants 1528153 and CNS-0917375, by DARPA under Agreement Number FA8750-16-C-0043, and by Grant “Ricerca di base 2017” of the Università degli Studi di Verona.

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Bonacina, M.P., Graham-Lengrand, S. & Shankar, N. Conflict-Driven Satisfiability for Theory Combination: Transition System and Completeness. J Autom Reasoning 64, 579–609 (2020). https://doi.org/10.1007/s10817-018-09510-y

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