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Combining Theories Sharing Dense Orders

The Nelson-Oppen combination method combines decision procedures for first-order theories satisfying certain conditions into a single decision procedure for the union theory. The Nelson-Oppen combination method can be applied only if the signatures of the combined theories are disjoint. Combination tableaux (C-tableaux) are an extension of Smullyan tableaux for combining first-order theories whose signatures may not be disjoint. C-tableaux are sound and complete, but not terminating in general. In this paper we show that, when we combine first-order theories that share the theory of dense order, C-tableaux can be made terminating without sacrificing completeness. Thus, C-tableaux provide a decision procedure for the combination of first-order theories sharing the theory of dense order.