C-tableaux
Abstract
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 method is restricted to the combination of stably infinite theories over disjoint signatures. In this report we present C-tableaux, an extension of Smullyan tableaux that generalizes the Nelson-Oppen method to the combination of arbitrary universal theories, not necessarily stably infinite and not necessarily over disjoint signatures. C-tableaux are sound and complete, but not terminating in general. Although C-tableaux do not provide a decidability result in general, in this report we describe two approaches that can be used in order to obtain decidability results using C-tableaux. Using the first approach, we are able to obtain a decidability result when combining theories that share the dense orders. Using the second approach, we are able to obtain a decidability result when combining theories whose union is stably finite.