Combining Decision Procedures for Theories in Sorted Logics

The Nelson-Oppen combination method combines decision procedures for theories satisfying certain conditions into a decision procedure for their union. While the method is known to be correct in the setting of unsorted first-order logic, some current implementations of it appear in tools that use a sorted input language. So far, however, there have been no theoretical results on the correctness of the method in a sorted setting, nor it is obvious that the method in fact lifts as is to logics with sorts. To bridge this gap between the existing theoretical results and the current implementations, we extend the Nelson-Oppen method to (order-)sorted logic and prove it correct under conditions similar to the original ones. From a theoretical standpoint, the extension is relevant because it provides a rigorous foundation for the application of the method in a sorted setting. From a practical standpoint, the extension has the considerable added benefits that in a sorted setting the method’s preconditions become easier to satify in practice, and the method’s nondeterminism is generally reduced.