Herbrand disjunctions are a means for reducing the problem ofwhether a first-oder formula is valid in an open theory T or not to theproblem whether an open formula, one of the so called Herbrand disjunctions,is T -valid or not. Nevertheless, the set of Herbrand disjunctions, which hasto be examined, is undecidable in general. Fore this reason the practicalvalue of Herbrand disjunctions has been estimated negatively .Relying on completeness proofs which are based on the algebraizationtechnique presented in [30], but taking a (...) more optimistic view, we show howHerbrand disjunctions become the base of a method for building in theoriesinto automatic theorem provers [26]. Surveying newer results we discusshow to treat heterogeneous theories [29] as well as practical implications ofdifferent normal form transformations. (shrink)

A growing number of applications of automated reasoning exhibitsthe necessity of flexible deduction systems. A deduction system should beable to execute inference rules which are appropriate to the given problem.One way to achieve this behavior is the integration of different calculi. Thisled to so called hybrid reasoning [22, 1, 10, 20] which means the integrationof a general purpose foreground reasoner with a specialized background reasoner. A typical task of a background reasoner is to perform special purposeinference rules according to a (...) built-in theory. The aim of this paper is togo a step further, i.e. to treat the background reasoner as a hybrid systemitself. The paper formulates sufficient criteria for the construction of complete calculi which enable reasoning under hybrid theories combined fromsub-theories. For this purpose we use a generic approach described in [20].This more detailed view on built-in theories is not covered by the knowngeneral approaches [1, 3, 6, 20] for building in theories into theorem provers.The approach is demonstrated by its application to the target calculi of thealgebraic translation [9] of multi-modal and extended multi-modal [7] logicto first-order logic. (shrink)