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Connection Calculus Theorem Proving with Multiple Built-in Theories

Gradually more applications of automated reasoning are discovered. This development has the consequence that deduction systems need to be increasingly flexible. They should exhibit a behavior appropriate to a given problem. One way to achieve this behavior is the integration of different systems or calculi. This leads to the so-called hybrid reasoning (Stickel, 1985; Frisch, 1991; Baumgartner, 1992; Petermann, 1993a) which describes the integration of a general purpose foreground reasoner with one specialized theory reasoner. The aim of this paper is to go a step further, i.e. to treat the theory reasoner as a hybrid system itself. The framework proposed below is suitable for building multiple theories into theorem provers. Those theories can be given syntactically but also semantically. Here, semantical reasoning is understood as reasoning, or rather computing, under a theory given by a class of models, whereas syntactical reasoning means reasoning under a theory given by first-order axioms. The presented approach is a generalization of previous attempts of combining syntactical reasoning under the empty theory with semantical reasoning (Burckert, 1994; Baumgartner and Stolzenburg, 1995), of combining different theories given syntactically (Petermann, 1997) or just theory (or hybrid) reasoning. The paper formulates sufficient criteria for the construction of complete calculi which enable reasoning under hybrid theories combined from sub-theories given semantically and those given syntactically and briefly reports experimental work.