Automated Model Building as Future Research Topic

Compared to the vast amount of work that has been dedicated to developing and reening proof search methods (not only) for classical rst-order logic, surprisingly little is known about algorithmic methods for Model Building. On the other hand nding and investigating models of abstract structures is at the very heart of mathematical activity. Consequently the value of models in Automated Deduction is widely admitted. The mismatch between research eeorts and the importance of the topic can be partly explained by a common misunderstanding. Given the fact that the set of satissable rst-order formulas is not recursively enumerable one tends to think that developing algorithms for detecting the existence of models and nally constructing descriptions of such models is futile. Certainly, the possibility to construct models for all satissable formulas of some fragment of a logic implies that the fragment constitutes a decidable class. But it is well documented by now that Automated Deduction methods, especially resolution variants, can be beneecially employed as decision procedures for many important classes of rst-order formulas (or sets of clauses, respectively). We refer to FLTZ93] for a summary of work along this line. Using hyperresolution based decision procedures, FL95] describes a method to extract nite representations of Herbrand models from saturated sets of clauses. This constitutes just one approach to Automated Model Building. Other approaches based on theorem proving methods have been taken by T. practically relevant and successful, did not deene a general algorithmic method. Tammet's approach, like ours, is based on resolution decision procedures. His method of nite Model Building applies to the monadic and Ackermann class and is based on the termination of an ordering reenement. In the resulting model description the interpretation of the function symbols is given completely, but the interpretation of predicate symbols is only partial. Moreover, Tammet 1