Affiliations: Department of Computer and Information Science, Linköping University, S-581 83 Linköping, Sweden, e-mail: patdo@ida.liu.se | Institute of Informatics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland, e-mail: witlu@mimuw.edu.pl | Institute of Informatics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland, e-mail: szalas@mimuw.edu.pl
Note: [] Supported in part by the Swedish Council for Engineering Sciences (TFR).
Note: [] Supported in part by KBN grant 3 P406 019 06.
Note: [] Supported in part by KBN grant 3 P406 019 06.
Abstract: Circumscription has been perceived as an elegant mathematical technique for modeling nonmonotonic and commonsense reasoning, but difficult to apply in practice due to the use of second-order formulas. One proposal for dealing with the computational problems is to identify classes of first-order formulas whose circumscription can be shown to be equivalent to a first-order formula. In previous work, we presented an algorithm which reduces certain classes of second-order circumscription axioms to logically equivalent first-order formulas. The basis for the algorithm is an elimination lemma due to Ackermann. In this paper, we capitalize on the use of a generalization of Ackermann's Lemma in order to deal with a subclass of universal formulas called semi-Horn formulas. Our results subsume previous results by Kolaitis and Papadimitriou regarding a characterization of circumscribed definite logic programs which are first-order expressible. The method for distinguishing which formulas are reducible is based on a boundedness criterion. The approach we use is to first reduce a circumscribed semi-Horn formula to a fixpoint formula which is reducible if the formula is bounded, otherwise not. In addition to a number of other extensions, we also present a fixpoint calculus which is shown to be sound and complete for bounded fixpoint formulas.
