A Reduction Result for Circumscribed Semi-Horn Formulas

Circumscription has been perceived as an elegant mathematical technique for modeling nonmonotonic and commonsense reasoning but di cult 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 rst order formulas whose cir cumscription can be shown to be equivalent to a rst order formula In previous work we presented an algorithm which reduces certain classes of second order circumscription axioms to logically equivalent rst order formulas The basis for the algorithm is an elimination lemma due to Ackermann In this paper we cap italize 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 cir cumscribed de nite logic programs which are rst order expressible The method for distinguishing which formulas are reducible is based on a boundedness crite rion The approach we use is to rst reduce a circumscribed semi Horn formula to a xpoint formula which is reducible if the formula is bounded otherwise not In addition to a number of other extensions we also present a xpoint calculus which is shown to be sound and complete for bounded xpoint formulas Accepted for publication Fundamenta Informaticae