About the computation of forgetting symbols and literals

Recently, the old logical notion of forgetting propositional symbols (or reducing the logical vocabulary) has been generalized to a new notion: forgetting literals. The aim was to help the automatic computation of various formalisms which are currently used in knowledge representation, particularly for nonmonotonic reasoning. We develop here a further generalization, allowing propositional symbols to vary while forgetting literals. We describe the new notion, on the syntactical and the semantical side. We provide various manipulations over the basic definitions involved, including for the original version, which hopefully should help improving again the efficiency of the computation. This work concerns especially circumscription, since it is known that one way of computing circumscription uses the forgetting of literals. Introduction The well-known notion of forgetting propositional symbols, which is known at least since a 1854 paper by Boole under the name “elimination of middle terms”, has been used for a long time in mathematical logic and in its applications for knowledge representation (see e.g. (Lin & Reiter 1994; Lin 2001; Su, Lv, & Zhang 2004)). It is a reduction of the vocabulary, thanks to the suppression of some propositional symbols. Let us consider the formula (bird ∧ ¬exceptional → flies) ∧ ¬exceptional. We may want to “forget” the symbol exceptional, considered here as “auxiliary”, then we get the formula