Forgetting literals with varying propositional symbols

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. We extend here this notion, by allowing propositional symbols to vary while forgetting literals. We describe the new notion, on the syntactical and the semantical side. We provide various different syntactical characterizations, in order to provide various methods for computing the notion introduced here. This confirms that one of the main interests of the notion of forgetting literals (original, and new version) is that it provides new kinds of methods of computation. Then, we show how to apply it to the computation of circumscription. This computation has been done before with standard literal forgetting, but here we show how introducing varying propositional symbols simplifies significantly the computation. We revisit a fifteen years old result about computing circumscription, showing that it can be improved in the same way. We provide hints in order to apply this forgetting method also to other logical formalisms.