A Hierarchy of Backward Translations: Applications to Modal Logics
Résumé
Methods for automated deduction for non classical logics can be broadly divided in two categories: the direct approach (construction of specific proof systems for the non classical logics) and the translation approach (translation between a source logic and a target logic). One of the advantages of the approach by translation lies on the use of a theorem prover dedicated to the target logic (for instance the classical first-order logic) in order to solve problems initially expressed in the source logic (for instance the modal logic S4). However the presentation of (partial) proofs in calculi of the source logic remains an aspect of the translation approach that has been generally neglected. A hierarchy of backward translations is given in the paper in order to present different types of information that can be recovered in the source logics.The backward translations are then classified with respect to the information retrieved from the target logics. Partial backward translations for some usual propositional modal logics are discussed, involving resolution and tableaux calculi. Different examples illustrate the construction of (partial) proofs in the source logics.