Nested Sequent Calculi and Theorem Proving for Normal Conditional Logics

In this paper we focus on proof methods and theorem proving for normal conditional logics, by describing nested sequent calculi as well as a theorem prover for them. Nested sequent calculi are a useful generalization of ordinary sequent calculi, where sequents are allowed to occur within sequents. Nested sequent calculi have been profitably employed in the area of (multi)-modal logic to obtain analytic and modular proof systems for these logics. In this work, we describe nested sequent calculi recently introduced for the basic conditional logic CK and some of its significant extensions. We also provide a calculus for Kraus Lehman Magidor cumulative logic C. The calculi are internal (a sequent can be directly translated into a formula), cut-free and analytic. Moreover, they can be used to design (sometimes optimal) decision procedures for the respective logics, and to obtain complexity upper bounds. Our calculi are an argument in favour of nested sequent calculi for modal logics and alike, showing their versatility and power. We also describe NESCOND, a Prolog implementation of nested sequent calculi mentioned above. NESCOND (NESted sequent calculi for propositional CONDitional logics) is inspired by the methodolody of leanTAP. The paper also shows some experimental results, witnessing that the performances of NESCOND are promising. NESCOND is available at http://www.di.unito.it/∼pozzato/nescond/