Abstract
Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules Gentzen’s original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system \({ G3K} \) of Negri for the modal logic \(K\) and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e., each rule application in \({ G3K} \) corresponds to a bipole—a pair of a positive and a negative phases—in \({ LKF} \). Since geometric axioms (when properly polarized) induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.
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Notes
- 1.
Note that, for simplicity, as in [14], we restrict to the case where only a single variable is bound to each existential quantifier.
- 2.
In fact, it is possible to show that every modal formula can be translated into a formula in the fragment of first-order logic which uses only two variables [2]. By the decidability of such a fragment, an easy proof of the decidability of propositional modal logic follows.
- 3.
Note that in \({ LKF} \) we consider one-sided sequents and the one we propose is in fact a polarization of the negation of the axiom.
- 4.
We note that in this way, we provide no information on which substitution term to use in case of existential quantifiers, and let such terms be reconstructed by the checker. In order to obtain a completely faithful encoding of the original \({ G3K} \) proof, the label term used for instantiating \(\lozenge \)-formulas should also be contained in the proof certificate and the expert predicate for the \(\exists \) should take that into account.
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Acknowledgments
This work was carried out during the tenure of an ERCIM Alain Bensoussan Fellowship Programme by the second author and was funded by the ERC Advanced Grant ProofCert.
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Miller, D., Volpe, M. (2015). Focused Labeled Proof Systems for Modal Logic. In: Davis, M., Fehnker, A., McIver, A., Voronkov, A. (eds) Logic for Programming, Artificial Intelligence, and Reasoning. LPAR 2015. Lecture Notes in Computer Science(), vol 9450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48899-7_19
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