Skip to Main content Skip to Navigation
Conference papers

Focused labeled proof systems for modal logic

Dale Miller 1, 2 Marco Volpe 1, 2
1 PARSIFAL - Proof search and reasoning with logic specifications
LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau], Inria Saclay - Ile de France
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 from 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.
Document type :
Conference papers
Complete list of metadata

Cited literature [19 references]  Display  Hide  Download

https://hal.inria.fr/hal-01213858
Contributor : Marco Volpe <>
Submitted on : Friday, October 9, 2015 - 2:56:46 PM
Last modification on : Friday, April 30, 2021 - 10:02:41 AM
Long-term archiving on: : Sunday, January 10, 2016 - 10:21:04 AM

File

MilVolLPAR15.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-01213858, version 1

Collections

Citation

Dale Miller, Marco Volpe. Focused labeled proof systems for modal logic. 20th International Conference on Logic for Programming, Artificial Intelligence and Reasoning (LPAR), Nov 2015, Suva, Fiji. ⟨hal-01213858⟩

Share

Metrics

Record views

523

Files downloads

420