Geodesic
Circular Proofs for the G\"odel--L\"ob Provability Logic
Matematičeskie zametki, Tome 96 (2014) no. 4, pp. 609-622

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Sequent calculus for the provability logic $\mathsf{GL}$ is considered, in which provability is based on the notion of a circular proof. Unlike ordinary derivations, circular proofs are represented by graphs allowed to contain cycles, rather than by finite trees. Using this notion, we obtain a syntactic proof of the Lyndon interpolation property for $\mathsf{GL}$.
Keywords: provability logic, sequent calculus, circular proof, the Gödel–Löb logic, the Lyndon interpolation property, split sequent. 
@article{MZM_2014_96_4_a11,
     author = {D. S. Shamkanov},
     title = {Circular {Proofs} for the {G\"odel--L\"ob} {Provability} {Logic}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {609--622},
     publisher = {mathdoc},
     volume = {96},
     number = {4},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_96_4_a11/}
}
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D. S. Shamkanov. Circular Proofs for the G\"odel--L\"ob Provability Logic. Matematičeskie zametki, Tome 96 (2014) no. 4, pp. 609-622. http://geodesic.mathdoc.fr/item/MZM_2014_96_4_a11/