Mathematical Notes

, Volume 96, Issue 3–4, pp 575–585

# Circular proofs for the Gödel-Löb provability logic

• D. S. Shamkanov
Article

## Abstract

Sequent calculus for the provability logic 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 GL.

## Keywords

provability logic sequent calculus circular proof the Gödel-Löb logic the Lyndon interpolation property split sequent

## References

1. 1.
S. Negri, “Proof analysis in modal logic,” J. Philos. Logic 34(5–6), 507–544 (2005).
2. 2.
F. Poggiolesi, “A purely syntactic and cut-free sequent calculus for the modal logic of provability,” Rev. Symb. Log. 2(4), 593–611 (2009).
3. 3.
R. Goré and R. Ramanayake, “Valentini’s cut-elimination for provability logic resolved,” Rev. Symb. Log. 5(2), 212–238 (2012).
4. 4.
J. Brotherston, N. Gorogiannis, and R. L. Petersen, “A generic cyclic theorem prover,” in Programming Languages and Systems, Lecture Notes in Comput. Sci. (Springer-Verlag, Berlin, 2012), Vol. 7705, pp. 350–367.
5. 5.
J. van Benthem, “Modal frame correspondences and fixed-points,” Studia Logica 83(1–3), 133–155 (2006).
6. 6.
A. Visser, “Löb’s logic meets the µ-calculus,” in Processes, Terms and Cycles: Steps on the Road to Infinity, Lecture Notes in Comput. Sci. (Springer-Verlag, Berlin, 2005), Vol. 3838, pp. 14–25.
7. 7.
L. Alberucci and A. Facchini, “On modal µ-calculus and Gödel-Löb logic,” Studia Logica 91(2), 145–169 (2009).
8. 8.
D. S. Shamkanov, “Interpolation properties for provability logics GL and GLP,” Trudy Mat. Inst. Steklov 274, 329–342 (2011) [Proc. Steklov Inst.Math. 274, 303–316 (2011)].
9. 9.
G. Sambin and S. Valentini, “A modal sequent calculus for a fragment of arithmetic,” Studia Logica 39(2–3), 245–256 (1980).
10. 10.
G. Sambin and S. Valentini, “The modal logic of provability. The sequential approach,” J. Philos. Logic 11(3), 311–342 (1982).
11. 11.
D. Leivant, “On the proof theory of the modal logic for arithmetic provability,” J. Symbolic Logic 46(3), 531–538 (1981).
12. 12.
W. W. Tait, “Normal derivability in classical logic,” in The Syntax and Semantics of Infinitary Languages, Lecture Notes in Math. (Springer-Verlag, Berlin, 1986), Vol. 72, pp. 204–236.Google Scholar
13. 13.
S. Valentini, “The modal logic of provability: cut-elimination,” J. Philos. Logic 12(4), 471–476 (1983).
14. 14.
J. Brotherston, Sequent Calculus Proof Systems for Inductive Definitions, PhD Thesis (Univ. of Edinburgh, Edinburgh, 2006).Google Scholar
15. 15.
B. Jacobs and J. Rutten, “A tutorial on (co)algebras and (co)induction,” Bull. EATCS 62, 222–259 (1997).
16. 16.
P. Lindström, “Provability logic-A short introduction,” Theoria 62(1–2), 19–61 (1996).
17. 17.
C. Smoryński, “Modal logic and self-reference,” in Handbook of Philosophical Logic, Vol. II. Extensions of Classical Logic, Synthese Lib. (Reidel, Dordrecht, 1984), Vol. 165, pp. 441–495.Google Scholar