Ruitenburg's Theorem via Duality and Bounded Bisimulations

Luigi Santocanale (1, 2) , Silvio [e-mail] Ghilardi (3)

1 LIS - Laboratoire d'Informatique et des Systèmes (LIS) (Marseille, Toulon)
2 LIRICA - Logique, Interaction, Raisonnement et Inférence, Complexité, Algèbre
3 UNIMI - Università degli Studi di Milano = University of Milan

For a given intuitionistic propositional formula A and a propositional variable x occurring in it, define the infinite sequence of formulae { A _i | i≥1} by letting A_1 be A and A_{i+1} be A(A_i/x). Ruitenburg's Theorem [8] says that the sequence { A _i } (modulo logical equivalence) is ultimately periodic with period 2, i.e. there is N ≥ 0 such that A N+2 ↔ A N is provable in intuitionistic propositional calculus. We give a semantic proof of this theorem, using duality techniques and bounded bisimulations ranks.

Mots clés: Ruitenburg's Theorem; Sheaf Duality; Bounded Bisimulations

Domaines: Logique [math.LO]; Logique en informatique [cs.LO]

Luigi Santocanale, Silvio [e-mail] Ghilardi. Ruitenburg's Theorem via Duality and Bounded Bisimulations. Advances in Modal Logic, Aug 2018, Bern, Switzerland. ⟨hal-01766636⟩