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For a given intuitionistic propositional formula A and a propositionalvariable x occurring in it, define the infinite sequence of formulae { A \_i |i$\ge$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 ultimatelyperiodic with period 2, i.e. there is N $\ge$ 0 such that A N+2$\leftrightarrow$ A N is provable in intuitionistic propositional calculus. Wegive a semantic proof of this theorem, using duality techniques and boundedbisimulations ranks.

Ruitenburg's Theorem via Duality and Bounded Bisimulations