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We present two sequent calculi for the modal  µ -calculus over  S5  and prove their completeness by using classical methods. One sequent calculus has an analytical cut rule and could be used for a decision procedure the other uses a modified version of the induction rule.We also provide a completeness theorem for Kozen's Axiomatization over  S5  without using the completeness result established byWalukiewicz for the modal  µ -calculus over arbitrary models.

Sequent Calculi for the Modal µ-Calculus over S5