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On the $$\mu$$-calculus over transitive and finite transitive frames. (English) Zbl 1208.68145
Summary: We prove that the modal $$\mu$$-calculus collapses to first order logic over the class of finite transitive frames. The proof is obtained by using some byproducts of a new proof of the collapse of the $$\mu$$-calculus to the alternation free fragment over the class of transitive frames.
Moreover, we prove that the modal $$\mu$$-calculus is Büchi and co-Büchi definable over the class of all models where, in a strongly connected component, vertexes are distinguishable by means of the propositions they satisfy.

##### MSC:
 68Q60 Specification and verification (program logics, model checking, etc.) 03B45 Modal logic (including the logic of norms)
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