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The modal $$\mu$$-calculus hierarchy over restricted classes of transition systems. (English) Zbl 1191.03012
It is well known that the alternation hierarchy of the modal $$\mu$$-calculus is strict, cf. [J. C. Bradfield, Theor. Comput. Sci. 195, No. 2, 133–153 (1998; Zbl 0915.03017)]. In this paper, the authors study the alternation hierarchy of the modal $$\mu$$-calculus over restricted classes of transition systems.
The main result of the paper is that the modal $$\mu$$-calculus hierarchy over transitive transition systems is strict. In order to obtain this result, the authors provide an explicit syntactical translation of the full modal $$\mu$$-calculus into the alternation-free fragment. Furthermore, the authors show that the modal $$\mu$$-calculus hierarchy over reflexive transition systems is strict as well. The proof is a modification of a proof of the strictness of the $$\mu$$-calculus hierarchy over the class of binary transition systems by A. Arnold [Theor. Inform. Appl. 33, No. 4–5, 329–339 (1999; Zbl 0945.68118)] and L. Alberucci [Lect. Notes Comput. Sci. 2500, 185–201, 365–376 (2002; Zbl 1021.03012)].
Other results in this article are finite model theorems for reflexive and for transitive transition systems and a proof of the fact that the modal $$\mu$$-calculus over the class of transitive and symmetric transition systems collapses to its purely modal fragment.

##### MSC:
 03B45 Modal logic (including the logic of norms) 03B70 Logic in computer science 05C57 Games on graphs (graph-theoretic aspects) 68Q85 Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)
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