## Simplifying the modal mu-calculus alternation hierarchy

**Julian C. Bradfield**
*Abstract:*

In a previous paper,
the strictness of the modal mu-calculus alternation hierarchy was
shown by transferring a hierarchy from arithmetic; the latter was a
minor corollary of a highly technical analysis of Robert Lubarsky.
In this paper, we give an entirely elementary proof of that
hierarchy; further, simple examples of strict alternation depth
*n* formulae can be constructed, which in turn give very
simple examples to separate the modal hierarchy. In addition, the
winning strategy formulae of parity games are shown to be such
examples.

**ECS-LFCS-98-385**.

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