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it is well known that the modal $\mu$-calculus is one of the most expressive temporal logics for expressing properties of trees/graphs, and that CTL* is strictly less expressive than the $\mu$-calculus.

Here I would like to ask for an example of $\mu$-calculus formula, as simple as possible, that is not expressible in CTL*, and hopefully for an explaination of its meaning (fixed-point formulas quickly become unreadable). Any good reference for a "concrete" simple example would also be great!

Thank you in advance

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Take a path property that is not first-order expressible, e.g. $$\nu x.p\wedge\Diamond\Diamond x$$ which says that there exists a path where the atomic proposition $p$ holds at every even position, and any valuation can be used on odd positions.

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  • $\begingroup$ thanks a lot for this simple answer. Could you also suggest a reference supporting this example? Thank you again $\endgroup$
    – LORE81
    Jan 22, 2013 at 8:46
  • $\begingroup$ Nice question & answer (+2). Have a look at cstheory.stackexchange.com/q/16186/6424. I gave the evenness example there too. Maybe some answer will refer to evenness, too. $\endgroup$
    – DaveBall aka user750378
    Jan 22, 2013 at 19:57
  • $\begingroup$ @LORE81: the ``$p$ at even positions'' example is a classic, that you can find for instance in the paper by Wolper pointed by @DaveBall in his question. It is not too difficult to prove directly by induction on LTL formulae; alternatively, you can construct the transition monoid and see that it is not aperiodic; finally you can try an Ehrenfeucht-Fraïssé argument, though it's long to spell out in full detail. $\endgroup$
    – Sylvain
    Jan 22, 2013 at 20:13

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