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2

I'm not sure whether it would work in your case, but in order to show succinctness results in modal/temporal logic (e.g. the fact the two-variable logic over words in exponentially more succinct than unary temporal logic) one can employ formula size games or Adler-Immerman games. Probably the most recent paper to read is by Lauri Hella and Miikka Vilander. ...


1

Your question is formally the same as this one: how many symbols $X$ are needed to write the polynomial $XX + XXXX + XXXXXX = XX(1 + XX(1 + XX))$? Well, you obtained $6$ as an upper bound. It is also a lower bound because any expression involving less than $6$ symbols $X$ would define a polynomial of degree $< 6$. For the general case, I would not be ...


4

Sorry to provide the same answer as for this question, but this was proved in [1, Theorem 3.1]. [1] J. Cohen, D. Perrin and J.-É. Pin, On the expressive power of temporal logic for finite words, J. Comput. System Sci. 46 (1993), 271-294.


5

As mentioned in the comments, the translation is shown in: Volker Diekert and Paul Gastin. "First-order definable languages." (2008) http://www.lsv.fr/Publis/PAPERS/PDF/DG-WT08.pdf And it goes via a characterization of $LTL$ as $FO[<]$.


4

An algebraic characterisation of the restricted temporal logic (fragment using only next and eventually, but not until) is given in [1]. The expressive power of this fragment is the set of regular languages whose syntactic monoid are locally $\cal L$-trivial. I am not sure of the result you are looking for, but this article probably contains enough material ...


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