# Defining long words succinctly in FO and MSO

We consider the length of the longest word definable in FO and MSO via a formula of size n. For both logics we obtain as an upper bound for this number an exponential tower of height linear in n. We prove this by counting types with respect to a fixed quantifier rank. As lower bounds we obtain for both FO and MSO an exponential tower of height in the order of a rational power of n. We show these lower bounds by giving concrete formulas defining word representations of levels of the cumulative hierarchy of sets. In addition, we consider the Löwenheim-Skolem and Hanf numbers of these logics on words and obtain similar bounds for these as well.

• 5 publications
• 4 publications
03/30/2019

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### A Fixed-Depth Size-Hierarchy Theorem for AC^0[⊕] via the Coin Problem

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### New Bounds on Antipowers in Binary Words

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### Homotopy height, grid-major height and graph-drawing height

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### Asymptotic bounds for the number of closed and privileged words

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### Learning Formulas in Finite Variable Logics

We consider grammar-restricted exact learning of formulas and terms in f...