# Does the first order theory of a finite structure have bounded quantifier rank?

Let $\mathfrak{A}$ be any finite structure. Does its first order theory $\mathfrak{T} := \mathfrak{TH}(\mathfrak{A})$ have bounded quantifier rank, in the sense that there is a $q\in\mathbb{N}$ such that for all $\varphi\in\mathfrak{T}$ with $qr(\varphi) > q$ there is a $\varphi'\in\mathfrak{T}$ with $qr(\varphi')\leq q$ and $\varphi'\equiv\varphi$ ?

• Isn't this a question for Mathoverflow rather than CS theory? – Andrej Bauer Aug 27 '17 at 21:10
• @Andrej, Finite model theory and descriptive complexity are also considered part of TCS. – Kaveh Aug 27 '17 at 21:37
• Excellent, so it's like Bob Harper said once: math is a special case of computer science. – Andrej Bauer Aug 28 '17 at 6:02
• Computer science is also a special case of math, and they are both also special cases of logic, and vice versa. – fhyve Sep 6 '17 at 1:53