Arithmetical Definability over Finite Structures
Troy Lee
Abstract:
Arithmetical definability has been extensively studied over the
natural numbers. In this paper, we take up the study of arithmetical
definability over finite structures, motivated by the correspondence
between uniform $\AC^0$ and $\FO(\PLUS,\TIMES)$. We prove finite
analogs of three classic results in arithmetical definability, namely
that $<$ and TIMES can first-order define PLUS, that $<$ and DIVIDES
can first-order define TIMES, and that $<$ and COPRIME can first-order
define TIMES.
The first result sharpens the known equivalence
${\FO(\PLUS,\TIMES)=}{\FO(\BIT)}$ to $\FO(<,\TIMES)=\FO(\BIT)$,
answering a question raised by Barrington et al. (LICS 2001) about the
Crane Beach Conjecture. Together with previous results on the Crane
Beach Conjecture, our results imply that $\FO(\PLUS)$ is strictly less
expressive than $\FO(<,\TIMES)=\FO(<,\DIVIDES)=\FO(<,\COPRIME)$. In
more colorful language, one could say this containment adds evidence
to the belief that multiplication is harder than addition.