Arithmetical definability over finite structures

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 and . 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 equivalence ! " # to $ % & ' ( $ " # , answering a question raised by Barrington et al. about the Crane Beach Conjecture. Together with previous results on the Crane Beach Conjecture, our results imply that ( ) * is strictly less expressive than $ % & ' + , ( & /. -! 0 1 , ( $ & ! )23 . In more colorful language, one could say that, for parallel computation, multiplication is harder than addition.