Arithmetical definability over finite structures

Abstract

AbstractArithmetical 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 AC0 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 equivalence FO(PLUS, TIMES) =FO(BIT) to FO(<, TIMES) = FO(BIT), 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 FO(PLUS) is strictly less expressive than FO(<, TIMES) = FO(<, DIVIDES) = FO(<,COPRIME). In more colorful language, one could say that, for parallel computation, multiplication is harder than addition.