PP-2002-04: Arithmetical Definability over Finite Structures

PP-2002-04: Lee, Troy (2002) Arithmetical Definability over Finite Structures. [Report]

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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.

Item Type: Report
Report Nr: PP-2002-04
Series Name: Prepublication (PP) Series
Year: 2002
Uncontrolled Keywords: Arithmetical Definability; Descriptive Complexity; Finite Model Theory
Date Deposited: 12 Oct 2016 14:36
Last Modified: 12 Oct 2016 14:36
URI: https://eprints.illc.uva.nl/id/eprint/68

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