PP-2002-04: Arithmetical Definability over Finite Structures

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

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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.

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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