MoL-2001-13: Is Multiplication Harder than Addition? Arithmetical Definability over Finite Structures

MoL-2001-13: Lee, Troy (2001) Is Multiplication Harder than Addition? Arithmetical Definability over Finite Structures. [Report]

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Abstract

Is Multiplication Harder than Addition? Arithmetical Definability over Finite Structures Troy Lee Abstract: In this thesis, we show that two definability results of Julia Robinson, namely that multiplication and the successor relation can first-order define addition, and that the divisibility relation and successor can first-order define multiplication, also hold over finite structures (where ordering is used instead of successor). The first result is obtained by showing that the BIT predicate can be defined with TIMES and ordering, thus FO(<,BIT)=FO(<,TIMES). Then the ``carry-look-ahead'' construction can be used to define PLUS with BIT. We also show that there is no first-order sentence with TIMES but without ordering which can define PLUS. A corollary is then that TIMES cannot first-order define ordering. Our results, together with recent results on the Crane Beach conjecture show that there is a language definable in FO(<,TIMES) but not in FO(<,PLUS). In other words, there is no first-order sentence with < and PLUS which can express TIMES.

Item Type: Report
Report Nr: MoL-2001-13
Series Name: Master of Logic Thesis (MoL) Series
Year: 2001
Date Deposited: 12 Oct 2016 14:38
Last Modified: 12 Oct 2016 14:38
URI: https://eprints.illc.uva.nl/id/eprint/726

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