The succinctness of first-order logic on linear orders

Succinctness is a natural measure for comparing the strength of different logics. Intuitively, a logic L_1 is more succinct than another logic L_2 if all properties that can be expressed in L_2 can be expressed in L_1 by
Succinctness is a natural measure for comparing the strength of different logics. Intuitively, a logic L_1 is more succinct than another logic L_2 if all properties that can be expressed in L_2 can be expressed in L_1 by formulas of (approximately) the same size, but some properties can be expressed in L_1 by (significantly) smaller formulas.
We study the succinctness of logics on linear orders. Our first theorem is concerned with the finite variable fragments of first-order logic. We prove that:
(i) Up to a polynomial factor, the 2- and the 3-variable fragments of first-order logic on linear orders have the same succinctness. (ii) The 4-variable fragment is exponentially more succinct than the 3-variable fragment. Our second main result compares the succinctness of first-order logic on linear orders with that of monadic second-order logic. We prove that the fragment of monadic second-order logic that has the same expressiveness as first-order logic on linear orders is non-elementarily more succinct than first-order logic.
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Metadaten
Author:Martin Grohe, Nicole Schweikardt
URN:urn:nbn:de:hebis:30:3-125535
DOI:http://dx.doi.org/10.2168/LMCS-1(1:6)2005
ISSN:1860-5974
ArXiv Id:http://arxiv.org/abs/arXiv:cs/0502047
Parent Title (English):Logical methods in computer science
Publisher:Department of Theoretical Computer Science, Technical University of Braunschweig
Place of publication:Braunschweig
Document Type:Article
Language:English
Date of Publication (online):2005/05/29
Date of first Publication:2005/05/29
Publishing Institution:Univ.-Bibliothek Frankfurt am Main
Release Date:2011/09/30
Tag:finite model theory; first-order logic; succinctness
Volume:2005
Issue:1:6
Pagenumber:25
First Page:1
Last Page:25
Institutes:Informatik
Dewey Decimal Classification:004 Datenverarbeitung; Informatik
Licence (German):License LogoCreative Commons - Namensnennung-Keine kommerzielle Nutzung-Weitergabe unter gleichen Bedingungen

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