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 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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Author:Martin Grohe, Nicole Schweikardt
URN:urn:nbn:de:hebis:30:3-125535
DOI:https://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):2011/09/30
Year of first Publication:2005
Publishing Institution:Universitätsbibliothek Johann Christian Senckenberg
Release Date:2011/09/30
Tag:Computer science; F.4.1; Logic in computer science; finite model theory; first-order logic; succinctness
Volume:1
Issue:1, Art. 6
Page Number:25
First Page:1
Last Page:25
Note:
This work is licensed under the Creative Commons Attribution-NoDerivs License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nd/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.
HeBIS-PPN:359279465
Institutes:Informatik und Mathematik / Informatik
Dewey Decimal Classification:0 Informatik, Informationswissenschaft, allgemeine Werke / 00 Informatik, Wissen, Systeme / 004 Datenverarbeitung; Informatik
Sammlungen:Universitätspublikationen
Licence (German):License LogoCreative Commons - Namensnennung-Keine Bearbeitung