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VOLUME 1, ISSUE 1, PAPER 6


The succinctness of first-order logic on linear orders

©Martin Grohe, Humboldt-Universität zu Berlin
©Nicole Schweikardt, Humboldt-Universität zu Berlin

Abstract
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.

Publication date: June 29, 2005

Full Text: PDF | PostScript
DOI: 10.2168/LMCS-1(1:6)2005

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