10.2168/LMCS-1(1:6)2005
Grohe, Martin
Martin
Grohe
Schweikardt, Nicole
Nicole
Schweikardt
The succinctness of first-order logic on linear orders
episciences.org
2005
Computer Science - Logic in Computer Science
F.4.1
contact@episciences.org
episciences.org
2004-11-04T00:00:00+01:00
2016-11-21T15:37:03+01:00
2005-06-29
eng
Journal article
https://lmcs.episciences.org/2276
arXiv:cs/0502047
1860-5974
PDF
1
Logical Methods in Computer Science ; Volume 1, Issue 1 ; 1860-5974
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.