  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.
