  We show that descriptive complexity's result extends in High Order Logic to
capture the expressivity of Turing Machine which have a finite number of
alternation and whose time or space is bounded by a finite tower of
exponential. Hence we have a logical characterisation of ELEMENTARY. We also
consider the expressivity of some fixed point operators and of monadic high
order logic.
  Finally, we show that Variable Order logic over finite structures contain the
Analytical Hierarchy.
