A Note on Higher Order and Variable Order Logic over Finite Models

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.