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

TL;DR

This paper extends descriptive complexity results to high order logic, providing a logical characterization of ELEMENTARY and exploring the expressivity of fixed point and monadic high order logic, including Variable Order logic.

Contribution

It demonstrates that high order logic captures ELEMENTARY complexity and analyzes the expressivity of fixed point operators and Variable Order logic over finite models.

Findings

High order logic characterizes ELEMENTARY complexity.

Variable Order logic encompasses the Analytical Hierarchy.

Fixed point operators' expressivity is analyzed in high order logic.

Abstract

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.