The succinctness of first-order logic on linear orders

TL;DR

This paper investigates the relative succinctness of various fragments of first-order logic on linear orders, revealing exponential and non-elementary differences in their expressive efficiencies.

Contribution

It provides the first detailed comparison of the succinctness of finite variable fragments and monadic second-order logic on linear orders, highlighting significant efficiency gaps.

Findings

2- and 3-variable fragments have similar succinctness up to polynomial factors

4-variable fragment is exponentially more succinct than 3-variable fragment

Monadic second-order logic is non-elementarily more succinct than first-order logic

Abstract

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.