A Logic that Captures $\beta$P on Ordered Structures

TL;DR

This paper introduces an extension of inflationary fixed-point logic with bounded second-order quantifiers, demonstrating it captures the class βP on ordered structures but not on all finite structures.

Contribution

It develops a new logical framework with bounded quantifiers and proves its expressive power aligns with βP on ordered structures, including a novel Ehrenfeucht-Fra"issé game.

Findings

The extended logic captures βP on ordered structures.

The capturing does not hold on all finite structures.

A new Ehrenfeucht-Fra"issé game is designed for this logic.

Abstract

We extend the inflationary fixed-point logic, IFP, with a new kind of second-order quantifiers which have (poly-)logarithmic bounds. We prove that on ordered structures the new logic $\exists^{\log^{\omega}}\text{IFP}$ captures the limited nondeterminism class $\beta\text{P}$. In order to study its expressive power, we also design a new version of Ehrenfeucht-Fra\"iss\'e game for this logic and show that our capturing result will not hold on the general case, i.e. on all the finite structures.