# Quantifier alternation in a class of recursively defined tree properties

## TL;DR

This paper introduces a recursive sequence of first-order properties on trees and rigorously determines their alternating quantifier depth using Ehrenfeucht-Fra"{i}ssé games, shedding light on the complexity of logical expressions.

## Contribution

It defines a new class of recursively constructed tree properties and provides a method to precisely compute their quantifier alternation depth.

## Key findings

- Determined the alternating quantifier depth for each property in the sequence.
- Established a recursive framework for analyzing logical complexity on trees.
- Applied Ehrenfeucht-Fra"{i}ssé} games to rigorously analyze logical properties.

## Abstract

Alternating quantifier depth is a natural measure of difficulty required to express first order logical sentences. We define a sequence of first order properties on rooted, locally finite trees in a recursive manner, and provide rigorous arguments for finding the alternating quantifier depth of each property in the sequence, using Ehrenfeucht-Fra\"{i}ss\'{e} games.