# A Crevice on the Crane Beach: Finite-Degree Predicates

## TL;DR

This paper explores the expressive power of first-order logic over words with finite-degree predicates, demonstrating its locality properties and the conditions under which it satisfies the Crane Beach Property, contrasting it with more expressive logics.

## Contribution

It shows that FO with order, MSB$_0$, and finite-degree predicates has the Crane Beach Property, revealing its locality and expressive limitations compared to FO with arbitrary predicates.

## Key findings

- FO[<, Fin] satisfies the Crane Beach Property.
- FO[<, Fin] exhibits locality despite expressive power.
- FO[<, Fin] can express a wide variety of languages.

## Abstract

First-order logic (FO) over words is shown to be equiexpressive with FO equipped with a restricted set of numerical predicates, namely the order, a binary predicate MSB$_0$, and the finite-degree predicates: FO[Arb] = FO[<, MSB$_0$, Fin]. The Crane Beach Property (CBP), introduced more than a decade ago, is true of a logic if all the expressible languages admitting a neutral letter are regular. Although it is known that FO[Arb] does not have the CBP, it is shown here that the (strong form of the) CBP holds for both FO[<, Fin] and FO[<, MSB$_0$]. Thus FO[<, Fin] exhibits a form of locality and the CBP, and can still express a wide variety of languages, while being one simple predicate away from the expressive power of FO[Arb]. The counting ability of FO[<, Fin] is studied as an application.