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Separating complexity classes of LCL problems on grids

Katalin Berlow, Anton Bernshteyn, Clark Lyons, Felix Weilacher

TL;DR

This work investigates the complexity of locally checkable labeling (LCL) problems on $\mathbb{Z}^n$ through descriptive set theory, ergodic theory, and computability, revealing separations among several natural complexity classes and producing counterexamples to conjectures. The main technical device is a toast construction, in particular rectangular toasts, which enables encoding LCLs as local constraints whose global solutions encode large scale combinatorial structures. The authors present a single LCL $\Pi$ on $\mathbb{Z}^n$ that lies in $\mathtt{MEASURE}(\mathbb{Z}^n)$ but not in $\mathtt{BAIRE}(\mathbb{Z}^n)$, is solvable on the free shift in the Baire sense but not by finitary i.i.d. methods, yet remains computable on free computable actions; extensions (CRT^+) yield even stronger separations. A companion sequel is announced to establish pairwise incomparability among $\mathtt{MEASURE}$, $\mathtt{COMPUTABLE}$, and $\mathtt{BaireSHIFT}$ for $\mathbb{Z}^n$, highlighting fundamental differences between $\mathbb{Z}^n$ and free groups. The results illuminate how local constraints (rectangular toasts) control global regularity properties across multiple definability paradigms and offer a unified framework for comparing complexity classes in descriptive combinatorics and beyond.

Abstract

We study the complexity of locally checkable labeling (LCL) problems on $\mathbb{Z}^n$ from the point of view of descriptive set theory, computability theory, and factors of i.i.d. Our results separate various complexity classes that were not previously known to be distinct and serve as counterexamples to a number of natural conjectures in the field.

Separating complexity classes of LCL problems on grids

TL;DR

This work investigates the complexity of locally checkable labeling (LCL) problems on through descriptive set theory, ergodic theory, and computability, revealing separations among several natural complexity classes and producing counterexamples to conjectures. The main technical device is a toast construction, in particular rectangular toasts, which enables encoding LCLs as local constraints whose global solutions encode large scale combinatorial structures. The authors present a single LCL on that lies in but not in , is solvable on the free shift in the Baire sense but not by finitary i.i.d. methods, yet remains computable on free computable actions; extensions (CRT^+) yield even stronger separations. A companion sequel is announced to establish pairwise incomparability among , , and for , highlighting fundamental differences between and free groups. The results illuminate how local constraints (rectangular toasts) control global regularity properties across multiple definability paradigms and offer a unified framework for comparing complexity classes in descriptive combinatorics and beyond.

Abstract

We study the complexity of locally checkable labeling (LCL) problems on from the point of view of descriptive set theory, computability theory, and factors of i.i.d. Our results separate various complexity classes that were not previously known to be distinct and serve as counterexamples to a number of natural conjectures in the field.

Paper Structure

This paper contains 23 sections, 34 theorems, 64 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Descriptive complexity of LCL problems
  3. Shift actions: the universal domain for descriptive combinatorics?
  4. (Finitary) factors of i.i.d.
  5. Computable combinatorics
  6. Summary, further results, and an overview of the remainder of the paper
  7. Rectangular toasts
  8. Basic definitions
  9. Rectangular toasts and measure theory
  10. Rectangular toasts and Baire category
  11. Rectangular toasts and shifts
  12. Rectangular toasts and finitary factors of i.i.d.
  13. Computable rectangular toasts
  14. LCL problems based on rectangular toasts
  15. The rectangular $q$-toast problem
  16. ...and 8 more sections

Key Result

Theorem 1.8

For every integer $n \geqslant 2$, we have As a consequence, $\mathtt{BAIRE}(\mathbb{Z}^n) \neq \mathtt{MEASURE}(\mathbb{Z}^n)$ and $\mathtt{BOREL}(\mathbb{Z}^n) \neq \mathtt{MEASURE}(\mathbb{Z}^n)$.

Figures (5)

  • Figure 1: Complexity classes of LCL problems on $\mathbb{Z}^n$ with $n \geqslant 2$. Here:
  • Figure 2: A toast for $\mathbb{Z}^2$.
  • Figure 3: A rectangular toast for $\mathbb{Z}^2$.
  • Figure 4: The labeling corresponding to the rectangular toast from Fig. \ref{['fig:waffle']}.
  • Figure 5: An illustration for the proof of Theorem \ref{['theo:main_extra']}. A rectangle $K \in \mathcal{R}$ and the two regions $D_1(K)$ and $D_2(K)$ in which the functions $h_1$ and $h_2$ respectively must equal ${1}$.

Theorems & Definitions (94)

  • Definition 1.1: LCL problems on a group
  • Example 1.2: Proper $k$-colorings of Cayley graphs
  • Definition 1.3: Solutions to LCL problems on group actions
  • Example 1.4: Proper $2$-coloring on $\mathbb{Z}$
  • Definition 1.5: Descriptive complexity classes
  • Theorem 1.8
  • Definition 1.9: Shift action
  • Theorem 1.10: Seward--Tucker-Drob STD, AB/Seward Ber_cont
  • Remark 1.11
  • Definition 1.12: $\mathtt{BaireSHIFT}$
  • ...and 84 more