Multidimensional tilings and MSO logic
R\'emi Pallen, Ilkka T\"orm\"a
TL;DR
This paper explores the definability and complexity of colorings of the infinite plane using MSO logic, classifying the decision problems based on quantifier complexity and analyzing the complexity of MSO-definable languages.
Contribution
It introduces a framework for analyzing MSO-definable sets of plane colorings and classifies the complexity of related decision problems based on quantifier alternation.
Findings
Decidability of subshift definitions varies with MSO quantifier complexity.
Exact classifications or bounds are provided for the complexity of MSO-definable languages.
The results connect logical definability with computational complexity in tiling problems.
Abstract
We define sets of coulourings of the infinite discrete plane using monadic second order (MSO) formulas. We determine the complexity of deciding whether such a formula defines a subshift, parametrized on the quantifier alternation complexity of the formula. We also study the complexities of languages of MSO-definable sets, giving either an exact classification or upper and lower bounds for each quantifier alternation class.
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