🤖 AI Summary
This paper investigates the computational complexity of monadic second-order (MSO) logic defining colorings on the infinite two-dimensional grid. **Problem:** It addresses (1) the decidability of whether an MSO formula defines a subshift, and (2) the complexity characterization of the associated language families. **Method:** For the first time, it systematically establishes an exact correspondence between the quantifier alternation depth of MSO formulas and the computational complexity of subshift definability, integrating subshift theory, automata theory, model theory of infinite graphs, and formal language complexity analysis. **Contribution/Results:** For each quantifier alternation class Σₙ and Πₙ, it provides tight complexity classifications—e.g., Σₙᵖ-completeness—for the subshift definability problem; moreover, it delivers complete and optimal upper and lower bounds for the class of languages definable by MSO over ℤ², thereby filling a fundamental gap in understanding the systematic relationship between MSO expressiveness and computational complexity on multidimensional discrete structures.
📝 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.