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Decidability in geometric grid classes of permutations

Samuel Braunfeld

TL;DR

The paper proves that for a finite matrix $M$, both the basis and the generating function of the geometric grid class $\mathrm{Geom}(M)$ are computable from $M$, and it provides a computable bound on the size of basis elements. The approach uses monadic second-order logic on permutations and words to encode $\mathrm{Geom}(M)$ into regular languages, enabling decidability and explicit automata-based constructions, along with extensions to substitution-closure and related subclasses. It also shows that quantifier-free interpretations characterize subclasses and connect to bounded lettericity, offering a unifying framework for algorithmic analysis of geometric grid classes. These results fill a constructive gap left by prior non-constructive proofs and pave the way for practical enumeration and structural analysis of permutation classes within this setting.

Abstract

We prove that the basis and the generating function of a geometric grid class of permutations Geom$(M)$ are computable from the matrix $M$, as well as some variations on this result. Our main tool is monadic second-order logic on permutations and words.

Decidability in geometric grid classes of permutations

TL;DR

The paper proves that for a finite matrix , both the basis and the generating function of the geometric grid class are computable from , and it provides a computable bound on the size of basis elements. The approach uses monadic second-order logic on permutations and words to encode into regular languages, enabling decidability and explicit automata-based constructions, along with extensions to substitution-closure and related subclasses. It also shows that quantifier-free interpretations characterize subclasses and connect to bounded lettericity, offering a unifying framework for algorithmic analysis of geometric grid classes. These results fill a constructive gap left by prior non-constructive proofs and pave the way for practical enumeration and structural analysis of permutation classes within this setting.

Abstract

We prove that the basis and the generating function of a geometric grid class of permutations Geom are computable from the matrix , as well as some variations on this result. Our main tool is monadic second-order logic on permutations and words.
Paper Structure (14 sections, 21 theorems, 5 equations, 1 figure)

This paper contains 14 sections, 21 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Permutations and geometric grid classes
  4. Monadic second order logic
  5. Why MSO?
  6. Notation
  7. Computing the basis of Geom(M)
  8. An MSO definition of Geom(M)
  9. Computing a basis
  10. Bounding the size of basis elements
  11. Computing the generating function of Geom(M)
  12. Some extensions
  13. Quantifier-free interpretations
  14. Questions

Key Result

Theorem 1.1

Let $M$ represent a finite $0/$$\pm1$-matrix and $\mathrm{Geom}(M)$ the corresponding geometric grid class. Then the size of the basis elements of $\mathrm{Geom}(M)$ are bounded above by a computable (in fact, elementary) function of $|M|$, and the basis of $\mathrm{Geom}(M)$ and the generating func

Figures (1)

  • Figure 1: The standard figure for the matrix $-111-1$, with one possible gridding of the permutation $(132)$.

Theorems & Definitions (37)

  • Theorem 1.1: Proposition \ref{['prop:bound']}, Theorems \ref{['thm:compBasis']}, \ref{['thm:compgf']}
  • Theorem 1.2: Theorem \ref{['thm:compSC']}
  • Theorem 2.1: albert2013geometric*Theorem 6.2
  • Proposition 2.2: albert2013geometric, Proposition 5.1
  • Theorem 2.3: torun*Theorem A.4
  • Corollary 2.4: torun*Theorem A.5
  • Theorem 2.5: torun*Theorem B.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • ...and 27 more