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Rank-decreasing transductions

Mikołaj Bojańczyk, Pierre Ohlmann

TL;DR

This work characterises rank-decreasing graph-to-graph transductions by a purely structural criterion, showing that for inputs of bounded treewidth, rank-decreasing is equivalent to being contained in a cmso-definable transduction. The authors prove the easy direction (definable implies rank-decreasing) in full generality, and the harder direction (rank-decreasing implies sub-definable) first for inputs with equality-only, then for trees, and finally for graphs of bounded treewidth using definable tree decompositions. Central tools include generalized cut-rank notions via type matrices, asymptotic rank equivalences, and a tree-automata–driven reconstruction method that makes the output recoverable by MSO reasoning. The results bridge logical definability and structural graph parameters, with modulo counting highlighted as essential and several conjectures proposed to extend the theory beyond bounded treewidth, potentially impacting recognisability versus definability in broader structural classes.

Abstract

We propose to study transformations on graphs, and more generally structures, by looking at how the cut-rank (as introduced by Oum) of subsets is affected when going from the input structure to the output structure. We consider transformations in which the underlying sets are the same for both the input and output, and so the cut-ranks of subsets can be easily compared. The purpose of this paper is to give a characterisation of logically defined transductions that is expressed in purely structural terms, without referring to logic: transformations which decrease the cut-rank, in the asymptotic sense, are exactly those that can be defined in monadic second-order logic. This characterisation assumes that the transduction has inputs of bounded treewidth; we also show that the characterisation fails in the absence of any assumptions.

Rank-decreasing transductions

TL;DR

This work characterises rank-decreasing graph-to-graph transductions by a purely structural criterion, showing that for inputs of bounded treewidth, rank-decreasing is equivalent to being contained in a cmso-definable transduction. The authors prove the easy direction (definable implies rank-decreasing) in full generality, and the harder direction (rank-decreasing implies sub-definable) first for inputs with equality-only, then for trees, and finally for graphs of bounded treewidth using definable tree decompositions. Central tools include generalized cut-rank notions via type matrices, asymptotic rank equivalences, and a tree-automata–driven reconstruction method that makes the output recoverable by MSO reasoning. The results bridge logical definability and structural graph parameters, with modulo counting highlighted as essential and several conjectures proposed to extend the theory beyond bounded treewidth, potentially impacting recognisability versus definability in broader structural classes.

Abstract

We propose to study transformations on graphs, and more generally structures, by looking at how the cut-rank (as introduced by Oum) of subsets is affected when going from the input structure to the output structure. We consider transformations in which the underlying sets are the same for both the input and output, and so the cut-ranks of subsets can be easily compared. The purpose of this paper is to give a characterisation of logically defined transductions that is expressed in purely structural terms, without referring to logic: transformations which decrease the cut-rank, in the asymptotic sense, are exactly those that can be defined in monadic second-order logic. This characterisation assumes that the transduction has inputs of bounded treewidth; we also show that the characterisation fails in the absence of any assumptions.
Paper Structure (27 sections, 31 theorems, 23 equations)

This paper contains 27 sections, 31 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Sub-definable implies rank-decreasing
  4. Structures and logic
  5. Cut-rank for general structures
  6. Asymptotic equivalence
  7. Equivalent variants of cut-rank
  8. Rank-decreasing implies sub-definable
  9. Inputs with equality only
  10. Inputs are trees
  11. Trees and their representation
  12. Transductions that input trees
  13. A criterion for being rank-decreasing.
  14. Transductions that input graphs of bounded treewidth
  15. Some conjectures
  16. ...and 12 more sections

Key Result

Theorem 2.5

Let $\alpha$ be a graph-to-graph transduction with inputs of bounded treewidth. Then $\alpha$ is rank-decreasing if and only if it is sub-definable.

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2: Rank-decreasing transduction
  • Definition 2.3: Definable graph-to-graph transductions
  • Definition 2.4: Sub-definable
  • Theorem 2.5
  • Definition 3.1: Cut-rank and type matrix
  • Lemma 3.2
  • Theorem 3.3
  • Corollary 3.4
  • Lemma 3.5
  • ...and 41 more