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The Power-Set Construction for Tree Algebras

Achim Blumensath

TL;DR

This work investigates how the upwards-closed power-set monad $\mathbb{U}$ interacts with tree monads used in algebraic language theory for infinite trees. It proves a unique distributive law $\mathbb{M} \mathbb{U} \Rightarrow \mathbb{U} \mathbb{M}$ exists for linear polynomial monads (in particular, linear trees) but not for the non-linear tree monad $\mathbb{T}^\times$, establishing a sharp dividing line between these cases. To address non-linear trees, the authors develop a partial lift of $\mathbb{U}$ to free $\mathbb{T}^\times$-algebras via unravelling, enabling applications to substitutions and regular expressions for infinite trees. They further extend the framework to infinite sorts and provide a detailed treatment of variable actions, unravellings, and related algebraic structures, including a regular-expression calculus grounded in parity-automata arguments. The results illuminate when power-set operations lift to tree languages and suggest directions for broader monadic generalizations and quotients.

Abstract

We study power-set operations on classes of trees and tree algebras. Our main result consists of a distributive law between the tree monad and the upwards-closed power-set monad, in the case where all trees are assumed to be linear. For non-linear ones, we prove that such a distributive law does not exist.

The Power-Set Construction for Tree Algebras

TL;DR

This work investigates how the upwards-closed power-set monad interacts with tree monads used in algebraic language theory for infinite trees. It proves a unique distributive law exists for linear polynomial monads (in particular, linear trees) but not for the non-linear tree monad , establishing a sharp dividing line between these cases. To address non-linear trees, the authors develop a partial lift of to free -algebras via unravelling, enabling applications to substitutions and regular expressions for infinite trees. They further extend the framework to infinite sorts and provide a detailed treatment of variable actions, unravellings, and related algebraic structures, including a regular-expression calculus grounded in parity-automata arguments. The results illuminate when power-set operations lift to tree languages and suggest directions for broader monadic generalizations and quotients.

Abstract

We study power-set operations on classes of trees and tree algebras. Our main result consists of a distributive law between the tree monad and the upwards-closed power-set monad, in the case where all trees are assumed to be linear. For non-linear ones, we prove that such a distributive law does not exist.
Paper Structure (11 sections, 39 theorems, 250 equations, 1 figure)

This paper contains 11 sections, 39 theorems, 250 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Monads for trees
  3. Polynomial functors
  4. Non-linear trees
  5. Infinite sorts
  6. The action on the variables
  7. Graphs and unravellings
  8. A partial distributive law
  9. Substitutions
  10. Regular expressions for infinite trees
  11. Conclusion

Key Result

Proposition 2.2

$\langle\mathbb R,\mathrm{flat},\mathrm{sing}\rangle$ forms a monad on $\mathsf{Pos}^\Xi$.

Figures (1)

  • Figure 1: The flattening operation: $g$ and $\mathrm{flat}(g)$ (edge directions not shown to reduce noise)

Theorems & Definitions (113)

  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • Corollary 2.6
  • Definition 2.7
  • Proposition 2.8
  • Example 2.9
  • Definition 2.10
  • ...and 103 more