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Profinite trees, through Lawvere theories and the lambda-calculus

Vincent Moreau

TL;DR

The paper develops a topological, algebraic framework to study regular languages of finite trees by extending profinite methods from words to trees via clones. Using codensity monads and Lawvere-theory semantics, it defines a profinite completion of clones and shows it generalizes both the ultrafilter monad and the profinite completion of monoids. When applied to free clones on a ranked alphabet, this yields profinite trees, and the authors prove a precise isomorphism between profinite trees and a fragment of the profinite λ-calculus, establishing a solid bridge between tree automata/topology and higher-order λ-theory. The results rely on parametricity arguments to relate profinite trees to profinite λ-terms, yielding a robust, topological account of regular tree languages and inviting future work on pseudovarieties and broader algebraic structures. Overall, the work provides a cohesive synthesis of clone-theoretic, category-theoretic, and λ-calculus perspectives to advance profinite methods in higher-order language theory.

Abstract

The starting point of algebraic language theory is that regular languages of finite words are exactly those recognized by finite monoids. This finiteness condition gives rise to a topological space whose points, called profinite words, encode the limiting behavior of words with respect to finite monoids. In this work, we move from words and monoids to trees and clones, the algebraic structures underlying deterministic bottom-up tree automata. Using the categorical notion of codensity monad, we introduce a profinite completion for clones. We prove that this construction on clones simultaneously generalizes the ultrafilter monad on sets and the profinite completion of monoids. When applied to free clones on a ranked alphabet, the profinite completion of clones yields a notion of profinite tree, providing a topological approach to regular languages of finite trees. We prove that these profinite trees coincide with a well-identified fragment of the profinite lambda-calculus.

Profinite trees, through Lawvere theories and the lambda-calculus

TL;DR

The paper develops a topological, algebraic framework to study regular languages of finite trees by extending profinite methods from words to trees via clones. Using codensity monads and Lawvere-theory semantics, it defines a profinite completion of clones and shows it generalizes both the ultrafilter monad and the profinite completion of monoids. When applied to free clones on a ranked alphabet, this yields profinite trees, and the authors prove a precise isomorphism between profinite trees and a fragment of the profinite λ-calculus, establishing a solid bridge between tree automata/topology and higher-order λ-theory. The results rely on parametricity arguments to relate profinite trees to profinite λ-terms, yielding a robust, topological account of regular tree languages and inviting future work on pseudovarieties and broader algebraic structures. Overall, the work provides a cohesive synthesis of clone-theoretic, category-theoretic, and λ-calculus perspectives to advance profinite methods in higher-order language theory.

Abstract

The starting point of algebraic language theory is that regular languages of finite words are exactly those recognized by finite monoids. This finiteness condition gives rise to a topological space whose points, called profinite words, encode the limiting behavior of words with respect to finite monoids. In this work, we move from words and monoids to trees and clones, the algebraic structures underlying deterministic bottom-up tree automata. Using the categorical notion of codensity monad, we introduce a profinite completion for clones. We prove that this construction on clones simultaneously generalizes the ultrafilter monad on sets and the profinite completion of monoids. When applied to free clones on a ranked alphabet, the profinite completion of clones yields a notion of profinite tree, providing a topological approach to regular languages of finite trees. We prove that these profinite trees coincide with a well-identified fragment of the profinite lambda-calculus.
Paper Structure (11 sections, 15 theorems, 117 equations)

This paper contains 11 sections, 15 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Plan of the article
  3. Related work
  4. Notations
  5. Abstract clones, tree automata, and monoids and sets
  6. A functorial viewpoint on clones and related structures
  7. Profinite completions of clones
  8. Relating the profinite completions
  9. Profinite trees and parametricity
  10. Profinite trees and the profinite λ-calculus
  11. Conclusion

Key Result

Proposition 7

We have an adjunction \begin{tikzcd}[ampersand replacement=\&] \Mon \&\& \Clone \arrow[""{name=0, anchor=center, inner sep=0}, "\Act", shift left=2, from=1-1, to=1-3] \arrow[""{name=1, anchor=center, inner sep=0}, "{C_1\,\mapsfrom\,C}", shift left=2, from=1-3, to=

Theorems & Definitions (50)

  • Definition 1
  • Remark 2
  • Definition 3
  • Definition 4
  • Remark 5
  • Definition 6
  • Proposition 7
  • Definition 8
  • Proposition 9
  • Definition 10
  • ...and 40 more