Paper: Problems with Fixpoints of Polynomials of Polynomials (at LICS 2026)

Authors: Cécilia Pradic Ian Price

Open access: https://doi.org/10.4230/LIPIcs.LICS.2026.77

Abstract

Motivated by applications in computable analysis, we study fixpoints of certain endofunctors over categories of containers. More specifically, we focus on fibred endofunctors over the fibrewise opposite of the codomain fibration that can be themselves be represented by families of polynomial endofunctors. In this setting, we show how to compute initial algebras, terminal coalgebras and another kind of fixpoint ζ. We then explore a number of examples of derived operators inspired by Weihrauch complexity and the usual construction of the free polynomial monad. We introduce ζ-expressions as the syntax of μ-bicomplete categories, extended with ζ-binders and parallel products, which thus have a natural denotation in containers. By interpreting certain ζ-expressions in a category of type-2 computable maps, we are able to capture a number of meaningful Weihrauch degrees, ranging from closed choice on {0,1} to determinacy of infinite parity games, via an "answerable part" operator.

BibTeX

  @InProceedings{PradicPrice-ProblemswithFixpoin,
    author = 	 {Cécilia Pradic and Ian Price},
    title = 	 {Problems with Fixpoints of Polynomials of Polynomials},
    booktitle =  {Proceedings of the Forty-First Annual Symposium on Logic in Computer Science (LICS 2026)},
    year =	 {2026},
    month =	 {July}, 
    pages =      {77:1--77:27},
    location =   {Lisbon, Portugal}, 
    publisher =	 {Schloss Dagstuhl -- Leibniz-Zentrum für Informatik},
    doi =        {10.4230/LIPIcs.LICS.2026.77}
  }