[Paper Review] Finitary Corecursion for the Infinitary Lambda Calculus

Stefan Milius, Thorsten Wißmann|arXiv (Cornell University)|May 28, 2015

Logic, programming, and type systems1 references4 citations

TL;DR

This paper establishes that the rational fixpoint of the functor for the infinitary lambda calculus on nominal sets consists precisely of rational lambda-trees—those with finitely many subtrees up to isomorphism—thereby enabling a finitary corecursion principle for defining operations like substitution and normal form computations on these terms.

ABSTRACT

Kurz et al. have recently shown that infinite $λ$-trees with finitely many free variables modulo $α$-equivalence form a final coalgebra for a functor on the category of nominal sets. Here we investigate the rational fixpoint of that functor. We prove that it is formed by all rational $λ$-trees, i.e. those $λ$-trees which have only finitely many subtrees (up to isomorphism). This yields a corecursion principle that allows the definition of operations such as substitution on rational $λ$-trees.

Motivation & Objective

• To characterize the rational fixpoint of the functor $ L_{eta} $ for the infinitary lambda calculus in the category of nominal sets.
• To show that this rational fixpoint corresponds exactly to rational lambda-trees modulo $\alpha$-equivalence.
• To establish a finitary corecursion principle for operations on rational lambda-terms.
• To demonstrate that coinductive definitions such as substitution and normal form computation restrict to rational trees.
• To extend the coalgebraic framework for variable binding to include finitary coinductive reasoning via the rational fixpoint.

Proposed method

• Leverages the theory of nominal sets to formalize variable binding and $\alpha$-equivalence in the lambda calculus.
• Uses the final coalgebra for $ L_{\alpha} $, established by Kurz et al., as a foundation for infinitary lambda-terms.
• Characterizes the rational fixpoint as the final locally orbit-finite $ L_{\alpha} $-coalgebra.
• Proves that the rational fixpoint is carried by the set of all rational $\lambda$-trees modulo $\alpha$-equivalence.
• Applies the finality of the rational fixpoint to derive a corecursion principle for defining operations on rational trees.
• Applies the principle to coinductive definitions of substitution and normal form computations (e.g., Böhm, Levy-Longo, Berarducci trees).

Experimental results

Research questions

• RQ1 What is the structure of the rational fixpoint of the functor $ L_{\alpha} $ on the category of nominal sets?
• RQ2 Which $\lambda$-trees are captured by the rational fixpoint of $ L_{\alpha} $, and how do they relate to finitary behavior?
• RQ3 Can coinductive definitions such as substitution and normal form computation be restricted to rational $\lambda$-trees?
• RQ4 How does the rational fixpoint provide a finitary corecursion principle in the context of infinitary $\lambda$-calculus with binding?
• RQ5 Is there a formal connection between the rational fixpoint results in nominal sets and those in presheaf categories, as seen in related work?

Key findings

• The rational fixpoint of the functor $ L_{\alpha} $ on nominal sets is precisely the set of all rational $\lambda$-trees modulo $\alpha$-equivalence.
• This characterization yields a finitary corecursion principle for defining operations on rational $\lambda$-terms.
• The coinductive definition of substitution on infinitary $\lambda$-terms, as given by Kurz et al., restricts to rational trees.
• Normal form computations such as the Böhm, Levy-Longo, and Berarducci trees can be coinductively defined on rational $\lambda$-terms.
• The rational fixpoint captures all behaviors of finitely presentable $ L_{\alpha} $-coalgebras, generalizing concepts like regular trees and rational streams.
• The results suggest a formal connection between the nominal set and presheaf-based approaches to rational and infinitary $\lambda$-terms, though this remains to be formally established.