Guard Your Daggers and Traces: On The Equational Properties of Guarded (Co-)recursion

TL;DR

This paper explores the equational properties of guarded (co-)recursion, generalizing iteration theories and establishing a correspondence between guarded trace and fixpoint operators across various models.

Contribution

It introduces axioms for guarded fixpoint operators, generalizes Conway axioms to guarded recursion, and establishes a one-to-one correspondence between guarded trace and fixpoint operators.

Findings

Guarded fixpoint axioms generalize classical iteration theories.

A unique dagger operation satisfies all Conway axioms in guarded settings.

Guarded trace and fixpoint operators are categorically equivalent.

Abstract

Motivated by the recent interest in models of guarded (co-)recursion we study its equational properties. We formulate axioms for guarded fixpoint operators generalizing the axioms of iteration theories of Bloom and Esik. Models of these axioms include both standard (e.g., cpo-based) models of iteration theories and models of guarded recursion such as complete metric spaces or the topos of trees studied by Birkedal et al. We show that the standard result on the satisfaction of all Conway axioms by a unique dagger operation generalizes to the guarded setting. We also introduce the notion of guarded trace operator on a category, and we prove that guarded trace and guarded fixpoint operators are in one-to-one correspondence. Our results are intended as first steps leading to the description of classifying theories for guarded recursion and hence completeness results involving our axioms of guarded fixpoint operators in future work.