Guarded and Unguarded Iteration for Generalized Processes

TL;DR

This paper introduces an abstract framework for guarded and unguarded iteration in monads, showing that unguarded iteration models can be derived from guarded ones via quotienting, unifying different approaches to recursive computation.

Contribution

It formalizes an abstract notion of guardedness in monads and characterizes Elgot monads as quotients of guarded iterative monads, unifying guarded and unguarded iteration models.

Findings

Elgot monads are exactly the iteration-congruent retracts of guarded iterative monads.

Models of unguarded iteration can be obtained by quotienting guarded models.

Provides a unified categorical framework for recursive definitions in computation.

Abstract

Models of iterated computation, such as (completely) iterative monads, often depend on a notion of guardedness, which guarantees unique solvability of recursive equations and requires roughly that recursive calls happen only under certain guarding operations. On the other hand, many models of iteration do admit unguarded iteration. Solutions are then no longer unique, and in general not even determined as least or greatest fixpoints, being instead governed by quasi-equational axioms. Monads that support unguarded iteration in this sense are called (complete) Elgot monads. Here, we propose to equip (Kleisli categories of) monads with an abstract notion of guardedness and then require solvability of abstractly guarded recursive equations; examples of such abstractly guarded pre-iterative monads include both iterative monads and Elgot monads, the latter by deeming any recursive definition to be abstractly guarded. Our main result is then that Elgot monads are precisely the iteration-congruent retracts of abstractly guarded iterative monads, the latter being defined as admitting unique solutions of abstractly guarded recursive equations; in other words, models of unguarded iteration come about by quotienting models of guarded iteration.