Unguarded Recursion on Coinductive Resumptions

Abstract

We study a model of side-effecting processes obtained by starting from a monad modelling base effects and adjoining free operations using a cofree coalgebra construction; one thus arrives at what one may think of as types of non-wellfounded side-effecting trees, generalizing the infinite resumption monad. Correspondingly, the arising monad transformer has been termed the coinductive generalized resumption transformer. Monads of this kind have received some attention in the recent literature; in particular, it has been shown that they admit guarded iteration. Here, we show that they also admit unguarded iteration, i.e. form complete Elgot monads, provided that the underlying base effect supports unguarded iteration. Moreover, we provide a universal characterization of the coinductive resumption monad transformer in terms of coproducts of complete Elgot monads.T Σ X = νγ. T (X + Σγ) for each object X, assuming enough structure on T , Σ, and the base category. Inhabitants of T Σ X are understood as (possibly) nonterminating processes that proceed in steps, where each step produces side-effects specified by T (e.g. writing to shared global memory, nondeterminism) and performs communication actions specified by Σ. E.g. in the simplest case, Σ is of the form a × (−) b , which may be understood as reading inputs of type b and writing outputs of type a. CC Creative Commons2 GONCHAROV, RAUCH, SCHRÖDER, AND JAKOB UNGUARDED RECURSION ON COINDUCTIVE RESUMPTIONS 3Section 5, and recall the coinductive generalized resumption transformer in Section 6, showing in particular that it preserves strength. Sections 7 and 8 contain our main results, showing that the coinductive generalized resumption transformer preserves complete Elgotness and can be seen as freely extending complete Elgot monads with communication actions. We discuss related work in Section 9, and conclude in Section 10. Example: Unrestricted While LoopsWe proceed to discuss examples, aimed mainly at illustrating the benefits of not being restricted to guarded equations in recursive definitions thanks to complete Elgotness of coinductive resumption monads (Theorem 7.1). We work with the intuitive understanding of monads, T Σ , guardedness, and complete iterativity provided in the introduction, and briefly explain the requisite categorical notation regarding strong monads and distributive categories along the way, deferring a more formal treatment to Sections 4 and 6.Our first example is a simple while-language with actions proposed by Rutten, given by the grammar P, Q : and, following Piróg and Gibbons [32], interpreted in the Kleisli category of a monad M. Here, A ranges over atomic actions interpreted as Kleisli morphisms A : n → M n for some fixed object n, and b over atomic predicates, interpreted as Kleisli morphisms b : n → M (1 + 1) (where we read the left-hand summand as 'false' and the right-hand one as 'true', and 1 denotes the terminal object). We say that A is of output type if A : n → M n has the form A = (M fst) τ id n , p for some p : n → M 1, where fs...

A Metalanguage for Guarded Iteration

“…This monad is therefore both guarded iterative in the former sense, but only guarded Elgot in the latter sense, for under total iteration, the fixpoints f : are no longer unique. This setup is analysed more generally in detail in previous work [19,23].…”

“…Our denotational semantics is generic, and is parametrized by two orthogonal features: a notion of computation, given in terms of a strong monad, and a notion of axiomatic guardedness, which serves to support guarded iteration. The notion of guardedness can range from vacuous guardedness (inducing trivial iteration, which unfolds at most once) to total guardedness (supported by monads equipped with a total iteration operator, specifically Elgot monads); the latter case covers classical denotational semantics, since any monad in a category of domains is Elgot [22].…”

Complete Elgot Monads and Coalgebraic Resumptions

“…It is shown in [14] that an ω-continuous monad is a complete Elgot monad with e † calculated as the least fixed point of the map f → [η, f ] ⋄ e. This yields the powerset monad P, the Maybe-monad (--+1), or the nondeterministic state monad P(--×S) S as examples of complete Elgot monads on Set. The lifting monad (--) ⊥ is a complete Elgot monad on the category of complete partial orders without bottom.…”

“…The lifting monad (--) ⊥ is a complete Elgot monad on the category of complete partial orders without bottom. [14] that whenever the functor T Σ defined by (⋆) exists, it yields the free complete Elgot monad on Σ (note that the original T is the free complete Elgot monads on Σ being the constant functor on the initial object of C). On Set (more generally, on any hyperextensive category [3]) the initial complete Elgot monad T is the Maybe-monad --+1.…”

“…P ω is not a complete Elgot monad, but the countable powerset monad P ω1 is). The central result of the recent work [14] is that whenever T is a complete Elgot monad then so is the transformed monad (⋆). In particular, this allows for solving recursive equations over processes (in the sense of Example 1) whenever recursive equations over T are solvable.…”