Complete Elgot Monads and Coalgebraic Resumptions

Goncharov, Sergey; Milius, Stefan; Rauch, Christoph

doi:10.1016/j.entcs.2016.09.036

Cited by 6 publications

(4 citation statements)

References 28 publications

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“…There is an concept of complete Elgot algebra [3] complementing complete Elgot monads. It has been shown that the algebras of complete Elgot monads are complete Elgot algebras satisfying additional conditions [18]; the precise relationship between complete Elgot monads and complete Elgot algebras remains to be determined, possibly using our results on iteration-congruent retracts of monads with iteration [21].…”

Section: Discussionmentioning

confidence: 96%

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Unguarded Recursion on Coinductive Resumptions

Goncharov

Rauch

Schröder

2015

Electronic Notes in Theoretical Computer Science

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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...

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Section: Discussionmentioning

confidence: 96%

“…It has recently been shown [16,18] that dinaturality, previously standardly included in axiomatizations of iteration [12], is in fact derivable from the other axioms in Definition 5.1. We record this for future reference:…”

Section: Complete Elgot Monadsmentioning

confidence: 99%

Unguarded Recursion on Coinductive Resumptions

Goncharov

Rauch

Schröder

2015

Electronic Notes in Theoretical Computer Science

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“…If C " Set, then DX " pX ˆN `1q does turn out to be Elgot [15] (but applying Theorem 5.15 to D qua Elgot monad yields a different p--q ν -algebra structure than the initial one), and L is, in this case, a retract of D in MonpCq ν . The situation is more intricate in categories with a nonclassical internal logic, for which D is mainly intended.…”

Section: Complete Elgot Monads and Iteration Congruencesmentioning

confidence: 99%

Guarded and Unguarded Iteration for Generalized Processes

Goncharov¹,

Schröder²,

Rauch³

et al. 2019

Logical Methods in Computer Science ; Volume 15

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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.

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“…Like P, this monad is order-enriched and thus supports a total iteration operator via least fixpoints (see e.g. [18]). From this, a guarded iteration operator is obtained by restricting to the guarded category with f : X Ñ 2 PpAct ‹ ˆpY `Zqq iff f factors through the map…”

Section: (Complete Finite Traces)mentioning

confidence: 99%

A Metalanguage for Guarded Iteration

Goncharov

Rauch

Schröder

2018

Theoretical Aspects of Computing – ICTAC 2018

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Notions of guardedness serve to delineate admissible recursive definitions in various settings in a compositional manner. In recent work, we have introduced an axiomatic notion of guardedness in symmetric monoidal categories, which serves as a unifying framework for various examples from program semantics, process algebra, and beyond. In the present paper, we propose a generic metalanguage for guarded iteration based on combining this notion with the fine-grain callby-value paradigm, which we intend as a unifying programming language for guarded and unguarded iteration in the presence of computational effects. We give a generic (categorical) semantics of this language over a suitable class of strong monads supporting guarded iteration, and show it to be in touch with the standard operational behaviour of iteration by giving a concrete big-step operational semantics for a certain specific instance of the metalanguage and establishing soundness and (computational) adequacy for this case.

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Complete Elgot Monads and Coalgebraic Resumptions

Cited by 6 publications

References 28 publications

Unguarded Recursion on Coinductive Resumptions

Unguarded Recursion on Coinductive Resumptions

Guarded and Unguarded Iteration for Generalized Processes

A Metalanguage for Guarded Iteration

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