Fibrational Induction Meets Effects

Abstract

Abstract. This paper provides several induction rules that can be used to prove properties of effectful data types. Our results are semantic in nature and build upon Hermida and Jacobs' fibrational formulation of induction for polynomial data types and its extension to all inductive data types by Ghani, Johann, and Fumex. An effectful data type µ(T F ) is built from a functor F that describes data, and a monad T that computes effects. Our main contribution is to derive induction rules that are generic over all functors F and monads T such that µ(T F ) exists. Along the way, we also derive a principle of definition by structural recursion for effectful data types that is similarly generic. Our induction rule is also generic over the kinds of properties to be proved: like the work on which we build, we work in a general fibrational setting and so can accommodate very general notions of properties, rather than just those of particular syntactic forms. We give examples exploiting the generality of our results, and show how our results specialize to those in the literature, particularly those of Filinski and Støvring.

The Coinductive Resumption Monad

“…Different programming patterns that involve resumptions were discussed by Claessen [14], Kiselyov [28], Harrison [23], and the present authors [38]. Interleaving of data and effects in the algebraic context was also studied by Filinski and Støvring [17], and Atkey et al [10].…”

Interleaving data and effects

“…A further line of future work lies in deeper investigation of the categorical properties of the category of f -and-m-algebras. In the present work, we constructed the category of f -and-m-algebras and showed that it had initial objects from first principles, while in Atkey et al's previous work (Atkey et al, 2012), this was demonstrated by constructing an adjunction between the category of f -and-m-algebras and the category of ( f • m)-algebras. An anonymous reviewer has pointed out the interesting property that the category of f -and-m-algebras is isomorphic to the category of ((FreeM f ) + m)-EilenbergMoore algebras, showing that the monad coproduct construction in Section 8 has a deeper significance.…”

“…The concept of initial f -and-m-algebras is originally due to Filinski and Støvring in the specific setting of Cpo (the category of complete partial orders and continuous functions) (Filinski & Støvring, 2007), and was subsequently extended to a general category-theoretic setting for arbitrary functors f by Atkey, Ghani, Jacobs, and Johann (Atkey et al, 2012). In this article, we aim to introduce the concept of initial f -and-m-algebras to a general functional programming audience and show how they can be used to structure and reason about functional programs in practice, without the heavy category-theoretic prerequisites of Atkey et al's work.…”

“…To solve the problems we have identified with the direct use of f -algebras, we use the concept of f -and-m-algebras, originally introduced by Filinski and Støvring (Filinski & Støvring, 2007), and generalised to arbitrary functors by Atkey, Ghani, Jacobs and Johann (Atkey et al, 2012). As the name may imply, f -and-m-algebras are simultaneously falgebras and m-algebras.…”

Relating Computational Effects by ⊤ ⊤-Lifting

“…(Fibration) 1 For any I ∈ C, Y ∈ E and f : I → pY, there exists X ∈ E above I such that f : X→Y and the following property holds: for any Z ∈ E and g : pZ…”