New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic

Crole, Roy L.; Pitts, Andrew M.

doi:10.1016/0890-5401(92)90018-b

Cited by 46 publications

(56 citation statements)

References 7 publications

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“…Filinski and Støvring do, however, give a principle of proof by rigid induction for such datatypes that is a variant of those of both Lehmann and Smyth (Lehman & Smyth, 1981) and Crole and Pitts (Crole & Pitts, 1992). More generally, their development supports the same kind of reasoning principles, again in the specific category and for the specific functors with which they work, that we show arbitrary initial f -and-m-algebras to support.…”

Section: Related Workmentioning

confidence: 89%

Interleaving data and effects

Atkey

Johann

2015

J. Funct. Prog.

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The study of programming with and reasoning about inductive datatypes such as lists and trees has benefited from the simple categorical principle of initial algebras. In initial algebra semantics, each inductive datatype is represented by an initial f -algebra for an appropriate functor f . The initial algebra principle then supports the straightforward derivation of definitional principles and proof principles for these datatypes. This technique has been expanded to a whole methodology of structured functional programming, often called origami programming.In this article we show how to extend initial algebra semantics from pure inductive datatypes to inductive datatypes interleaved with computational effects. Inductive datatypes interleaved with effects arise naturally in many computational settings. For example, incrementally reading characters from a file generates a list of characters interleaved with input/output actions, and lazily constructed infinite values can be represented by pure data interleaved with the possibility of non-terminating computation. Straightforward application of initial algebra techniques to effectful datatypes leads either to unsound conclusions if we ignore the possibility of effects, or to unnecessarily complicated reasoning because the pure and effectful concerns must be considered simultaneously. We show how pure and effectful concerns can be separated using the abstraction of initial f -and-m-algebras, where the functor f describes the pure part of a datatype and the monad m describes the interleaved effects. Because initial f -and-m-algebras are the analogue for the effectful setting of initial f -algebras, they support the extension of the standard definitional and proof principles to the effectful setting.Initial f -and-m-algebras are originally due to Filinski and Støvring, who studied them in the category Cpo. They were subsequently generalised to arbitrary categories by Atkey, Ghani, Jacobs, and Johann in a FoSSaCS 2012 paper. In this article we aim to introduce the general concept of initial f -and-m-algebras to a general functional programming audience.

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Section: Related Workmentioning

confidence: 89%

Interleaving data and effects

Atkey

Johann

2015

J. Funct. Prog.

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“…Moggi introduced equational theories for let-categories in [13]. The equational theories corresponding to categories with fixpoint objects were introduced in [4].…”

Section: Introductionmentioning

confidence: 99%

On fixpoint objects and gluing constructions

Crole

1996

Appl Categor Struct

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This article has two parts: in the first part, we present some general results about fixpoint objects. The minimal categorical structure required to model soundly the equational type theory which combines higher order recursion and computation types (introduced by Crole and Pitts (1992)) is shown to be precisely a let-category possessing a fixpoint object. Functional completeness for such categories is developed. We also prove that categories with fixpoint operators do not necessarily have a fixpoint object.In the second part, we extend Freyd's gluing construction for cartesian closed categories to cartesian closed let-categories, and observe that this extension does not obviously apply to categories possessing fixpoint objects. We solve this problem by giving a new gluing construction for a limited class of categories with fixpoint objects; this is the main result of the paper. We use this category-theoretic construction to prove a type-theoretic conservative extension result.Mathematics Subject Classification (1991). 18C10.

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“…Crole and Pitts [3] use this rule to give a fixpoint induction rule for effectful computations, generalising the usual notion of Scott induction. Filinski and Støvring's induction principle, which we have generalised in this paper, extends these rules to handle the interleaving of data and effects.…”

Section: Conclusion Related Work and Future Workmentioning

confidence: 99%

Fibrational Induction Meets Effects

Atkey

Ghani

Jacobs

et al. 2012

Foundations of Software Science and Computational Structures

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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.

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New foundations for fixpoint computations: FIX-hyperdoctrines and the FIX-logic

Cited by 46 publications

References 7 publications

Interleaving data and effects

Interleaving data and effects

On fixpoint objects and gluing constructions

Fibrational Induction Meets Effects

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