Free theorems involving type constructor classes

Voigtländer, Janis

doi:10.1145/1596550.1596577

“…The naturality law specifies that malias is a natural transformation. Although we do not give a formal proof, we argue that the law follows from the parametricity of the malias type signature and can be obtained using the method described by Voigtländer [25].…”

Section: The Meaning Of Malias Lawsmentioning

confidence: 89%

Evaluation strategies for monadic computations

Petříček

¹

2012

Electron. Proc. Theor. Comput. Sci.

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Monads have become a powerful tool for structuring effectful computations in functional programming, because they make the order of effects explicit. When translating pure code to a monadic version, we need to specify evaluation order explicitly. Two standard translations give call-by-value and call-by-name semantics. The resulting programs have different structure and types, which makes revisiting the choice difficult. In this paper, we translate pure code to monadic using an additional operation malias that abstracts out the evaluation strategy. The malias operation is based on computational comonads; we use a categorical framework to specify the laws that are required to hold about the operation. For any monad, we show implementations of malias that give call-by-value and call-by-name semantics. Although we do not give call-by-need semantics for all monads, we show how to turn certain monads into an extended monad with call-by-need semantics, which partly answers an open question. Moreover, using our unified translation, it is possible to change the evaluation strategy of functional code translated to the monadic form without changing its structure or types

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“…• We believe that for a large range of datatypes T (including all regular datatypes), the naturality properties of traverse (but not 'naturality' in the datatype) actually hold for all polymorphic functions of type Applicative m ⇒ (a → m b) → T a → m (T b); they should be free theorems (Wadler 1989;Voigtländer 2009). …”

Section: Discussionmentioning

confidence: 99%

Understanding idiomatic traversals backwards and forwards

Bird

¹

,

Gibbons

²

,

Mehner

³

et al. 2013

Proceedings of the 2013 ACM SIGPLAN Symposium on Haskell

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We present new ways of reasoning about a particular class of effectful Haskell programs, namely those expressed as idiomatic traversals. Starting out with a specific problem about labelling and unlabelling binary trees, we extract a general inversion law, applicable to any monad, relating a traversal over the elements of an arbitrary traversable type to a traversal that goes in the opposite direction. This law can be invoked to show that, in a suitable sense, unlabelling is the inverse of labelling. The inversion law, as well as a number of other properties of idiomatic traversals, is a corollary of a more general theorem characterising traversable functors as finitary containers: an arbitrary traversable object can be decomposed uniquely into shape and contents, and traversal be understood in terms of those. Proof of the theorem involves the properties of traversal in a special idiom related to the free applicative functor.Life can only be understood backwards; but it must be lived forwards.-Søren Kierkegaard

show abstract

“…• We believe that for a large range of datatypes T (including all regular datatypes), the naturality properties of traverse (but not 'naturality' in the datatype) actually hold for all polymorphic functions of type Applicative m ⇒ (a → m b) → T a → m (T b); they should be free theorems (Wadler 1989;Voigtländer 2009). • We conjecture the maybe somewhat surprising fact that, in Set, the monadic variant of traverse's type is no more accommodating than the idiomatic one.…”

Section: Discussionmentioning

confidence: 99%

Understanding idiomatic traversals backwards and forwards

Bird

¹

,

Gibbons

²

,

Mehner

³

et al. 2013

Self Cite

View full text Add to dashboard Cite

We present new ways of reasoning about a particular class of effectful Haskell programs, namely those expressed as idiomatic traversals. Starting out with a specific problem about labelling and unlabelling binary trees, we extract a general inversion law, applicable to any monad, relating a traversal over the elements of an arbitrary traversable type to a traversal that goes in the opposite direction. This law can be invoked to show that, in a suitable sense, unlabelling is the inverse of labelling. The inversion law, as well as a number of other properties of idiomatic traversals, is a corollary of a more general theorem characterising traversable functors as finitary containers: an arbitrary traversable object can be decomposed uniquely into shape and contents, and traversal be understood in terms of those. Proof of the theorem involves the properties of traversal in a special idiom related to the free applicative functor.

show abstract

Free theorems involving type constructor classes

Cited by 19 publications

References 25 publications

Evaluation strategies for monadic computations

Evaluation strategies for monadic computations

Understanding idiomatic traversals backwards and forwards

Understanding idiomatic traversals backwards and forwards

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