Two fresh papers from the Edinburgh theory stable:
- Lindley, Wadler & Yallop, 2008. The Arrow Calculus, (Functional Pearl) (submitted to ICFP).
- Lindley, Wadler & Yallop, 2008. Idioms are oblivious, arrows are meticulous, monads are promiscuous (submitted to MSFP)
We revisit the connection between three notions of computation: Moggi’s monads, Hughes’s arrows and
McBride and Paterson’s idioms (also called applicative functors ). We show that idioms are equivalent to
arrows that satisfy the type isomorphism A ∼> B ≅ 1 ∼> (A -> B) and that monads are equivalent to arrows
that satisfy the type isomorphism A ∼> B ≅A → (1 ∼> B). Further, idioms embed into arrows and arrows embed into monads.
The first paper introduce a reformulation of the Power/Thielecke/Paterson/McBride axiomatisation of arrows, which the authors argue is more natural, and shows that arrows generalise both monads and idioms. The second paper studies the relationships between the three formalisations in more formal depth; in particular the results about applicative functors struck me as significant.