Applicative functors other than monads and ZipList?

Question

Two well-known examples of applicatives are monads and ziplists. Are there any other examples?

Answer 1

Swierstra and Duponcheel defined an efficient style of parser, this parser was pretty much the early poster child for Arrows, but it doesn't need anything from Arrow that it can't get from Applicative. However, Applicatives hadn't been coined at the time.

Effectively it computes 'FIRST' sets for an LL(1) parser and uses that to do smarter branch selection. However, you can't compute these sets when you work monadically.

This is perhaps not a terribly fair example, because the Swierstra/Duponcheel parser admits a blending of static and dynamic parsers, and it is only the static parser that is limited to being applicative.

With observable sharing you can carry their parser design farther and also compute 'FOLLOW' sets (as long as you are careful to not build an infinite context free grammar.). This yields nice asymptotic guarantees for parsing context free grammars, which are not available to you when parsing using monadic (context sensitive) parser designs.

Also interesting perhaps, is to consider the structures for which the <*> of applicative is available, but not the pure. Many comonads admit a (<*>)-like definition, which respects the structure of the comonad, but don't have a reasonable definition for 'pure'. My semigroupoids package and the myriad packages that depend upon it explore this idea further.