Is it possible to express chainl1 using applicative?

Question

Is it possible to express the chainl1 combinator from Parsec not using the Monad instance defined by parsec?

chainl1 p op =
  do x <- p
     rest x
  where
    rest x = do f <- op
                y <- p
                rest (f x y)
          <|> return x

Answer 1

Yes, it is:

chainl1 p op = foldl (flip ($)) <$> p <*> many (flip <$> op <*> p)

The idea is that you have to parse p (op p)* and evaluate it as (...(((p) op p) op p)...).

It might help to expand the definition a bit:

chainl1 p op = foldl (\x f -> f x) <$> p <*> many ((\f y -> flip f y) <$> op <*> p)

As the pairs of op and p are parsed, the results are applied immediately, but because p is the right operand of op, it needs a flip.

So, the result type of many (flip <$> op <*> p) is f [a -> a]. This list of functions is then applied from left to right on an initial value of p by foldl.