Alternative exercise

Haskell Programming from first principles - aka haskellbook "misses" a simple, yet hopefully insightful exercise about Alternative (introduced in Parser Combinators chapter). I have discovered this one, while working on a real world project:

Exercise Write a function with following signature:

count' :: Alternative f
       => Int  -- ^ minimum count
       -> Int  -- ^ maximum count
       -> f a
       -> f [a]

f could be a regular expression RE from regex-applicative package, in that case count' n m re should work as (re){n,m} in a traditional regexp syntax. In other words accept n to m occurrences of re.

lambda> "a" =~ count' 2 3 (sym 'a')
Nothing

lambda> "aa" =~ count' 2 3 (sym 'a')
Just "aa"

lambda> "aaa" =~ count' 2 3 (sym 'a')
Just "aaa"

lambda> "aaaa" =~ count' 2 3 (sym 'a')
Nothing

lambda> "ab" =~ count' 2 3 (sym 'a')
Nothing

## Notes, hints and follow-ups

IF the exercise feels to hard to crack directly, there are few notes, hints and also few follow-ups.

### Negative numbers

If n < m, e.g. count' 2 1, you may return empty, or accept exactly 2 values. FWIW: JavaScript throws an exception: Invalid regular expression: /^a{3,2}$/: numbers out of order in {}, let's not do that.

Also in the following sub-exercises, treat negative numbers properly: write

foo n | n < 0 = ... -- not n == 0, or foo 0
      | otherwise = ... foo (n - 1)

This will save you from the infinite recursion when negative numbers are passed in.

### some and many

The definition of many and some in base library are slightly complicated:

-- | One or more.
some :: f a -> f [a]
some v = some_v
  where
    many_v = some_v <|> pure []
    some_v = (fmap (:) v) <*> many_v

-- | Zero or more.
many :: f a -> f [a]
many v = many_v
  where
    many_v = some_v <|> pure []
    some_v = (fmap (:) v) <*> many_v

Write many as a one-liner, using <$>, <*> and <|>.

Note: <$> and <*> bind tighter than <|>, so you can write expressions like fun <$> with <*> parsers <|> foo <$> bar <*> quux, which is grouped as (fun <$> with <*> parsers) <|> foo <$> bar <*> quux).

### Exact count

A small step towards the goal is to write simpler function first:

count :: Alternative f
      => Int  -- ^ exact count
      -> f a
      -> f [a]

That should accept the exact amount of parses. count n re = count' n n re ≈ (re){n} in traditional regex notation.

lambda> "aaa" =~ count 3 (sym 'a')
Just "aaa"

### upto

Another variant with the same type-signature (!):

upto :: Alternative f
     => Int  -- ^ up to this amount
     -> f a
     -> f [a]

This should accept upto some amount of parses. upto n re = count' 0 n re ≈ (re){n} in traditional regex notation.

### Divide and conquer

Now, when you have defined count and upto, you can define count' as simply as:

count' n m p = (++) <$> count n p <*> upto (m - n) p

GHC is smart, but not super-smart. Let's help it. Inline definition of count, removing the use of (++) from above definition.

### Order of alternatives

Virtually every parser combinator library is left-biased, in other word the left hand side of <|> is preferred. Explain what's the behavioral difference between

pure [] <|> some p
some p <|> pure []

Analyze your count', count and upto definitions, and try examples above with your backtracking parser-combinator library of choice (e.g. trifecta). Do they work as expected?

### Counting other Alternatives

There are also other Alternatives than parsers. For example [] and STM. Explain to your colleague what count' does for [] and STM. If there are other interesting Alternatives, where count' does something neat, please tell me! (e.g. via Twitter).

## Solution

Fortunately, parser-combinators package by Mark Karpov provides count and count', so you could use them in your code, and check the source for one way of solving this exercise.

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