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Attempts to split the computation, giving access to the first result. Satisfies the following laws:

msplit empty          == pure Nothing
msplit (pure a <|> m) == pure (Just (a, m))

Fair disjunction. It is possible for a logical computation to have an infinite number of potential results, for instance:

odds = pure 1 <|> fmap (+ 2) odds

Such computations can cause problems in some circumstances. Consider:

two = do x <- odds <|> pure 2
         if even x then pure x else empty

>>> observe two
...never completes...

Such a computation may never consider pure 2, and therefore even observe two will never return any results. By contrast, using interleave in place of <|> ensures fair consideration of both branches of a disjunction.

fairTwo = do x <- odds `interleave` pure 2
             if even x then pure x else empty

>>> observe fairTwo
2

Note that even with interleave this computation will never terminate after returning 2: only the first value can be safely observed, after which each odd value becomes empty (equivalent to Prolog's fail) which does not stop the evaluation but indicates there is no value to return yet.

Unlike <|>, interleave is not associative:

>>> let x = [1,2,3]; y = [4,5,6]; z = [7,8,9] :: [Int]
>>> x `interleave` y
[1,4,2,5,3,6]
>>> (x `interleave` y) `interleave` z
[1,7,4,8,2,9,5,3,6]
>>> y `interleave` z
[4,7,5,8,6,9]
>>> x `interleave` (y `interleave` z)
[1,4,2,7,3,5,8,6,9]

Fair conjunction. Similarly to the previous function, consider the distributivity law, naturally expected from MonadPlus:

(a <|> b) >>= k = (a >>= k) <|> (b >>= k)

If a >>= k can backtrack arbitrarily many times, b >>= k may never be considered. In logic statements, "backtracking" is the process of discarding the current possible solution value and returning to a previous decision point where a new value can be obtained and tried. For example:

>>> do { x <- pure 0 <|> pure 1 <|> pure 2; if even x then pure x else empty } :: [Int]
[0,2]

Here, the x value can be produced three times, where <|> represents the decision points of that production. The subsequent if statement specifies empty (fail) if x is odd, causing it to be discarded and a return to an <|> decision point to get the next x.

The statement "a >>= k can backtrack arbitrarily many times" means that the computation is resulting in empty and that a has an infinite number of <|> applications to return to. This is called a conjunctive computation because the logic for a and k must both succeed (i.e. pure a value instead of empty).

Similar to the way interleave allows both branches of a disjunctive computation, the >>- operator takes care to consider both branches of a conjunctive computation.

Consider the operation:

odds = pure 1 <|> fmap (2 +) odds

oddsPlus n = odds >>= \a -> pure (a + n)

g = do x <- (pure 0 <|> pure 1) >>= oddsPlus
       if even x then pure x else empty

>>> observeMany 3 g
...never completes...

This will never produce any value because all values produced by the do program come from the pure 1 driven operation (adding one to the sequence of odd values, resulting in the even values that are allowed by the test in the second line), but the pure 0 input to oddsPlus generates an infinite number of empty failures so the even values generated by the pure 1 alternative are never seen. Using interleave here instead of <|> does not help due to the aforementioned distributivity law.

Also note that the do notation desugars to >>= bind operations, so the following would also fail:

do a <- pure 0 <|> pure 1
   x <- oddsPlus a
   if even x then pure x else empty

The solution is to use the >>- in place of the normal monadic bind operation >>= when fairness between alternative productions is needed in a conjunction of statements (rules):

h = do x <- (pure 0 <|> pure 1) >>- oddsPlus
       if even x then pure x else empty

>>> observeMany 3 h
[2,4,6]

However, a bit of care is needed when using >>- because, unlike >>=, it is not associative. For example:

>>> let m = [2,7] :: [Int]
>>> let k x = [x, x + 1]
>>> let h x = [x, x * 2]
>>> m >>= (\x -> k x >>= h)
[2,4,3,6,7,14,8,16]
>>> (m >>= k) >>= h -- same as above
[2,4,3,6,7,14,8,16]
>>> m >>- (\x -> k x >>- h)
[2,7,3,8,4,14,6,16]
>>> (m >>- k) >>- h -- central elements are different
[2,7,4,3,14,8,6,16]

This means that the following will be productive:

(pure 0 <|> pure 1) >>-
  oddsPlus >>-
    \x -> if even x then pure x else empty

Which is equivalent to

((pure 0 <|> pure 1) >>- oddsPlus) >>-
  (\x -> if even x then pure x else empty)

But the following will not be productive:

(pure 0 <|> pure 1) >>-
  (\a -> (oddsPlus a >>- \x -> if even x then pure x else empty))

Since do notation desugaring results in the latter, the RebindableSyntax language pragma cannot easily be used either. Instead, it is recommended to carefully use explicit >>- only when needed.

Pruning. Selects one result out of many. Useful for when multiple results of a computation will be equivalent, or should be treated as such.

As an example, here's a way to determine if a number is composite (has non-trivial integer divisors, i.e. not a prime number):

choose = foldr ((<|>) . pure) empty

divisors n = do a <- choose [2..n-1]
                b <- choose [2..n-1]
                guard (a * b == n)
                pure (a, b)

composite_ v = do _ <- divisors v
                  pure "Composite"

While this works as intended, it actually does too much work:

>>> observeAll (composite_ 20)
["Composite", "Composite", "Composite", "Composite"]

Because there are multiple divisors of 20, and they can also occur in either order:

>>> observeAll (divisors 20)
[(2,10), (4,5), (5,4), (10,2)]

Clearly one could just use observe here to get the first non-prime result, but if the call to composite is in the middle of other logic code then use once instead.

composite v = do _ <- once (divisors v)
                 pure "Composite"

>>> observeAll (composite 20)
["Composite"]

Inverts a logic computation. If m succeeds with at least one value, lnot m fails. If m fails, then lnot m succeeds with the value ().

For example, evaluating if a number is prime can be based on the failure to find divisors of a number:

choose = foldr ((<|>) . pure) empty

divisors n = do d <- choose [2..n-1]
                guard (n `rem` d == 0)
                pure d

prime v = do _ <- lnot (divisors v)
             pure True

>>> observeAll (prime 20)
[]
>>> observeAll (prime 19)
[True]

Here if divisors never succeeds, then the lnot will succeed and the number will be declared as prime.

Logical conditional. The equivalent of Prolog's soft-cut. If its first argument succeeds at all, then the results will be fed into the success branch. Otherwise, the failure branch is taken. The failure branch is never considered if the first argument has any successes. The ifte function satisfies the following laws:

ifte (pure a) th el       == th a
ifte empty th el          == el
ifte (pure a <|> m) th el == th a <|> (m >>= th)

For example, the previous prime function returned nothing if the number was not prime, but if it should return False instead, the following can be used:

choose = foldr ((<|>) . pure) empty

divisors n = do d <- choose [2..n-1]
                guard (n `rem` d == 0)
                pure d

prime v = once (ifte (divisors v)
                  (const (pure True))
                  (pure False))

>>> observeAll (prime 20)
[False]
>>> observeAll (prime 19)
[True]

Notice that this cannot be done with a simple if-then-else because divisors either generates values or it does not, so there's no "false" condition to check with a simple if statement.