Paramorphism

A paramorphism (from Greek παρά, meaning "close together") is an extension of the concept of catamorphism first introduced by Lambert Meertens [1] to deal with a form which “eats its argument and keeps it too”,[2][3] as exemplified by the factorial function. Its categorical dual is the apomorphism.

It is a more convenient version of catamorphism in that it gives the combining step function immediate access not only to the result value recursively computed from each recursive subobject, but the original subobject itself as well.

Example Haskell implementation, for lists:

cata :: (a ->       b  -> b) -> b -> [a] ->  b
para :: (a -> ([a], b) -> b) -> b -> [a] ->  b
ana  :: (b -> (a,            b))  ->  b  -> [a]
apo  :: (b -> (a, Either [a] b))  ->  b  -> [a]

cata f b (a:as) = f a     (cata f b as)
cata _ b []     = b

para f b (a:as) = f a (as, para f b as)
para _ b []     = b

ana  u b = case u b of (a,       b') -> a : ana u b'

apo  u b = case u b of (a, Right b') -> a : apo u b'
                       (a, Left  as) -> a : as

See also

References

  1. ↑ Meertens, Lambert (1992). "Paramorphisms". CiteSeerX 10.1.1.19.4825 Freely accessible.
  2. ↑ Philip Wadler.Views: A way for pattern matching to cohabit with data abstraction. Technical Report 34, Programming Methodology Group, University of Göteburg and Chalmers University of Technology, March 1987.
  3. ↑ Meijer, Erik; Fokkinga, Maarten; Paterson, Ross (1991). "Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire". CiteSeerX 10.1.1.41.125 Freely accessible.

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