Hylomorphism (computer science)
In computer science, and in particular functional programming, a hylomorphism is a recursive function, corresponding to the composition of an anamorphism (which first builds a set of results; also known as 'unfolding') followed by a catamorphism (which then folds these results into a final return value). Fusion of these two recursive computations into a single recursive pattern then avoids building the intermediate data structure. This is an example of deforestation, a program optimization strategy. A related type of function is a metamorphism, which is a catamorphism followed by an anamorphism.
A hylomorphism can be defined in terms of its separate anamorphic and catamorphic parts.
The catamorphic part can be defined as a combination of an initial value for the fold and a binary operator used to perform the fold.
Thus a hylomorphism
may be defined (assuming appropriate definitions of & ).
An abbreviated notation for the above hylomorphism is .
Hylomorphisms in practice
Lists are common data structures as they naturally reflect linear computational processes. These processes arise in repeated (iterative) function calls. Therefore, it is sometimes necessary to generate a temporary list of intermediate results before reducing this list to a single result.
One example of a commonly encountered hylomorphism is the canonical factorial function.
factorial :: Integer -> Integer factorial n | n == 0 = 1 | n > 0 = n * factorial (n - 1)
In the previous example (written in Haskell, a purely functional programming language) it can be seen that this function, applied to any given valid input, will generate a linear call tree isomorphic to a list. For example, given n = 5 it will produce the following:
factorial 5 = 5 * (factorial 4) = 120 factorial 4 = 4 * (factorial 3) = 24 factorial 3 = 3 * (factorial 2) = 6 factorial 2 = 2 * (factorial 1) = 2 factorial 1 = 1 * (factorial 0) = 1 factorial 0 = 1
In this example, the anamorphic part of the process is the generation of the call tree which is isomorphic to the list
[1, 1, 2, 3, 4, 5]. The catamorphism, then, is the calculation of the product of the elements of this list. Thus, in the notation given above, the factorial function may be written as where and .
However, the term 'hylomorphism' does not apply solely to functions acting upon isomorphisms of lists. For example, a hylomorphism may also be defined by generating a non-linear call tree which is then collapsed. An example of such a function is the function to generate the nth term of the Fibonacci sequence.
fibonacci :: Integer -> Integer fibonacci n | n == 0 = 0 | n == 1 = 1 | n > 1 = fibonacci (n - 2) + fibonacci (n - 1)
This function, again applied to any valid input, will generate a call tree which is non-linear. In the example on the right, the call tree generated by applying the
fibonacci function to the input
- Morphisms of F-algebras
- Erik Meijer; Maarten Fokkinga; Ross Paterson (1991). "Functional Programming with Bananas, Lenses, Envelopes and Barbed Wire" (PDF). pp. 4, 5.