A tutorial on the universality and expressiveness of fold

In functional programming, fold is a standard operator that encapsulates a simple pattern of recursion for processing lists. This article is a tutorial on two key aspects of the fold operator for lists. First of all, we emphasize the use of the universal property of fold both as a proof principle that avoids the need for inductive proofs, and as a de(cid:12)nition principle that guides the transformation of recursive functions into de(cid:12)nitions using fold. Secondly, we show that even though the pattern of recursion encapsulated by fold is simple, in a language with tuples and functions as (cid:12)rst-class values the fold operator has greater expressive power than might (cid:12)rst be expected.


Introduction
Many programs that involve repetition are naturally expressed using some form of recursion, and properties proved of such programs using some form of induction.Indeed, in the functional approach to programming, recursion and induction are the primary tools for de ning and proving properties of programs.
Not surprisingly, many recursive programs will share a common pattern of recursion, and many inductive proofs will share a common pattern of induction.Repeating the same patterns again and again is tedious, time consuming, and prone to error.Such repetition can be avoided by introducing special recursion operators and proof principles that encapsulate the common patterns, allowing us to concentrate on the parts that are di erent for each application.
In functional programming, fold (also known as foldr) is a standard recursion operator that encapsulates a common pattern of recursion for processing lists.The fold operator comes equipped with a proof principle called universality, which encapsulates a common pattern of inductive proof concerning lists.Fold and its universal property together form the basis of a simple but powerful calculational theory of programs that process lists.This theory generalises from lists to a variety of other datatypes, but for simplicity we restrict our attention to lists.
This article is a tutorial on two key aspects of the fold operator for lists.First of all, we emphasize the use of the universal property of fold (together with the derived fusion property) both as proof principles that avoid the need for inductive proofs, and as de nition principles that guide the transformation of recursive functions into de nitions using fold.Secondly, we show that even though the pattern of recursion encapsulated by fold is simple, in a language with tuples and functions as rst-class values the fold operator has greater expressive power than might rst be expected, thus permitting the powerful universal and fusion properties of fold to be applied to a larger class of programs.The article concludes with a survey of other work on recursion operators that we do not have space to pursue here.
The article is aimed at a reader who is familiar with the basics of functional programming, say to the level of (Bird & Wadler, 1988;Bird, 1998).All programs in the article are written in Haskell (Peterson et al. , 1997), the standard lazy functional programming language.However, no special features of Haskell are used, and the ideas can easily be adapted to other functional languages.

The fold operator
The fold operator has its origins in recursion theory (Kleene, 1952), while the use of fold as a central concept in a programming language dates back to the reduction operator of APL (Iverson, 1962), and later to the insertion operator of FP (Backus, 1978).In Haskell, the fold operator for lists can be de ned as follows: fold :: That is, given a function f of type !! and a value v of type , the function fold f v processes a list of type ] to give a value of type by replacing the nil constructor ] at the end of the list by the value v, and each cons constructor (:) within the list by the function f.In this manner, the fold operator encapsulates a simple pattern of recursion for processing lists, in which the two constructors for lists are simply replaced by other values and functions.A number of familiar functions on lists have a simple de nition using fold.For example: sum :: Int] !Int product :: Int] !Int sum = fold (+) 0 product = fold ( ) 1 and :: Bool] !Bool or :: Bool] !Bool and = fold (^) True or = fold (_) False Recall that enclosing an in x operator in parentheses ( ) converts the operator into a pre x function.This notational device, called sectioning, is often useful when de ning simple functions using fold.If required, one of the arguments to the operator can also be enclosed in the parentheses.For example, the function (++) that appends two lists to give a single list can be de ned as follows: (++) :: ] !] !] (++ ys) = fold (:) ys In all our examples so far, the constructor (:) is replaced by a built-in function.However, in most applications of fold the constructor (:) will be replaced by a userde ned function, often de ned as a nameless function using the notation, as in the following de nitions of standard list-processing functions: length :: filter p = fold ( x xs !if p x then x : xs else xs) ] Programs written using fold can be less readable than programs written using explicit recursion, but can be constructed in a systematic manner, and are better suited to transformation and proof.For example, we will see later on in the article how the above de nition for map using fold can be constructed from the standard de nition using explicit recursion, and more importantly, how the de nition using fold simpli es the process of proving properties of the map function.

The universal property of fold
As with the fold operator itself, the universal property of fold also has its origins in recursion theory.The rst systematic use of the universal property in functional programming was by Malcolm (1990a), in his generalisation of Bird and Meerten's theory of lists (Bird, 1989;Meertens, 1983) to arbitrary regular datatypes.For nite lists, the universal property of fold can be stated as the following equivalence between two de nitions for a function g that processes lists: g ] = v g (x : xs) = f x (g xs) , g = fold f v In the right-to-left direction, substituting g = fold f v into the two equations for g gives the recursive de nition for fold.Conversely, in the left-to-right direction the two equations for g are precisely the assumptions required to show that g = fold f v using a simple proof by induction on nite lists (Bird, 1998).Taken as a whole, the universal property states that for nite lists the function fold f v is not just a solution to its de ning equations, but in fact the unique solution.
The key to the utility of the universal property is that it makes explicit the two assumptions required for a certain pattern of inductive proof.For speci c cases then, by verifying the two assumptions (which can typically be done without the need for induction) we can then appeal to the universal property to complete the inductive proof that g = fold f v.In this manner, the universal property of fold encapsulates a simple pattern of inductive proof concerning lists, just as the fold operator itself encapsulates a simple pattern of recursion for processing lists.
The universal property of fold can be generalised to handle partial and in nite lists (Bird, 1998), but for simplicity we only consider nite lists in this article.

Universality as a proof principle
The primary application of the universal property of fold is as a proof principle that avoids the need for inductive proofs.As a simple rst example, consider the following equation between functions that process a list of numbers: (+1) sum = fold (+) 1 The left-hand function sums a list and then increments the result.The right-hand function processes a list by replacing each (:) by the addition function (+) and the empty list ] by the constant 1.The equation asserts that these two functions always give the same result when applied to the same list.
To prove the above equation, we begin by observing that it matches the righthand side g = fold f v of the universal property of fold, with g = (+1) sum, f = (+), and v = 1.Hence, by appealing to the universal property, we conclude that the equation to be proved is equivalent to the following two equations: ((+1) sum) ] = 1 ((+1) sum) (x : xs) = (+) x (((+1) sum) xs) At rst sight, these may seem more complicated than the original equation.However, simplifying using the de nitions of composition and sectioning gives sum ] + 1 = 1 sum (x : xs) + 1 = x + (sum xs + 1) which can now be veri ed by simple calculations, shown here in two columns: sum ] + 1 sum (x : xs) + 1 = f De nition of sum g = f De nition of sum g 0 + 1 (x + sum xs) + 1 = f Arithmetic g = f Arithmetic g 1 x + (sum xs + 1) This completes the proof.Normally this proof would have required an explicit use of induction.However, in the above proof the use of induction has been encapsulated in the universal property of fold, with the result that the proof is reduced to a simpli cation step followed by two simple calculations.
In general, any two functions on lists that can be proved equal by induction can also be proved equal using the universal property of the fold operator, provided, of course, that the functions can be expressed using fold.The expressive power of the fold operator will be addressed later on in the article.

The fusion property of fold
Now let us generalise from the sum example and consider the following equation between functions that process a list of values: h fold g w = fold f v This pattern of equation occurs frequently when reasoning about programs written using fold.It is not true in general, but we can use the universal property of fold to calculate conditions under which the equation will indeed be true.The equation matches the right-hand side of the universal property, from which we conclude that the equation is equivalent to the following two equations: (h fold g w) ] = v (h fold g w) (x : xs) = f x ((h fold g w) xs) Simplifying using the de nition of composition gives h (fold g w ]) = v h (fold g w (x : xs)) = f x (h (fold g w xs)) which can now be further simpli ed by two calculations: , f De nition of fold g h (g x (fold g w xs)) = f x (h (fold g w xs)) ( f Generalising (fold g w xs) to a fresh variable y g h (g x y) = f x (h y) That is, using the universal property of fold we have calculated|without an explicit use of induction|two simple conditions that are together su cient to ensure for all nite lists that the composition of an arbitrary function and a fold can be fused together to give a single fold.Following this interpretation, this property is called the fusion property of the fold operator, and can be stated as follows: The rst systematic use of the fusion property in functional programming was again by Malcolm (1990a), generalising earlier work by Bird and Meertens (Bird, 1989;Meertens, 1983).As with the universal property, the primary application of the fusion property is as a proof principle that avoids the need for inductive proofs.In fact, for many practical examples the fusion property is often preferable to the universal property.As a simple rst example, consider again the equation: (+1) sum = fold (+) 1 In the previous section this equation was proved using the universal property of fold.However, the proof is simpler using the fusion property.First of all, we replace the function sum by its de nition using fold given earlier: (+1) fold (+) 0 = fold (+) 1 The equation now matches the conclusion of the fusion property, from which we conclude that the equation follows from the following two assumptions: (+1) 0 = 1 (+1) ((+) x y) = (+) x ((+1) y) Simplifying these equations using the de nition of sectioning gives 0 + 1 = 1 and (x + y) + 1 = x + (y + 1), which are true by simple properties of arithmetic.More generally, by replacing the use of addition in this example by an arbitrary in x operator that is associative, a simple application of fusion shows that: ( a) fold ( ) b = fold ( ) (b a) For a more interesting example, consider the following well-known equation, which asserts that the map operator distributes over function composition ( ): map f map g = map (f g) By replacing the second and third occurrences of the map operator in the equation by its de nition using fold given earlier, the equation can be rewritten in a form that matches the conclusion of the fusion property: map f fold ( x xs !g x : xs) ] = fold ( x xs !(f g) x : xs) ] Appealing to the fusion property and then simplifying gives the following two equations, which are trivially true by the de nitions of map and ( ): map f ] = ] map f (g x : y) = (f g) x : map f y In addition to the fusion property, there are a number of other useful properties of the fold operator that can be derived from the universal property (Bird, 1998).However, the fusion property su ces for many practical cases, and one can always revert to the full power of the universal property if fusion is not appropriate.

Universality as a de nition principle
As well as being used as a proof principle, the universal property of fold can also be used as a de nition principle that guides the transformation of recursive functions into de nitions using fold.As a simple rst example, consider the recursively de ned function sum that calculates the sum of a list of numbers: sum :: Int] !Int sum ] = 0 sum (x : xs) = x + sum xs Suppose now that we want to rede ne sum using fold.That is, we want to solve the equation sum = fold f v for a function f and a value v.We begin by observing that the equation matches the right-hand side of the universal property, from which we conclude that the equation is equivalent to the following two equations: sum ] = v sum (x : xs) = f x (sum xs) From the rst equation and the de nition of sum, it is immediate that v = 0. From the second equation, we calculate a de nition for f as follows: sum (x : xs) = f x (sum xs) , f De nition of sum g x + sum xs = f x (sum xs) ( f y Generalising (sum xs) to y g x + y = f x y , f Functions g f = (+) That is, using the universal property we have calculated that: sum = fold (+) 0 Note that the key step (y) above in calculating a de nition for f is the generalisation of the expression sum xs to a fresh variable y.In fact, such a generalisation step is not speci c to the sum function, but will be a key step in the transformation of any recursive function into a de nition using fold in this manner.
Of course, the sum example above is rather arti cial, because the de nition of sum using fold is immediate.However, there are many examples of functions whose de nition using fold is not so immediate.For example, consider the recursively de ned function map f that applies a function f to each element of a list: map :: To rede ne map f using fold we must solve the equation map f = fold g v for a function g and a value v.By appealing to the universal property, we conclude that this equation is equivalent to the following two equations: map f ] = v map f (x : xs) = g x (map f xs) From the rst equation and the de nition of map it is immediate that v = ].From the second equation, we calculate a de nition for g as follows: map f (x : xs) = g x (map f xs) , f De nition of map g f x : map f xs = g x (map f xs) ( f Generalising (map f xs) to ys g f x : ys = g x ys , f Functions g g = x ys !f x : ys That is, using the universal property we have calculated that: map f = fold ( x ys !f x : ys) ] In general, any function on lists that can be expressed using the fold operator can be transformed into such a de nition using the universal property of fold.
4 Increasing the power of fold: generating tuples As a simple rst example of the use of fold to generate tuples, consider the function sumlength that calculates the sum and length of a list of numbers: sumlength :: Int] !(Int; Int) sumlength xs = (sum xs; length xs) By a straightforward combination of the de nitions of the functions sum and length using fold given earlier, the function sumlength can be rede ned as a single application of fold that generates a pair of numbers from a list of numbers: sumlength = fold ( n (x; y) !(n + x; 1 + y)) (0; 0) This de nition is more e cient than the original de nition, because it only makes a single traversal over the argument list, rather than two separate traversals.Generalising from this example, any pair of applications of fold to the same list can always be combined to give a single application of fold that generates a pair, by appealing to the so-called `banana split' property of fold (Meijer, 1992).The strange name of this property derives from the fact that the fold operator is sometimes written using brackets (j j ) that resemble bananas, and the pairing operator is sometimes called split.Hence, their combination can be termed a banana split!As a more interesting example, let us consider the function dropWhile p that removes initial elements from a list while all the elements satisfy the predicate p: Unfortunately, the nal line above is not a valid de nition for f, because the variable xs occurs freely.In fact, it is not possible to rede ne dropWhile p directly using fold.However, it is possible indirectly, because the more general function that pairs up the result of applying dropWhile p to a list with the list itself can be rede ned using fold.By appealing to the universal property, we conclude that the equation dropWhile 0 p = fold f v is equivalent to the following two equations: dropWhile 0 p ] = v dropWhile 0 p (x : xs) = f x (dropWhile 0 p xs) A simple calculation from the rst equation gives v = ( ]; ]).From the second equation, we calculate a de nition for f as follows: Note that the nal line above is a valid de nition for f, because all the variables are bound.In summary, using the universal property we have calculated that: dropWhile 0 p = fold f v where f x (ys; xs) = (if p x then ys else x : xs; x : xs) v = ( ]; ]) This de nition satis es the equation dropWhile 0 p xs = (dropWhile p xs; xs), but does not make use of dropWhile in its de nition.Hence, the function dropWhile itself can now be rede ned simply by dropWhile p = fst dropWhile 0 p.
In conclusion, by rst generalising to a function dropWhile 0 that pairs the desired result with the argument list, we have now shown how the function dropWhile can be rede ned in terms of fold, as required.In fact, this result is an instance of a general theorem (Meertens, 1992) that states that any function on nite lists that is de ned by pairing the desired result with the argument list can always be rede ned in terms of fold, although not always in a way that does not make use of the original (possibly recursive) de nition for the function.

Primitive recursion
In this section we show that by using the tupling technique from the previous section, every primitive recursive function on lists can be rede ned in terms of fold.Let us begin by recalling that the fold operator captures the following simple pattern of recursion for de ning a function h that processes lists: h ] = v h (x : xs) = g x (h xs) Such functions can be rede ned by h = fold g v.We will generalise this pattern of recursion to primitive recursion in two steps.First of all, we introduce an extra argument y to the function h, which in the base case is processed by a new function f, and in the recursive case is passed unchanged to the functions g and h.That is, we now consider the following pattern of recursion for de ning a function h: h y ] = f y h y (x : xs) = g y x (h y xs) By simple observation, or a routine application of the universal property of fold, the function h y can be rede ned using fold as follows: h y = fold (g y) (f y) For the second step, we introduce the list xs as an extra argument to the auxiliary function g.That is, we now consider the following pattern for de ning h: h y ] = f y h y (x : xs) = g y x xs (h y xs) This pattern of recursion on lists is called primitive recursion (Kleene, 1952).Technically, the standard de nition of primitive recursion requires that the argument y is a nite sequence of arguments.However, because tuples are rst-class values in Haskell, treating the case of a single argument y is su cient.
In order to rede ne primitive recursive functions in terms of fold, we must solve the equation h y = fold i j for a function i and a value j.This is not possible directly, but is possible indirectly, because the more general function k y xs = (h y xs; xs) that pairs up the result of applying h y to a list with the list itself can be rede ned using fold.By appealing to the universal property of fold, we conclude that the equation k y = fold i j is equivalent to the following two equations: k y ] = j k y (x : xs) = i x (k y xs) A simple calculation from the rst equation gives j = (f y; ]).From the second equation, we calculate a de nition for i as follows: This de nition satis es the equation k y xs = (h y xs; xs), but does not make use of h in its de nition.Hence, the primitive recursive function h itself can now be rede ned simply by h y = fst k y.In conclusion, we have now shown how an arbitrary primitive recursive function on lists can be rede ned in terms of fold.
Note that the use of tupling to de ne primitive recursive functions in terms of fold is precisely the key to de ning the predecessor function for the Church numerals (Barendregt, 1984).Indeed, the intuition behind the representation of the natural numbers (or more generally, any inductive datatype) in the -calculus is the idea of representing each number by its fold operator.For example, the number 3 = succ (succ (succ zero)) is represented by the term f x !f (f (f x)), which is the fold operator for 3 in the sense that the arguments f and x can be viewed as the replacements for the succ and zero constructors respectively.

Using fold to generate functions
Having functions as rst-class values increases the power of primitive recursion, and hence the power of the fold operator.As a simple rst example of the use of fold to generate functions, the function compose that forms the composition of a list of functions can be de ned using fold by replacing each (:) in the list by the composition function ( ), and the empty list ] by the identity function id: compose :: !] ! ( ! ) compose = fold ( ) id As a more interesting example, let us consider the problem of summing a list of numbers.The natural de nition for such a function, sum = fold (+) 0, processes the numbers in the list in right-to-left order.However, it is also possible to de ne a function suml that processes the numbers in left-to-right order.The suml function is naturally de ned using an auxiliary function suml 0 that is itself de ned by explicit recursion and makes use of an accumulating parameter n: suml :: Int] !Int suml xs = suml 0 xs 0 where suml 0 ] n = n suml 0 (x : xs) n = suml 0 xs (n + x) Because the addition function (+) is associative and the constant 0 is unit for addition, the functions suml and sum always give the same result when applied to the same list.However, the function suml has the potential to be more e cient, because it can easily be modi ed to run in constant space (Bird, 1998).
Suppose now that we want to rede ne suml using the fold operator.This is not possible directly, but is possible indirectly, because the auxiliary function suml 0 :: Int] !(Int !Int) can be rede ned using fold.By appealing to the universal property, we conclude that the equation suml 0 = fold f v is equivalent to the following two equations: suml 0 ] = v suml 0 (x : xs) = f x (suml 0 xs) A simple calculation from the rst equation gives v = id.From the second equation, we calculate a de nition for the function f as follows: suml 0 (x : xs) = f x (suml 0 xs) , f Functions g suml 0 (x : xs) n = f x (suml 0 xs) n , f De nition of suml 0 g suml 0 xs (n + x) = f x (suml 0 xs) n ( f Generalising (suml 0 xs) to g g g (n + x) = f x g n , f Functions g f = x g ! ( n !g (n + x)) In summary, using the universal property we have calculated that: suml 0 = fold ( x g ! ( n !g (n + x))) id This de nition states that suml 0 processes a list by replacing the empty list ] by the identity function id on lists, and each constructor (:) by the function that takes a number x and a function g, and returns the function that takes an accumulator value n and returns the result of applying g to the new accumulator value n + x.
Note that the structuring of the arguments to suml 0 :: Int] !(Int !Int) is crucial to its de nition using fold.In particular, if the order of the two arguments is swapped or they are supplied as a pair, then the type of suml 0 means that it can no longer be de ned directly using fold.In general, some care regarding the structuring of arguments is required when aiming to rede ne functions using fold.Moreover, at rst sight one might imagine that fold can only be used to de ne functions that process the elements of lists in right-to-left order.However, as the de nition of suml 0 using fold shows, the order in which the elements are processed depends on the arguments of fold, not on fold itself.
In conclusion, by rst rede ning the auxiliary function suml 0 using fold, we have now shown how the function suml can be rede ned in terms of fold, as required: suml xs = fold ( x g ! ( n !g (n + x))) id xs 0 We end this section by remarking that the use of fold to generate functions provides an elegant technique for the implementation of `attribute grammars' in functional languages (Fokkinga et al., 1991;Swierstra et al., 1998).

The foldl operator
Now let us generalise from the suml example and consider the standard operator foldl that processes the elements of a list in left-to-right order by using a function f to combine values, and a value v as the starting value: Using this operator, suml can be rede ned simply by suml = foldl (+) 0. Many other functions can be de ned in a simple way using foldl.For example, the standard function reverse can rede ned using foldl as follows: reverse :: ] !] reverse = foldl ( xs x !x : xs) ] This de nition is more e cient than our original de nition using fold, because it avoids the use of the ine cient append operator (++) for lists.
A simple generalisation of the calculation in the previous section for the function suml shows how to rede ne the function foldl in terms of fold: foldl f v xs = fold ( x g ! ( a !g (f a x))) id xs v In contrast, it is not possible to rede ne fold in terms of foldl, due to the fact that foldl is strict in the tail of its list argument but fold is not.There are a number of useful `duality theorems' concerning fold and foldl, and also some guidelines for deciding which operator is best suited to particular applications (Bird, 1998).

Ackermann's function
For our nal example of the power of fold, consider the function ack that processes two lists of integers, and is de ned using explicit recursion as follows: ack :: Int] ! ( Int] !Int]) ack ] ys = 1 : ys ack (x : xs) ] = ack xs 1] ack (x : xs) (y : ys) = ack xs (ack (x : xs) ys) This is Ackermann's function, converted to operate on lists rather than natural numbers by representing each number n by a list with n arbitrary elements.This function is the classic example of a function that is not primitive recursion in a rstorder programming language.However, in a higher-order language such as Haskell, Ackermann's function is indeed primitive recursive (Reynolds, 1985).In this section we show how to calculate the de nition ack in terms of fold.
First of all, by appealing to the universal property of fold, the equation ack = fold f v is equivalent to the following two equations: ack ] = v ack (x : xs) = f x (ack xs) A simple calculation from the rst equation gives the de nition v = (1 :).From the second equation, proceeding in the normal manner does not result in a de nition for the function f, as the reader may wish to verify.However, progress can be made by rst using fold to rede ne the function ack (x : xs) on the left-hand side of the second equation.By appealing to the universal property, the equation ack (x : xs) = fold g w is equivalent to the following two equations: ack (x : xs) ] = w ack (x : xs) (y : ys) = g y (ack (x : xs) ys) In summary, using the universal property twice we have calculated that: ack = fold ( x g !fold ( y !g) (g 1])) (1 :)

Other work on recursion operators
In this nal section we brie y survey a selection of other work on recursion operators that we did not have space to pursue in this article.
Fold for regular datatypes.The fold operator is not speci c to lists, but can be generalised in a uniform way to `regular' datatypes.Indeed, using ideas from category theory, a single fold operator can be de ned that can be used with any regular datatype (Malcolm, 1990b;Meijer et al., 1991;Sheard & Fegaras, 1993).
Fold for nested datatypes.The fold operator can also be generalised in a natural way to `nested' datatypes.However, the resulting operator appears to be too general to be widely useful.Finding solutions to this problem is the subject of current research (Bird & Meertens, 1998;Jones & Blampied, 1998).
Fold for functional datatypes.Generalising the fold operator to datatypes that involve functions gives rise to technical problems, due to the contravariant nature of function types.Using ideas from category theory, a fold operator can be de ned that works for such datatypes (Meijer & Hutton, 1995), but the the use of this operator is not well understood, and practical applications are lacking.However, a simpler but less general solution has given rise to some interesting applications concerning cyclic structures (Fegaras & Sheard, 1996).
Monadic fold.In a series of in uential articles, Wadler showed how pure functional programs that require imperative features such as state and exceptions can be modelled using monads (Wadler, 1990;Wadler, 1992a;Wadler, 1992b).Building on this work, the notion of a `monadic fold' combines the use of fold operators to structure the processing of recursive values with the use of monads to structure the use of imperative features (Fokkinga, 1994;Meijer & Jeuring, 1995).
Relational fold.The fold operator can also be generalised in a natural way from functions to relations.This generalisation supports the use of fold as a speci cation construct, in addition to its use as a programming construct.For example, a relational fold is used in the circuit design calculus Ruby (Jones & Sheeran, 1990;Jones, 1990), the Eindhoven spec calculus (Aarts et al., 1992), and in a recent textbook on the algebra of programming (Bird & de Moor, 1997).
Other recursion operators.The fold operator is not the only useful recursion operator.For example, the dual operator unfold for constructing rather than processing recursive values has been used for speci cation purposes (Jones, 1990;Bird & de Moor, 1997), to program reactive systems (Kieburtz, 1998), to program operational semantics (Hutton, 1998), and is the subject of current research.Other interesting recursion operators include the so-called paramorphisms (Meertens, 1992), hylomorphisms (Meijer, 1992), and zygomorphisms (Malcolm, 1990a).
Automatic program transformation.Writing programs using recursion operators can simplify the process of optimisation during compilation.For example, eliminating the use of intermediate data structures in programs (deforestation) in considerably simpli ed when programs are written using recursion operators rather than general recursion (Wadler, 1981;Launchbury & Sheard, 1995;Takano & Meijer, 1995).A generic system for transforming programs written using recursion operators is currently under development (de Moor & Sittampalan, 1998).
Polytypic programming.De ning programs that are not speci c to particular datatypes has given rise to a new eld, called polytypic programming (Backhouse et al., 1998).Formally, a polytypic program is one that is parameterised by one or more datatypes.Polytypic programs have already been de ned for a number of applications, including pattern matching (Jeuring, 1995), uni cation (Jansson & Jeuring, 1998), and various optimisation problems (Bird & de Moor, 1997).
Programming languages.A number of experimental programming languages have been developed that focus on the use of recursion operators rather than general recursion.Examples include the algebraic design language ADL (Kieburtz & Lewis, 1994), the categorical programminglanguage Charity (Cockett & Fukushima,1992), and the polytypic programming language PolyP (Jansson & Jeuring, 1997).
7 Acknowledgements I would like to thank Erik Meijer and the members of the Languages and Programming group in Nottingham for many hours of interesting discussions about fold.I am also grateful to Roland Backhouse, Mark P.Jones, Philip Wadler, and the anonymous JFP referees for their detailed comments on the article, which led to a substantial improvement in both the content and presentation.This work is supported by Engineering and Physical Sciences Research Council (EPSRC) research grant GR/L74491, Structured Recursive Programming.
x : xs) = if p x then dropWhile p xs else x : xs Suppose now that we want to rede ne dropWhile p using the fold operator.By appealing to the universal property, we conclude that the equation dropWhile p = fold f v is equivalent to the following two equations:dropWhile p ] = v dropWhile p (x : xs) = f x (dropWhile p xs) From the rst equation it is immediate that v = ].From the second equation, we attempt to calculate a de nition for f in the normal manner: dropWhile p (x : xs) = f x (dropWhile p xs) , f De nition of dropWhile g if p x then dropWhile p xs else x : xs = f x (dropWhile p xs) ( f Generalising (dropWhile p xs) to ys g if p x then ys else x : xs = f x ys xs = (dropWhile p xs; xs) dropWhile 0 p (x : xs) = f x (dropWhile 0 p xs) , f De nition of dropWhile 0 g (dropWhile p (x : xs); x : xs) = f x (dropWhile p xs; xs) , f De nition of dropWhile g (if p x then dropWhile p xs else x : xs; x : xs) = f x (dropWhile p xs; xs) ( f Generalising (dropWhile p xs) to ys g (if p x then ys else x : xs; x : xs) = f x (ys; xs) k y (x : xs) = i x (k y xs) , f De nition of k g (h y (x : xs); x : xs) = i x (h y xs; xs) , f De nition of h g (g y x xs (h y xs); x : xs) = i x (h y xs; xs) ( f Generalising (h y xs) to z g (g y x xs z; x : xs) = i x (z; xs) In summary, using the universal property we have calculated that: k foldl