Graham Hutton’s 1999 monograph: *A tutorial on the universality and expressiveness of fold* [Hutton99] provides a fascinating and eminently readable analysis of an elementary reduction operator from functional programming: *fold*. Hutton’s tutorial is punctuated by increasingly compelling examples, written in Haskell, of the use of *fold* to implement common list operations, including summation; concatenation; reversal; function composition; *left* folds; and concluding with the Ackermann function. The *fold* combinator is directly comparable to the C++ standard library function: `std::accumulate`

. In this article, I focus on the *fold* examples from Hutton, and present equivalent generic C++ incarnations of each; making thorough use of appropriate C++14 library and language features, including polymorphic lambda functions^{1}.

The four-parameter `accumulate`

function constructs a result from the repeated pairwise application of a given binary function, to elements supplied from both an iterator range; and a single “zero” value, having the type of the result. A canonical example is summation; the value returned from the call to `accumulate`

in listing 1 is 10.

int xs[]{1,2,3,4}; accumulate(cbegin(xs), cend(xs), 0, plus<>{}); |

Listing 1 |

Hutton’s first four examples use *fold* to implement arithmetic and logic operations using built-in functions to specify the relevant binary operations. Similar function objects are provided by the C++ standard library within the `<functional>`

header. The example above demonstrates the use of `plus<>`

for `operator+`

; while `operator*`

, `operator&&`

, and `operator||`

correspond to `multiplies<>`

,` logical_and<>`

, and `logical_or<>`

respectively. Note that C++14 conveniently defines a default template argument for such function objects; `std::plus<>`

invokes a specialisation which infers the relevant types from its arguments.

## Accommodating std::accumulate

A binary operator is said to be *associative* when an expression involving a sequence of its applications produces the same result, regardless of the order of evaluation. The four C++ function objects from the previous section all denote associative operations. Consider addition: both (1 + 2) + 3 and 1 + ( 2 + 3 ) produce the same result; 6. Operator precedence is irrelevant when faced with a ⊕ b ⊕ c; the question is whether to evaluate from the *left*; or from the *right*. In particular, which order does `accumulate`

use? It uses a left ordering.

An elementary *non-associative* binary operation is *subtraction*. The call to `accumulate`

in listing 2 would then produce a result equal to (( 0 - 1) - 2 ) - 3, i.e. - 6; evaluating from the left. In contrast, an evaluation ordered from the *right*, say 1- ( 2 - ( 3 - 0 )), produces a result of 2. Alas, the remaining examples from Hutton [Hutton99] firmly assume the fold operation evaluates from the *right*.

vector<double> xs{1,2,3}; accumulate(cbegin(xs), cend(xs), 0, minus<>{}); |

Listing 2 |

Producing a result from `accumulate`

equal to a *right fold* requires two interventions: we need the order of traversal across the container elements reversed; *and* for the order of the arguments given to the binary operation to be switched^{2}. Listing 3 defines such a wrapper for `accumulate`

, called `foldr`

. The C++14 functions `crbegin`

and `crend`

return `const`

iterators to the *reverse beginning* and *reverse end* of the container argument `c`

. Meanwhile, the `flip`

function, uses `std::bind`

to reverse the argument order for the binary function object; e.g. `flip(minus<>{})(0,1)`

evaluates to 1; not - 1.

template <typename F> auto flip(const F &f) { return bind(f,_2,_1); } template <typename F, typename T, typename C> T foldr(const F &f, const T &z, const C &c) { return accumulate(crbegin(c), crend(c), z, flip(f)); } |

Listing 3 |

The definition of `foldr`

in listing 3 removes the need to call `crbegin`

, `crend`

and `flip`

. It also allows a single container argument to drive the C++ *fold*; much as with C++ *ranges* [Niebler15]. This allows listings here to remain concise; while also facilitating a comparison of the syntactical differences between Haskell and C++. We can now invoke a *right fold*. Assuming ``C`

` creates an arbitrary standard sequence container, with inferred value type^{3}, the call to `foldr`

in listing 4 returns the integer 2.

foldr(minus<>{}, 0, C(1,2,3)) |

Listing 4 |

## Non-reducing folds

Using a *fold* to concatenate two containers first requires a small helper function, which should return a new container, by adding a single element to the front of an existing one. Haskell provides the `(:)`

operator for this job. Listing 5 defines this using its traditional Lisp name: “cons”.

auto cons = [=](auto x, auto xs) { decltype(xs) ys{x}; copy(begin(xs), end(xs), back_inserter(ys)); return ys; }; |

Listing 5 |

Like *subtraction*, the `cons`

function is *non-associative*; and *non-commutative*. `cons`

though, expects two different argument types. Listing 6 provides `foldr`

with `cons`

as the binary function, and an empty container as the “zero” or starting value; to define an identity *fold*. That is, `id(C(1,2,3))`

will return a container object of the same type; with the same 3 elements. Meanwhile, the genericity of C++ allows a similar invocation which only changes the container type: `foldr(cons, list<int>{}, C(1,2,3))`

.

auto id = [=](auto xs) { return foldr(cons, decltype(xs){}, xs); }; |

Listing 6 |

To append one container to another, listing 7 again uses `cons`

for `foldr`

’s first argument; while providing the second, a container, as its “zero” value. Note that while the elements of, say, `append(C('a'),C('b'))`

and `C('a','b')`

are equal, so too are they equal to `append(C<list>('a'),C<vector>('b'))`

; as the definition is sufficiently generic.

auto append = [=](auto xs, auto ys) { return foldr(cons, ys, xs); }; |

Listing 7 |

## Folding with lambda arguments

The three functions^{4} of listing 8 provide each corresponding `foldr`

invocation with a binary lambda function, as, like Haskell, no equivalents exist within the standard library. The `length`

function returns the size of its container argument, using a lambda function with unnamed first argument. Both `reverse`

and `map`

return a container^{5}; with `map`

utilising the closure aspect of lambda expressions to capture `f`

.

auto length = [=](auto xs){ return foldr( [=](auto, auto n){ return 1+n; }, 0, xs); }; auto reverse = [=](auto xs){ return foldr( [=](auto y, auto ys) { return append(ys,decltype(xs){y}); }, decltype(xs){}, xs); }; auto map = [=](auto f, auto xs){ return foldr( [=](auto y, auto ys){ return cons(f(y),ys); }, vector<decltype(f(*xs.begin()))>{}, xs); }; |

Listing 8 |

Tuples allow a single *fold* to perform more than one calculation. For example, listing 9 defines a function which returns both the size of a container, and the sum of its elements^{6}.

auto sumlength = [=](auto xs){ return foldr( [=](auto n, auto p){ return make_pair(n + p.first, 1 + p.second); }, make_pair(0,0), xs); }; |

Listing 9 |

## Functions from folds

The result of applying the *composition* of two functions `f`

and `g`

to an argument `x`

can be achieved in C++ with the expression: `f(g(x))`

. In Haskell an elementary binary operator, `(.)`

, can also be used; accepting two functions as arguments, and returning a *new* function representing their composition. In listing 10, the *fold*’s binary function argument is a comparable lambda expression for composition. The result of invoking `compose`

with a container of callable elements is a function object representing the chained composition of each function in sequence. The “zero” argument of `foldr`

uses a simple lambda identity function; though notice it is wrapped by the type of the container element: an instantiation of `std::function`

. Why? While the type of each lambda expression is unique, the type of each container element is the same. `std::function `

provides exactly the required homogeneity; each lambda accepting and returning say an `int`

, becomes simply `std::function<int(int)>`

. The “zero”, meanwhile, needs the same wrapper, as it provides the type of the *fold*’s result.

auto compose = [=](auto fs){ using fn_t = typename decltype(fs)::value_type; return foldr( [=](auto f, auto g) { return [=](auto x){ return f(g(x)); }; }, fn_t([=](auto x){ return x; }), fs); }; |

Listing 10 |

One of the most intriguing functions capable of definition by a *right fold*, such as our `foldr`

, is a *left fold*. Listing 11 provides such a definition. As before, an identity function is required for the *fold*’s starting value, and again this wild lambda needs the guiding hand of `std::function`

; though the type in question is calculated in a different manner. Unlike `compose`

, the function object returned by `foldr`

is invoked within `foldl`

; upon `z`

. Our journey has brought us full circle to a *left fold*, akin to `std::accumulate`

; an invocation such as `foldl(minus<>{}, 0, C(1,2,3))`

will produce -6; much as listing 2.

auto foldl = [=](auto f, auto z, auto xs){ using z_t = decltype(z); using fn_t = std::function<z_t(z_t)>; return foldr( [=](auto x, auto g){ return [=](auto a){ return g(f(a,x)); }; }, fn_t([=](auto x){ return x; }),xs)(z); }; |

Listing 11 |

One last comment regarding *left* and *right* folds: should you ever be in the embarrassing situation of being uncertain of the handedness of your *fold* definition, the expression in listing 12 could be useful. Simply replace `fold`

with either `foldr`

or `foldl`

; for a `true`

or `false`

evaluation respectively.

fold([](auto x, auto){ return x; }, false, C(true)) |

Listing 12 |

Our final fold example, and so too in [Hutton99], is Ackermann’s function [Ackermann28]. Ackermann’s function is commonly cited as a recursive function which is not *primitive recursive* in a first-order programming language. John Reynolds, however, demonstrated [Reynolds85] that the function *is* primitive recursive in a higher-order programming language. The C++ implementation in listing 13 includes similar idioms to previous examples, but is given additional treatment to avoid the use of *currying* seen in Hutton’s definition. While the binary lambda function provided to the innermost *fold* in the curried original appears unary, λ *y *→ *g*, the C++ version must be uncurried: λ *y* *as* → *g(as)*. Note too, that these Ackermann *folds* encode the traditional natural number arguments within the size of the input and output container values.

auto ack = [=](auto xs, auto ys){ using ys_t = decltype(ys); using fn_t = std::function<ys_t(ys_t)>; return [=](auto zs){ return foldr( [=](auto, auto g){ return [=](auto ws){ return foldr( [=](auto, auto as){ return g(as); }, g(ys_t{1}), ws ); }; }, fn_t([=](auto bs){ return cons(1,bs); }), zs ); }(xs)(ys); }; |

Listing 13 |

## Summary

C++ lambda functions, including the polymorphic variety now available in C++14, allow the generic *fold* operation supported by `std::accumulate`

to extend well beyond simple reductions. While a complex *fold* can be less readable or idiomatic than the traditional form, the approach can be refined to construct and transform programs, as well as prove specific program properties; while building on established software engineering principles of reuse and abstraction.

The article places less emphasis on performance considerations, instead focusing on pedagogy and algorithmic aspects; while maintaining parity with the equivalent Haskell, with consistent use of `auto`

for type inference.

C++17 will introduce *fold expressions* [Sutton14]. Here, a *finite* set of operators, will share the brevity of the *comma* in pack expansion; consider `*`

versus `std::multiplies<>{}`

. One restriction is the variadic template pack length must be known at compile-time.

## Appendix

All examples, along with code for `dropWhile`

, `filter`

and other folds are available at https://bitbucket.org/pgk/accumulate.

## References

[Ackermann28] W. Ackermann. *Zum Hilbertschen Aufbau der reellen Zahlen* Mathematische Annalen, 1928.

[Bird88] R. Bird and P. Wadler.* Introduction to Functional Programming* Prentice Hall, 1988.

[Gill93] A. Gill, J. Launchbury and S.P. Jones. *A Short Cut to Deforestation*. ACM Press, 1993.

[Hutton99] G. Hutton. *A tutorial on the universality and expressiveness of fold* Cambridge University Press, 1999.

[Niebler15] E. Niebler. *Working Draft, C++ extensions for Range*s. The C++ Standards Committee, 2015.

[Reynolds85] J.C. Reynolds. *Three approaches to type structure* Springer-Verlag, 1985.

[Sutton14] A. Sutton and R. Smith. *Folding expressions*. The C++ Standards Committee, 2014.

- Note that most of these fold examples can also be found in [Bird88] or [Gill93].
- This is described as the third duality theorem in [Bird88, p68].
- Assume a
`vector`

default; i.e.`C(1,2)`

becomes:`C<vector,int>(1,2)`

. - A definition of
`filter`

from [Hutton99] appears in the appendix. - Note that
`map`

returns a`vector`

object here solely for brevity. - A similar tuple-based definition of
`dropWhile`

appears in the appendix.

## Overload Journal #130 - December 2015 + Programming Topics

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