» Using Scheme as a metaphor for template metaprogramming

» March 6, 2015 | cpp development english | Adrian Kummerländer

Back in January I looked at compile time computation in C++ based on handling lists in a functional fashion using a mixture between templates, generic lambda expressions and constexpr functions. The conclusion of the appropriate article was that the inherent restrictions of the approach taken in ConstList, namely the missing guarantee on compile time evaluation, inability to make return types depend on actual values and lambda expressions being unable to be declared as constant make viewing types as values and templates as functions the superior approach. This article describes how this approach works out in practice.

While ConstList turned out to be of limited use in actually performing compile time computations, its list manipulation and query functionality was already inspired by how lists are handled in LISP respectively its more minimalistic dialect Scheme, especially by the functionality described in the latter’s SRFI-1.
When I started developing a new library porting this basic concept to the type as value and templates as functions approach called TypeAsValue it quickly turned out that a Scheme like paradigm maps quite well to template metaprogramming. This was initially very surprising as I did not expect that C++ templates would actually feel like a - admittedly rather verbose - functional programming language if used in a certain way.

// (define sum
//         (fold +
//               0
//               (iota 5 2 2))) => 30
using sum = tav::Fold<

As we can see compile time computations expressed using this approach are more or less direct mappings of their Scheme equivalent if we overlook the need to explicitly declare types, the different syntax used for defining bindings as well as its immutability.

While TypeAsValue started out as a direct reimplementation of my previous attempt I am happy to say that the conclusions drawn concerning the superiority of a stricly template metaprogramming based implementation held true and enabled the implementation of equivalents for large parts of the Scheme list library. This includes actual content dependent list manipulations such as tav::Filter, which were impossible to implement in ConstList, in addition to e.g. a compile time implementation of Quick Sort.

Types as values

The desire to express values in terms of types restricts the set of usable types to integral types as only those types may be used as template parameters. According to the standard1 this includes all integer types i.e. all non-floating-point types such as bool, char and int. In case of TypeAsValue all values are expressed as specializations of std::integral_constant wrapped in template aliases to simplify their declaration.

using answer = tav::Int<42>;       // std::integral_constant<int, 42>
using letter = tav::Char<'A'>;     // std::integral_constant<char, 'A'>
using truth  = tav::Boolean<true>; // std::integral_constant<bool, true>

This need to explicitly declare all types because deduction during template resolution is not feasible marks one of the instances where the Scheme metaphor does not hold true. Luckily this is not a bad thing as the goal is after all not to develop a exact replica of Scheme in terms of template metaprogramming but to enable compile time computations in a Scheme like fashion. In this context not disregarding the C++ type system is an advantage, especially since it should be possible to enable type deduction where required using a tav::Any like std::integral_constant constructor.

Obviously expressing single values as types is not enough, we also require at least an equivalent for Scheme’s fundamental pair type, on top of which more complex structures such as lists and trees may be built.

template <
    typename CAR,
    typename CDR
struct Pair : detail::pair_tag {
    typedef CAR car;
    typedef CDR cdr;

    typedef Pair<CAR, CDR> type;

As we can see expressing a pair type in terms of a type template is very straight forward. Note that the recursive type definition will be discussed further in the next section on templates as functions. Each tav::Pair specialization derives from detail::pair_tag to simplify verification of values as pairs in tav::IsPair. The naming of the parameters as CAR and CDR is a reference to pair types being constructed using tav::Cons analogously to Scheme, where the pair (1 . 2) may be constructed using (cons 1 2).

To summarize the type concept employed in TypeAsValue we can say that all actual values are stored in std::integral_constant specializations that enable extraction in a template context via their constant value member. Those types are then aggregated into structures using the tav::Pair template. This means that we can easily provide functions to work on these values in the form of type templates and their parameters as will be discussed in the following section.

Templates as functions

template <
    typename X,
    typename Y
using Multiply = std::integral_constant<
    decltype(X::value * Y::value),
    X::value * Y::value

As we can see basic functionality such as a function respectively template to multiply a number by another is easily implemented in terms of an alias for a value type specialization, including automatic result type deduction using decltype. This also applies to higher order functionality which can be expressed only using other templates provided by the library such as the tav::Every list query.

// (define (every predicate list)
//         (fold (lambda (x y) (and x y))
//               #t
//               (map predicate list)))
template <
    template<typename> class Predicate,
    typename                 List
using Every = Fold<
    Map<Predicate, List>

If we ignore the need to explicitly declare the predicate as a template template parameter i.e. as a function this example is very simmilar to its Scheme equivalent. Concerning the function used to fold the list it is actually less verbose than the Scheme version of tav::Every as we can directly pass tav::And instead of wrapping it in a lambda expression as is required in Scheme 2.

Sadly the actual implementations of e.g. tav::Fold are not as easily expressed. The recursive nature of folding a list or even constructing its underlying pair based structure using the variadic tav::List requires partial template specializations which are not allowed for template aliases3.

template <
    template<typename, typename> class Function,
    typename                           Initial,
    typename                           Pair
struct fold_pair {
    typedef Function<
        Eval<fold_pair<Function, Initial, Cdr<Pair>>>
    > type;

template <
    template<typename, typename> class Function,
    typename                           Initial
struct fold_pair<Function, Initial, void> {
    typedef Initial type;

While the listing of tav::detail::fold_pair shows how the partial specialization of its tav::Pair parameter allows recursion until the list ends4, it also forces us to define its actual type in terms of a public type member typedef. Such a member type definition can lead to cluttering the code with typename keywords which is why TypeAsValue employs a simple tav::Eval template alias to hide all typename *::type constructs.

template <
    template<typename, typename> class Function,
    typename                           Initial,
    typename                           List
using Fold = Eval<detail::fold_pair<Function, Initial, List>>;

Hiding member type defintions behind template aliases enables most higher functionality and applications built using its functions to be written in a reasonably minimalistic - Scheme like - fashion as we can see in e.g. the listing of tav::Every. This also explains why tav::Pair recursively defines its own type, as we would otherwise have to be quite careful where to resolve it.


Not all programs are sensibly expressed as a nested chain of function calls if we want to reuse some values, separate functionality into appropriately named functions or hide complexity via private bindings. While in Scheme one would use let and the like for this purpose, such functionality is not easily replicated in the context of template metaprogramming. This is why TypeAsValue uses an alternate and more C++ like solution to this problem by simply making use of the standard object oriented aspects of the language.

// (define (quick_sort comparator sequence)
//   (if (null-list? sequence)
//     (list)
//     (let* ([index      (quotient (length sequence) 2)]
//            [pivot      (nth index sequence)]
//            [partitions (partition (lambda (x) (comparator pivot x))
//                                   (delete-nth index sequence))]
//            [lhs        (car partitions)]
//            [rhs        (cdr partitions)])
//       (concatenate (list (quick_sort comparator lhs)
//                          (list pivot)
//                          (quick_sort comparator rhs))))))
template <
    template<typename, typename> class Comparator,
    typename                           Sequence
class quick_sort {
        using index = Divide<Length<Sequence>, Size<2>>;
        using pivot = Nth<index, Sequence>;

        using partitions = Partition<
            Apply<Comparator, pivot, _0>::template function,
            DeleteNth<index, Sequence>

        using lhs = Car<partitions>;
        using rhs = Cdr<partitions>;

        using type = Concatenate<
            Eval<quick_sort<Comparator, lhs>>,
            Eval<quick_sort<Comparator, rhs>>

template <template<typename, typename> class Comparator>
struct quick_sort<Comparator, void> {
    typedef void type;

Above we can see the complete listing of the Quick Sort implementation employed by the tav::Sort template alias. At first glance this looks rather different from the corresponding Scheme program but if we look closer one could argue that it is a equivalent of a let binding where the private section of tav::detail::quick_sort contains the bound constants and the public section contains the body.

Partial function application

While partial application of the comparator function in the Scheme Quick Sort implementation is achieved via anonymous functions I have not as of yet thought of a good way to implement full support for lambda expressions in TypeAsValue. This is why there is currently only support for single level partial function evaluation using tav::Apply.

To put it simply tav::Apply implements a std::bind analog for usage in a template metaprogramming context. This means that it returns a new template for a given template including all its arguments where some or all of those arguments may be placeholders. In this context a placeholder is a specialization of the tav::detail::placeholder template on an arbitrary integer value that represents the index of the argument to the returned template the placeholder should resolve to.

// (define result
//         (map (lambda (x) (+ 2 x))
//              (list 1 2 3)))
// => (list 3 4 5)
using result = tav::Map<
    tav::Apply<tav::Add, tav::Int<2>, tav::_0>::template function,
    tav::List<tav::Int<1>, tav::Int<2>, tav::Int<3>>

Note that tav::_0 is just an alias to tav::detail::placeholder<0>. The tav::Apply template automatically selects a matching implementation as its base class depending on the count of placeholder arguments using tav::Cond. If there are more than two placeholder arguments it returns a generic variadic template as its function alias whereas there is a explicit version for one, two or zero placeholders. The zero placeholder variant of tav::Apply is useful if function application has to be deferred as is required for e.g. the handling of invalid values.

As we can see this kind of partial function application obviously is no full replacement for actual template based lambda expressions but it serves as a good enough solution until a nice approach to enabling placeholders as arguments in nested templates can be developed. Alternatively one could argue that the scenarios where one could use lambda expressions are better served by defining local templates and bindings as described in the previous section to e.g. increase readability by actually naming functions.


While the above listings and explanations should provide a basic overview about what I mean by using Scheme as a metaphor for template metaprogramming, some actual examples of how TypeAsValue may be used to perform some concrete compile time computations are certainly useful.

For this reason the library includes two example applications, namely a implementation of the Sieve of Eratosthenes and a Turing machine 5 simulator. To further support the analogy between functional and template metaprogramming both of these examples as well as all test cases are documented using their respective Scheme equivalent.

Both of these examples are certainly not what TypeAsValue might be used for in practice. I imagine the most viable real word use case for template metaprogramming based compile time computation to be the implementation of complex template based abstractions. For example it could serve as a utility to express base class selections and complex assertions of static parameters in a easily readable and hopefully relatable manner.


While the Scheme metaphor for template metaprogramming in C++ certainly has its limits, especially in the area of anonymous functions, I think that this article as well as the actual implementation of TypeAsValue are proof that it holds up quite well in many circumstances. As stated in the introduction to this article I was very surprised how close template metaprogramming can feel to a real functional programming language.

All listings in this article as well as the TypeAsValue library itself are freely available under the terms of the MIT license on Github and cgit. Feel free to check them out and contribute - I am especially interested in practical solutions to providing better partial function application support or even full compile time lambda expressions that do not require the whole library to be designed around this concept.

Finally I want to reference the Boost MPL library which supports everything and more than the solution described in this article without relying on any modern language features such as variadic templates. All non-experimental projects should probably use it instead of TypeAsValue, not the least because of greater portability.

  1. ISO C++ Standard draft, N3797, § 3.9.1 Fundamental types, Section 7↩︎

  2. and needs to be wrapped in anonymous function in Scheme because it is not a function but a macro↩︎

  3. ISO C++ Standard draft, N3797, § 14.5 Template declarations, Section 3↩︎

  4. TypeAsValue currently represents Scheme’s empty list '() as void↩︎

  5. Previously proofed in C++ Templates are Turing Complete (2003)↩︎