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The C++ Source
A Deeper Look at Metafunctions
by David Abrahams and Aleksey Gurtovoy
August 23, 2004

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3.5  Lambda Details

Now that you have an idea of the semantics of MPL's lambda facility, let's formalize that understanding and look at things a little more deeply.

3.5.1  Placeholders

The definition of "placeholder" may surprise you:


A placeholder is a metafunction class of the form mpl::arg<X>.  Implementation

The convenient names _1, _2,... _5 are actually typedefs for specializations of mpl::arg that simply select the Nth argument for any N. [6] The implementation of placeholders looks something like this:

namespace boost { namespace mpl { namespace placeholders {

template <int N> struct arg; // forward declarations
struct void_;

template <>
struct arg<1>
    template <
      class A1, class A2 = void_, ... class Am = void_>
    struct apply
        typedef A1 type; // return the first argument
typedef arg<1> _1;

template <>
struct arg<2>
    template <
      class A1, class A2, class A3 = void_, ...class Am = void_
    struct apply
        typedef A2 type; // return the second argument
typedef arg<2> _2;

more specializations and typedefs...


Remember that invoking a metafunction class is the same as invoking its nested apply metafunction. When a placeholder in a lambda expression is evaluated, it is invoked on the expression's actual arguments, returning just one of them. The results are then substituted back into the lambda expression and the evaluation process continues.  The Unnamed Placeholder

There's one special placeholder, known as the unnamed placeholder, that we haven't yet defined:

namespace boost { namespace mpl { namespace placeholders {

typedef arg<-1> _; // the unnamed placeholder


The details of its implementation aren't important; all you really need to know about the unnamed placeholder is that it gets special treatment. When a lambda expression is being transformed into a metafunction class by mpl::lambda,

the nth appearance of the unnamed placeholder in a given template specialization is replaced with _n.

So, for example, every row of Table 3.1 below contains two equivalent lambda expressions.

Unnamed Placeholder Semantics
  , boost::add_pointer<_>
  , boost::add_pointer<_1>
 , mpl::minus<_,_>
 , mpl::minus<_1,_2>

Especially when used in simple lambda expressions, the unnamed placeholder often eliminates just enough syntactic "noise" to significantly improve readability.

3.5.2  Placeholder Expression Definition

Now that you know just what placeholder means, we can define placeholder expression:


A placeholder expression is either:

  • a placeholder


  • a template specialization with at least one argument that is a placeholder expression.

In other words, a placeholder expression always involves a placeholder.

3.5.3  Lambda and Non-Metafunction Templates

There is just one detail of placeholder expressions that we haven't discussed yet. MPL uses a special rule to make it easier to integrate ordinary templates into metaprograms: After all of the placeholders have been replaced with actual arguments, if the resulting template specialization X doesn't have a nested ::type, the result is just X itself.

For example, mpl::apply<std::vector<_>, T> is always just std::vector<T>. If it weren't for this behavior, we would have to build trivial metafunctions to create ordinary template specializations in lambda expressions:

// trivial std::vector generator
template<class U> 
struct make_vector { typedef std::vector<U> type; };

typedef mpl::apply<make_vector<_>, T>::type vector_of_t;

Instead, we can simply write:

typedef mpl::apply<std::vector<_>, T>::type vector_of_t;

3.5.4  The Importance of Being Lazy

Recall the definition of always_int from the previous chapter:

struct always_int
    typedef int type;

Nullary metafunctions might not seem very important at first, since something like add_pointer<int> could be replaced by int* in any lambda expression where it appears. Not all nullary metafunctions are that simple, though:

struct add_pointer_f
    template <class T>
    struct apply : boost::add_pointer<T> {};
typedef mpl::vector<int, char*, double&> seq;
typedef mpl::transform<seq, add_pointer_f> calc_ptr_seq;

Note that calc_ptr_seq is a nullary metafunction, since it has transform's nested ::type. A C++ template is not instantiated until we actually "look inside it," though. Just naming calc_ptr_seq does not cause it to be evaluated, since we haven't accessed its ::type yet.

Metafunctions can be invoked lazily, rather than immediately upon supplying all of their arguments. We can use lazy evaluation to improve compilation time when a metafunction result is only going to be used conditionally. We can sometimes also avoid contorting program structure by naming an invalid computation without actually performing it. That's what we've done with calc_ptr_seq above, since you can't legally form double&*. Laziness and all of its virtues will be a recurring theme throughout this book.

3.6  Details

By now you should have a fairly complete view of the fundamental concepts and language of both template metaprogramming in general and of the Boost Metaprogramming Library. This section reviews the highlights.

Metafunction forwarding.
The technique of using public derivation to supply the nested type of a metafunction by accessing the one provided by its base class.
Metafunction class.
The most basic way to formulate a compile-time function so that it can be treated as polymorphic metadata; that is, as a type. A metafunction class is a class with a nested metafunction called apply.
Most of this book's examples will use the Boost Metaprogramming Library. Like the Boost type traits headers, MPL headers follow a simple convention:

#include <boost/mpl/component-name.hpp>

If the component's name ends in an underscore, however, the corresponding MPL header name does not include the trailing underscore. For example, mpl::bool_ can be found in <boost/mpl/bool.hpp>. Where the library deviates from this convention, we'll be sure to point it out to you.

Higher-order function.
A function that operates on or returns a function. Making metafunctions polymorphic with other metadata is a key ingredient in higher-order metaprogramming.
Lambda expression.
Simply put, a lambda expression is callable metadata. Without some form of callable metadata, higher-order metafunctions would be impossible. Lambda expressions have two basic forms: metafunction classes and placeholder expressions.
Placeholder expression.

A kind of lambda expression that, through the use of placeholders, enables in-place partial metafunction application and metafunction composition. As you will see throughout this book, these features give us the truly amazing ability to build up almost any kind of complex type computation from more primitive metafunctions, right at its point of use:

// find the position of a type x in some_sequence such that:
//         x is convertible to 'int'
//      && x is not 'char'
//      && x is not a floating type
typedef mpl::find_if<
    , mpl::and_<
        , mpl::not_<boost::is_same<_1,char> >
        , mpl::not_<boost::is_float<_1> >
    >::type iter;

Placeholder expressions make good on the promise of algorithm reuse without forcing us to write new metafunction classes. The corresponding capability is often sorely missed in the runtime world of the STL, since it is often much easier to write a loop by hand than it is to use standard algorithms, despite their correctness and efficiency advantages.

The lambda metafunction.
A metafunction that transforms a lambda expression into a corresponding metafunction class. For detailed information on lambda and the lambda evaluation process, please see the MPL reference manual.
The apply metafunction.
A metafunction that invokes its first argument, which must be a lambda expression, on its remaining arguments. In general, to invoke a lambda expression, you should always pass it to mpl::apply along with the arguments you want to apply it to in lieu of using lambda and invoking the result "manually."
Lazy evaluation.
A strategy of delaying computation until its result is required, thereby avoiding any unneccessary computation and any associated unneccessary errors. Metafunctions are only invoked when we access their nested ::types, so we can supply all of their arguments without performing any computation and delay evaluation to the last possible moment.


  1. Divisors just contribute negative exponents, since 1/x = x-1.
  2. In case you're wondering, the same approach could have been applied to plus_f, but since it's a little subtle, we introduced the straightforward but verbose formulation first.
  3. Users of EDG-based compilers should consult Appendix C for a caveat about metafunction forwarding. You can tell whether you have an EDG compiler by checking the preprocessor symbol __EDG_VERSION__, which is defined by all EDG-based compilers.
  4. See for an in-depth treatment, including a reference to Church's paper proving that the equivalence of lambda expressions is in general not decidable.
  5. See the Configuration Macros section of the MPL reference manual for a description of how to change the maximum number of arguments handled by mpl::apply.
  6. MPL provides five placeholders by default. See the Configuration Macros section of MPL reference manual for a description of how to change the number of placeholders provided.

Talk Back!

Discuss this article in the Articles Forum topic, A Deeper Look at Metafunctions.


David Abrahams and Aleksey Gurtovoy are the authors of Template Metaprogramming, which can be pre-ordered from on at:

About the Authors

David Abrahams is a leading authority on C++ software development. His company, Boost Consulting (, provides support and development services for the open-source Boost C++ libraries, and delivers professional training in the practice of software construction. As a founding member of Boost (, David has worked with some of the best developers from all over the world in designing and building widely-used, reliable, maintainable software components. David has been an ANSI/ISO C++ committee member since 1996, where he is best known for developing the theory, and implementation of exception safety in the C++ standard.

Aleksey Gurtovoy is a technical lead at MetaCommunications, Inc, and a contributing member of the C++ Boost community. He holds a MS degree in Computer Science from Krasnoyarsk Technical State University, Russia. He can be reached at

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