Isn’t the template argument (the signature) of std::function part of its type?

The problem is that both function<int()> and function<int(int)> are constructible from the same function. This is what the constructor declaration of std::function looks like in VS2010:

template<class _Fx>
function(_Fx _Func, typename _Not_integral<!_Is_integral<_Fx>::value, int>::_Type = 0);

Ignoring the SFINAE part, it is constructible from pretty much anything.
std::/boost::function employ a technique called type erasure, to allow arbitary objects/functions to be passed in, so long they satisfy the signature when being called. One drawback from that is, that you get an error in the deepest part of the implementation (where the saved function is being called) when supplying an object which can’t be called like the signature wants it to, instead of in the constructor.

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The problem can be illustrated with this little class:

template<class Signature>
class myfunc{
public:
    template<class Func>
    myfunc(Func a_func){
        // ...
    }
};

Now, when the compiler searches for valid functions for the overload set, it tries to convert the arguments if no perfect fitting function exists. The conversion can happen through the constructor of the parameter of the function, or through a conversion operator of the argument given to the function. In our case, it’s the former.
The compiler tries the first overload of a. To make it viable, it needs to make a conversion. To convert a int(*)() to a myfunc<int()>, it tries the constructor of myfunc. Being a template that takes anything, the conversion naturally succeeds.
Now it tries the same with the second overload. The constructor still being the same and still taking anything given to it, the conversion works too.
Being left with 2 functions in the overload set, the compiler is a sad panda and doesn’t know what to do, so it simply says the call is ambigious.

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So in the end, the Signature part of the template does belong to the type when making declarations/definitions, but doesn’t when you want to construct an object.

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Edit:
With all my attention on answering the title-question, I totally forgot about your second question. 🙁

Can I circumvent it or will I have to keep the (annoying) explicit casts?

Afaik, you have 3 options.

• Keep the cast

• Make a function object of the appropriate type and pass that

  function<int()> fx = x;
function<int(int)> fy = y;
a(fx);
a(fy);

• Hide the tedious casting in a function and use TMP to get the right signature

The TMP (template metaprogramming) version is quite verbose and with boilerplate code, but it hides the casting from the client. An example version can be found here, which relies on the get_signature metafunction that is partially specialized on function pointer types (and provides a nice example how pattern matching can work in C++):

template<class F>
struct get_signature;

template<class R>
struct get_signature<R(*)()>{
  typedef R type();
};

template<class R, class A1>
struct get_signature<R(*)(A1)>{
  typedef R type(A1);
};

Of course, this needs to be extended for the number of arguments you want to support, but that is done once and then buried in a "get_signature.h" header. 🙂

Another option I consider but immediatly discarded was SFINAE, which would introduce even more boilerplate code than the TMP version.

So, yeah, that are the options that I know of. Hope one of them works for you. 🙂