To better understand the C++ type system, I have endeavored to write a pointer wrapper class which propagates constness similar to std::experimental::propagate_const:
template <typename Pointee> class Ptr {
public:
  Ptr() = delete;
  explicit Ptr(Pointee *);
  Ptr(const Ptr<Pointee> &) = delete;
  Ptr(Ptr<Pointee> &&);
  Ptr<Pointee> &operator=(const Ptr<Pointee> &) = delete;
  Ptr<Pointee> &operator=(Ptr<Pointee> &&);
  ~Ptr()  = default;
  const Pointee *operator->() const;
  Pointee *operator->();
  const Pointee &operator*() const;
  Pointee &operator*();
private:
  Pointee *mPtr;
};
The wrapper is intended to provide near raw pointer like behavior while also enforcing a kind of 'deep' const correctness and guarding against unintentional aliasing.
To this end, the copy constructor and copy assignment operator are deleted:
However, the above design has two unfortunate consequences.
const Ptr<int> allocateImmutableInt(int val) { return Ptr<int>(new int(val)); }
void foo() {
  Ptr<int> immutableInt = allocateImmutableInt(0); // Initializes non-const Ptr from const Ptr 
  *immutableInt = 100;  // Oops, changed value of 'immutable' object
}
The first problem can be partially solved by introducing a move ctor that accepts a const rvalue reference (although this feels somewhat strange and non-idiomatic):
  Ptr(const Ptr<Pointee> &&);
However, this actually worsens the second problem. Now a const Ptr can be move constructed into a non-const Ptr even without mandatory move/copy-elision. As far as I can tell, to get around this issue we would need a so called 'const constructor', ie a constructor which can only be invoked to produce a const object:
  Ptr(const Ptr<Pointee>&&) const;
Even if c++ supported such a constructor, the second problem would still remain, as c++17 specifically ignores cv-qualification and the viability of constructors when deciding if mandatory move/copy-elision can be applied when initializing an object. There does not currently appear to be a way to ask c++ to check if a copy/move would be viable before applying mandatory copy/move elision to object initialization.
As far as I can tell, std::experimental::propagate_const suffers from these same issues. I am wondering if I have encountered a fundamental limitation of c++, or if I am designing the Ptr wrapper incorrectly? I am aware that these issues can likely be eliminated by creating two types, a Ptr for non-const access and ConstPtr for const-only access. However, this defeats the purpose of creating a const-propagating wrapper in the first place.
Perhaps I have just stumbled upon the reason why both an iterator type and a const_iterator type exist.
You are looking at problems that do not really exist.
- To prevent non-const access of a const pointed-to object by copying a const Ptr into a non-const Ptr.
 
This should not be a goal, as it goes against the idea of propagating const.
There are two aspects to propagating const. First, when the pointer is const-qualified, the object is also const-qualified. This aspect you have covered. Second, when the pointer is not const-qualified, the object uses its natural qualification. That is, if you can copy a const Ptr into a non-const Ptr, then that change propagates to the object, potentially making the object also non-const. This is desired propagation, not something to prevent.
Keep in mind a major use case for const propagation: class members. Propagating const for a member pointer helps ensure const-correctness by making the pointed-to data const in const-qualified member functions. Your imagined problems are not applicable to this use case. Don't make the situation more complex than it needs to be.
I am aware that these issues can likely be eliminated by creating two types, a Ptr for non-const access and ConstPtr for const-only access.
This is not necessary. If the object is supposed to remain const even when the pointer is not const, then the type should be Ptr<const T> instead of Ptr<T>.
As an example, your "immutable int" should look more like the following.
Ptr<const int> allocateImmutableInt(int val) { return Ptr<const int>(new int(val)); }
    ^^^^^                                                 ^^^^^
The const has been moved to qualify the int, making the int immutable regardless of the const-qualification of the Ptr.
Furthermore, you might note that new int(val) returns an int* that gets implicitly converted to const int* for your constructor.  You might want to replicate this implicit conversion for Ptr. A constructor like Ptr(Ptr<std::remove_const<Pointee>> &&) would do the trick, as long as it is defined only when Pointee is const-qualified (to avoid having a conflict with the regular move constructor).
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