1. Brief History
This paper started as a proposal by David Krauss, N4543[1], from 2015 and there has been an open issue in the LEWG bugzilla requesting such a facility since 2014[2].
Since then, the paper has gone through 4 revisions and has been considered in small groups in LEWG multiple times. Gradual feedback has led to the conservative proposal seen here. The mostrecent draft prior to this was a latepaper written and presented by Ryan McDougall in LEWGI in San Diego[3]. It included multiple references to implementations of moveonly functions and made a strong case for the importance of a moveonly form of std::function .
Feebdack given was encouragement for targetting C++20.
An updated version of that paper was presented on Saturday at the end of the San Diego meeting. Poll results from that presentation are presented after the overview in this document.
The revision was presented in Kona, receiving additional polls and feedback, and was to be forwarded to LWG pending updates reflecting additional poll results. Those changes have been applied to the wording in this paper. Polls from the LEWG Kona review are also provided in this document.
2. Overview
This conservative any_invocable is intended to be the same as std::function , with the exceptions of the following:

It is moveonly.

It does not have the constcorrectness bug of std::function detailed in n4348.[4]

It provides support for cv/ref/noexcept qualified function types.

It does not have the target_type and target accessors (direction requested by users and implementors).

Invocation has strong preconditions.
3. Specification
The following is relative to N4820.[5]
Add <any_invocable> to [tab:headers.cpp] — C++ library headers
Add the following entry to [tab:support.ft]:
Macro Name  Value  Header(s) 

__cpp_lib_any_invocable  xxxxxxL  <any_invocable> 
Make it so that <functional> is guaranteed to include <any_invocable> by adding the following at the start of the synopsis of [functional.syn]:
20.14.1 Header <functional> synopsis [functional.syn] #include <any_invocable> // see [inv.syn] namespace std {
Insert the following section at the end of Function Objects [function.objects], where SECTION is a placeholder for the root of the section numbering:
SECTION Storage for any callable [inv] 1 This subclause describes components that C++ programs may use to perform operations on callable objects of a discriminated type. SECTION.1 Header <any_invocable> synopsis [inv.syn] 1 The header <any_invocable> provides partial specializations of any_invocable for each combination of the possible replacements of the placeholders cv, ref, and noex where: (1.1) — cv is either const or empty. (1.2) — ref is either &, &&, or empty. (1.3) — noex is either true or false. 2 For each of the possible combinations of the placeholders mentioned above, there is a placeholder invquals defined as follows: (2.1) — If ref is empty, let invquals be cv& (2.2) — otherwise, let invquals be cv ref. namespace std { template<class Sig> class any_invocable; // never defined template<class R, class... ArgTypes> class any_invocable<R(ArgTypes...) cv ref noexcept(noex)>; } SECTION.2 Class <any_invocable> [inv.class] namespace std { template<class Sig> class any_invocable; // never defined template<class R, class... ArgTypes> class any_invocable<R(ArgTypes...) cv ref noexcept(noex)> { public: using result_type = R; // SECTION.3, construct/copy/destroy any_invocable() noexcept; any_invocable(nullptr_t) noexcept; any_invocable(any_invocable&&) noexcept; template<class F> any_invocable(F&&); template<class T, class... Args> explicit any_invocable(in_place_type_t<T>, Args&&...); template<class T, class U, class... Args> explicit any_invocable(in_place_type_t<T>, initializer_list<U>, Args&&...); any_invocable& operator=(any_invocable&&); any_invocable& operator=(nullptr_t) noexcept; template<class F> any_invocable& operator=(F&&); template<class F> any_invocable& operator=(reference_wrapper<F>) noexcept; ~any_invocable(); // SECTION.4, any_invocable modifiers void swap(any_invocable&) noexcept; // SECTION.5, any_invocable capacity explicit operator bool() const noexcept; // SECTION.6, any_invocable invocation R operator()(ArgTypes...) cv ref noexcept(noex); // SECTION.7, null pointer comparisons friend bool operator==(const any_invocable&, nullptr_t) noexcept; // SECTION.8, specialized algorithms friend void swap(any_invocable&, any_invocable&) noexcept; }; } 1 The any_invocable class template provides polymorphic wrappers that generalize the notion of a callable object (20.14.2). These wrappers can store, move, and call arbitrary callable objects (20.14.2), given a call signature (20.14.2), allowing functions to be firstclass objects. 2 Implementations are encouraged to avoid the use of dynamically allocated memory for a small contained value. However, any such smallobject optimization can only be applied to types T for which is_nothrow_move_constructible_v<T> is true. SECTION.3 Constructors and destructor [inv.con] any_invocable() noexcept; any_invocable(nullptr_t) noexcept; 1 Ensures: !*this is true. any_invocable(any_invocable&& f) noexcept; 2 Ensures: If !f is true, *this has no target; otherwise, the target of *this is equivalent to the target of f before the construction, and f is in a valid state with an unspecified value. template<class F> any_invocable(F&& f); 3 Let VT be decay_t<F>. 4 Constraints: (4.1) — remove_cvref_t<F> is not the same type as any_invocable, and (4.2) — remove_cvref_t<F> is not a specialization of in_place_type_t, and (4.3) — is_constructible_v<VT, F> is true, and (4.4) — is_invocable_r_v<R, VT invquals, ArgTypes...> is true, and (4.5) — !noex  is_nothrow_invocable_r_v<R, VT invquals, ArgTypes...> is true. 5 Expects: VT meets the Cpp17Destructible requirements, and if is_move_constructible_v<VT> is true, VT meets the Cpp17MoveConstructible requirements. 6 Ensures: !*this is true if any of the following hold: (6.1) — f is a null function pointer value, or (6.2) — f is a null member pointer value, or (6.3) — remove_cvref_t<F> is a specialization of the any_invocable class template, and !f is true. 7 Otherwise, *this targets an object of type VT initialized with std::forward<F>(f) by directnonlistinitialization. 8 Throws: Does not throw exceptions when VT is a function pointer or a reference_wrapper<T> for some T. Otherwise, may throw bad_alloc or any exception thrown by the expression VT(std::forward<F>(f)). template<class T, class... Args> explicit any_invocable(in_place_type_t<T>, Args&&... args); 9 Let VT be decay_t<T>. 10 Constraints: (10.1) — is_constructible_v<VT, Args...> is true, and (10.2) — is_invocable_r_v<R, VT invquals, ArgTypes...> is true, and (10.3) — !noex  is_nothrow_invocable_r_v<R, VT invquals, ArgTypes...> is true. 11 Mandates: VT is the same type as T. 12 Expects: VT meets the Cpp17Destructible requirements, and if is_move_constructible_v<VT> is true, VT meets the Cpp17MoveConstructible requirements. 13 Ensures: *this targets an object of type VT initialized with std::forward<Args>(args)... by directnonlistinitialization. template<class T, class U, class... Args> explicit any_invocable(in_place_type_t<T>, initializer_list<U> ilist, Args&&... args); 14 Let VT be decay_t<T>. 15 Constraints: (15.1) — is_constructible_v<VT, initializer_list<U>&, ArgTypes...> is true, and (15.2) — is_invocable_r_v<R, VT invquals, ArgTypes...> is true, and (15.3) — !noex  is_nothrow_invocable_r_v<R, VT invquals, ArgTypes...> is true. 16 Mandates: VT is the same type as T. 17 Expects: VT meets the Cpp17Destructible requirements, and if is_move_constructible_v<VT> is true, VT meets the Cpp17MoveConstructible requirements. 18 Ensures: *this targets an object of type VT initialized with ilist, std::forward<ArgTypes>(args)... by directnonlistinitialization. any_invocable& operator=(any_invocable&& f); 19 Effects: Replaces the target object of *this with the target object of f. 20 Ensures: If !f is true, *this has no target; otherwise, the target of *this is equivalent to the target of f before the construction, and f is in a valid state with an unspecified value. 21 Returns: *this. any_invocable& operator=(nullptr_t) noexcept; 22 Effects: If *this != nullptr is true, destroys the target object of this. 23 Ensures: !*this is true. 24 Returns: *this. template<class F> any_invocable& operator=(F&& f); 25 Effects: Equivalent to: any_invocable(std::forward<F>(f)).swap(*this); return *this; template<class F> any_invocable& operator=(reference_wrapper<F> f) noexcept; 26 Effects: Equivalent to: any_invocable(f).swap(*this); return *this; ~any_invocable(); 27 Effects: If *this != nullptr is true, destroys the target object of this. SECTION.4 Modifiers [inv.mod] void swap(any_invocable& other) noexcept; 1 Effects: Exchanges the targets of *this and other. SECTION.5 Capacity [inv.cap] explicit operator bool() const noexcept; 1 Returns: true if *this has a target, otherwise false. SECTION.6 Invocation [inv.inv] R operator()(ArgTypes... args) cv ref noexcept(noex); 1 Expects: bool(*this) is true. 2 Effects: Equivalent to: return INVOKE<R>(static_cast<F invquals>(f), std::forward<ArgTypes>(args)...);, where F is the type of the target object of *this and f is the target object of *this. SECTION.7 Null pointer comparison functions [inv.nullptr] bool operator==(const any_invocable& f, nullptr_t) noexcept; 1 Returns: !f. SECTION.8 Specialized algorithms [inv.alg] void swap(any_invocable& f1, any_invocable& f2) noexcept; 1 Effects: Equivalent to: f1.swap(f2).
4. Polls from LEWG San Diego Review (2018)
4.1. Support func(), func() const, func() &&
SF F N A SA 6 6 2 1 0
4.2. Support func() && only
SF F N A SA 2 2 7 1 1
4.3. Remove target/target_type
SF F N A SA 12 5 0 0 0
4.4. Require more stuff (noexcept, const&&, ...)
SF F N A SA 0 1 8 6 0
Note that the final poll (require more stuff) was not due to members being against the design, but because we could easily add those facilities in a later standard without any breakage.
4.5. Name Options
There was one final poll, which brought us to the name any_invocable .
3 unique_function 3 move_function 2 move_only_function 7 movable_function 8 mfunction 10 any_invocable 8 mofun 8 mofunc 0 callback 4 mvfunction 2 func 0 callable 2 any_function
5. Polls from LEWG Kona Review (2019)
5.1. We want to spend time on this now in order to include it in C++20
SF F N A SA 8 8 2 0 0
5.2. Add support for func() const& and func()&
SF F N A SA 0 8 7 0 0
5.3. Add support for func() noexcept (x all of the above)
SF F N A SA 2 12 2 0 0
5.4. Include the option for CTAD
SF F N A SA 0 1 5 9 0
5.5. Name: callable vs any invocable
SC C N AI SAI 3 2 3 5 6
5.6. any_invocable vs invocable
SAI AI N I SI 3 7 2 5 1
5.7. Header choice
7 < functional > 11 < any_invocable > 11 < invocable > 3 < () >
5.8. Can get std::function from <any_invocable>
SF F N A SA 0 1 4 4 7
5.9. Can get std::function from <invocable>
SF F N A SA 1 3 6 3 2
Decide on <any_invocable>. Unanimous for <functional> to pull it in, even if in its own header.
5.10. Remove the nullcheck in the call operator and throwing of bad_function_call
SF F N A SA 8 2 1 0 0
5.11. Remove the nullcheck in constructors that are not nullptr_t
std :: any_callable ac = my_ptr_object ; if ( ac ) { /* true even if my_ptr is nullptr */ }
SF F N A SA 0 2 2 4 3
5.12. Perfect forwarding for converting constructor instead of byvalue
Unanimous
5.13. Forward to LWG for C++20
SF F N A SA 8 5 0 0 0
6. Implementation Experience
There are many implementations of a moveonly std::function with a design that is similar to this. What is presented is a conservative subset of those implementations. The changes suggested in LEWG, though minimal, have not been used in a large codebase.
Previous revisions of this paper have included publicly accessible moveonly function implementations, notably including implementations in HPX, Folly, and LLVM.
7. Acknowledgments
Thanks to Tomasz Kamiński, Tim Song, and Nevin Liber for suggestions on wording simplifications.
8. References
[1]: David Krauss: N4543 "A polymorphic wrapper for all Callable objects" http://www.openstd.org/jtc1/sc22/wg21/docs/papers/2015/n4543.pdf
[2]: Geoffrey Romer: "Bug 34  Need typeerased wrappers for moveonly callable objects" https://issues.isocpp.org/show_bug.cgi?id=34
[3]: Ryan McDougall: P0288R2 "The Need for std::unique_function" https://wg21.link/p0288r2
[4]: Geoffrey Romer: N4348 "Making std::function safe for concurrency" www.openstd.org/jtc1/sc22/wg21/docs/papers/2015/n4348.html
[5]: Richard Smith: N4820 "Working Draft, Standard for Programming Language C++" http://www.openstd.org/jtc1/sc22/wg21/docs/papers/2019/n4820.pdf