Concepts can’t do quantifiers

People frequently ask how to do things like this in C++20:

• Make a concept Renderer that is satisfied if and only if t.render(u) is valid for all Printable types U. (From /r/cpp.)

• Make a concept ValidSize that is satisfied if and only if t is convertible to some integral type U. (From the standard.)

The key words above are “all” and “some.” In mathematics, these are known as the universal and existential quantifiers:

• Renderer<T> ⇔ ∀UPrintable: t.render(u) is well-formed

• ValidSize<T> ⇔ ∃UIntegral: U u = t; is well-formed

C++20 Concepts do not support either quantifier.

Universal quantification

Consider this code:

struct Apple {};

template<class T>
concept Pommivorous = requires(T t, Apple a) {
{ t.eat(a) };
};

struct Horse {
void eat(Apple);
};
static_assert(Pommivorous<Horse>);

struct Goat {
template<class T>
void eat(T);
};
static_assert(Pommivorous<Goat>);

Pommivorous says pretty much everything there is to say about Horse. But Goat is more interesting. A Goat can eat literally anything. Can we make a concept Omnivorous that is satisfied by classes like Goat that can eat anything (but not by classes like Horse)?

No, we cannot, because the notion of “eat anything” involves a universal quantifier! The best we can do is fake it, by doing something like this:

template<class T>
concept Omnivorous = requires (T t, Apple a,
Banana b, int c, std::string d) {
{ t.eat(a) };
{ t.eat(b) };
{ t.eat(c) };
{ t.eat(d) };
};

Any animal that can eat apples, bananas, ints, and strings is probably omnivorous, right? Sure. But we haven’t actually expressed a universal quantifier in this code. We’ve just listed off a finite set of special cases. The C++ compiler fundamentally cannot deal with the abstract idea of “any class at all.” C++ can deal only in concrete examples, and only a finite number of them. It can’t help us test a proposition for all possible classes.

By the way, the same is true for any infinite or open set of types. We can’t make a concept Omnivorous that means “eats everything ever,” and equally, we can’t make a concept that means “eats every type that satisfies Vegetable,” nor can we make one that means “eats all pointer types.” For a closed set, such as “eats all integral types,” we can do it, but only by listing out and testing every member of the set, one by one. (C++20 has exactly 16 integral types.)

But we can test for “all possible values”?

“Wait a minute,” you might say. “Consider this code…”

template<class T>
concept Munchicus = requires(T t, int i) {
{ t.eat(i) };
};

struct Digitus {
void eat(int);
};
static_assert(Munchicus<Digitus>);

“Doesn’t this code correctly convey the meaning that a Digitus can eat any int value? That is, t.eat(1), t.eat(2), t.eat(3),… These are all well-formed, according to concept Munchicus. So how can you say that C++ can’t deal with universal quantification?”

That’s a good point! Leaving aside some pedantic quibbles,[1] what you say is true: C++ can deal with universal quantification in some areas. Specifically, C++ believes that what’s true for one int (say, i) is going to hold true for all ints (say, 1, 2, 3,…). The whole idea of type-checking an expression is based on this fundamental intuition that one value of type int behaves pretty much the same as any other. In fact, this is pretty much what we mean when we talk about “type” in any programming language! We can let the expression i stand in for “1, or 2, or 3,…” precisely because all these expressions share the same bundle of behaviors, a.k.a. the same type.

C++’s static type system is essentially a vocabulary for talking about the ways in which one int “is like” any other int.

C++ has no vocabulary for talking about the ways in which one whole type “is like” any other whole type. Type-checking permits (in fact, requires) us to make sure our value-manipulating code preserves certain properties universally quantified over all possible values of a given type. But it’s difficult even to imagine a mechanism that would ensure our type-manipulating code preserves certain properties universally quantified over all possible types!

One possible analogue of type-checking is known as “definition checking.” See “Concept definition-checking and its woes” (2019-07-22).

To put it another way: The part of any programming language that can deal with universal quantifiers is the part that we refer to as the “type system.” Concepts lie essentially outside the bounds of C++’s type system. One easy way to tell: We don’t refer to concepts as “types.” :)

Pedantic quibbles

The pedantic quibbles mentioned above include that value categories are tricky:

struct Timidus {
void eat(int&);
};
static_assert(Munchicus<Timidus>);  // yet t.eat(42) is ill-formed

and that the expression 0 doesn’t always mean 0:

struct Assistus {
void eat(long);
void eat(void*);
};
static_assert(Munchicus<Assistus>);  // yet t.eat(0) is ill-formed

These specific quirks of C++ are unrelated to the point of this blog post, which is that concepts can’t quantify over types in general.

Existential quantification

Consider this code:

struct Apple {};
struct Carrot {};

struct Horse {
void eat(Apple);
void eat(Carrot);
};

struct Goat {
template<class T>
void eat(T);
};

struct Tree {
void photosynthesize();
};

Can we make a concept Eater that is satisfied by classes like Goat and Horse, but not by Tree? Essentially, we want to match any class that can eat something (but without specifying up front what that something is).

No, we cannot, because the notion of “eat something” involves an existential quantifier! It’s equivalent to saying: “Class T satisfies Eater if-and-only-if there exists some class U such that T eats U.”

There’s not even any half-decent way to fake the existential quantifier. It reminds me of that one Richard Dawkins quote:

We are all atheists about most of the gods that humanity has ever believed in. Some of us just go one god further.

If someone claims to believe in all the gods, you can ask them if they believe in ten or twenty specific gods you happen to know about, and if they say yes to all of the above, then that’s at least some evidence that their claim might be correct. But if somebody claims only to believe in some god, you might ask them about twenty specific gods — Zeus, Brahma, Odin, Marduk, Bes, Jehovah — and their answer might be “no” every single time. Distinguishing a monotheist from an atheist is surprisingly difficult!

In practice, the best way to fake this up is to require the purported theist to tell you the name of (one of) their god(s). In our Eater example, we might write:

template<class T>
concept Eater = requires(T t, T::food_type u) {
{ t.eat(u) };
};

struct Horse {
using food_type = Apple;
void eat(Apple);
void eat(Carrot);
};

struct Fox {
void eat(Chicken);
};

Under this regime, Horse is considered an Eater because it exposes a member typedef food_type and a member function eat that takes that type as a parameter. Fox has an eat method, but is not considered an Eater, because it does not expose its food_type.

In this example, the fact that Horse can also eat Carrot is effectively irrelevant. But if you imagine that Apple is std::string and Carrot is const char* (or std::string_view), you can maybe see how such an overload set could be useful in practice.

Footnote on SFINAE-friendliness

Both in the universal and in the existential case, beware of the fact that not all failure modes are SFINAE-friendly. Godbolt:

struct Human {
void eat(Apple);
template<class T=void>
auto eat(Glass) { T t; }
};

Even though Human is capable of eating something (namely Apple), it is problematic to test this by feeding arbitrary items to the Human, because merely testing the well-formedness of t.eat(Glass{}) for Human t will trigger a hard error.

This caveat applies to both the existential quantifier and the universal quantifier; it is dangerous to evaluate Eats<Human, Glass> regardless of whether you plan to || the result or && it.

Techniques for faking quantifiers

C++ Concepts don’t support universal or existential quantifiers. There is no hack I can teach you that will change this fundamental fact. But here are some tricks you can use to change the question you’re asking from one that Concepts can’t solve, to one that it can.

If your problem is of the form “T holds relationship R with all types” or “T can be R’ed with any type,” it likely involves a universal quantifier. Try to recast it using phrases like “T holds relationship R with int (or any specific, concrete type)” or “T can be R’ed with any type on this concrete list.”

If your problem is of the form “T holds relationship R with some type” or “there exists some type U such that T can be R’ed with U,” it likely involves an existential quantifier. Try to recast it in a form such as “T provides a member typedef T::foo_type, and T can be R’ed with T::foo_type.”

The following two mathematical identities are sometimes known as De Morgan’s laws for quantifiers.

• $$\neg\exists x\in X:P(x)$$ is equivalent to $$\forall x\in X:\neg P(x)$$

• $$\neg\forall x\in X:Q(x)$$ is equivalent to $$\exists x\in X:\neg Q(x)$$

If your problem is of the form “T does not hold relationship R with any other type” or “there is no type such that…,” it likely involves a universal quantifier in disguise. First use the mathematical identities above to rewrite it in positive form (e.g. “T holds relationship not-R with all other types”), and then recast it by applying the previous techniques (e.g. “T holds relationship not-R with int”).

Posted 2020-08-10