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2. Regular types and other concepts

We don’t know how to program yet

The field grew very fast. We never had time to stop and think. It used to be that people who were computer scientists were programmers. When Dijkstra1 gave his Turing award speech he called himself a “humble programmer”. When Ken Thompson, in his lecture “Reflection on Trusting Trust”, talks of himself he says, “I am a programmer”.

Then programmers disappeared. Everybody became a computer scientist or an architect. People say, “no I’m not a programmer, I’m an architect”. I guess they build buildings like Frank Lloyd Wright (joke). What it means is they don’t have any idea how to write code. (Clearly, I just insulted several communities).

A couple of years ago I was at a party with Mike Burrows who did the original AltaVista search engine and is now at Google. He told me, he cannot imagine how the guys at Google write anything at all. So, I’m not even going to mention Yahoo (joke). I did interview at Facebook and let me tell you what I saw of that place… That was five years ago. Maybe they improved but I don’t think so. Places tend to to decline.

You might say: “Who are you to tell us?” I am as guilty as any of you. Given the chance, I write very bad code. One thing which prevented me from being like everybody else was that some of my code had to go through thousands of people looking at it and telling me what they think about it. That is very helpful.

From early on I actually had this idea that programming is a very great discipline. I don’t mean that I’m very great, or anything like that. It’s that the discipline is a great discipline. I always wanted to learn to write ultimate code and I still think it’s possible. I disagree with many of the luminaries in the field who claim it’s an art, meaning that there are some gifted people like Mozart, and others who are just like us. I actually claim no, it’s a discipline.

This is why I want to discuss everything I write with you. If we together agree on it and we start practicing it maybe it will turn into a discipline. You say, “well, Alex what chance is there? There is the world and there is you”. Yes, there are no chances but you still have to do what you believe is right.

Well, plus they pay me (joke).

The motivation for concepts

Remember what I said about these awful “maps and maps of sets” and things like that? The fact that they actually work is kind of amazing2. It’s a testament to how flexible they are. You might think you can just put any type in a map, but actually there are requirements on the type, certain properties which are required for these containers to function.

Let, us think about what my task was when I started working on the C++ STL. It was to define standard data structures which will work for any reasonable subset of types. What is a reasonable subset of types? This is a little bit tricky but it’s of paramount importance.

My duty was to discover what will make a type work with any standard container. I knew that for sure, no matter what I do, there is one type which should work in every container: int. So, a regular type has to be like int, but not completely like it because it should work for double. Pointers are perhaps the most important type, even more important than int. It has to work for pointer.

To understand all this, we’re going to become a little bit theoretical. None of this stuff actually works unless you understand at least a little bit of theory. We will call such reasonable types Regular. What we will do is formally define a set of operators that all Regular types must have, along with a set of requirements on those operators3. The goal is that Regular types (those that obey the requirements) will behave sufficiently like int, double, and the rest. Such a definition is referred to as a concept, which we will talk about more later.

Once we understand Regular then we will understand what algorithms are allowed to do; use the operations which are defined on Regular types. Whatever is a natural idiomatic expression in C should be a natural idiomatic expression for Regular types.

Closure property for containers

Furthermore, I realized that I have to make my own containers be like built-in types. I had to close the loop. I had to provide a bunch of type constructors. A vector is a type constructor. It takes the type T and constructs a type out of type T. Out of type T you get type vector<T>. It’s a different type.

We want these constructors to be closed4. If you start with a regular type and keep applying these constructors you remain Regular.

Semiregular types

Leading up to Regular we will define Semiregular which is a bit weaker. All Semiregular types must have the following operations:

Copy constructor

We should be able to write:

T a(b); 

Or equivalently

T a = b;

They are not the same in general but they are the same if b is of type T.

How do we define the semantics of this operation? Could we use an equivalence relation? Consider the relation that always returns true. It is an equivalence relation. In other words, the relation R(a, b) = t, satisfies:

But, this is a wrong equivalence relation. We actually want something way stronger. We want equality. Given a notion of equality, we can define some axioms for our copy constructor.

Axiom: After a is copy constructed from b, we have a == b. Whatever our meaning of equality.

Let’s think about what a copy is. It is something which is equal to the original, but not identical to it.

Axiom: After a is copy constructed from b they have distinct identity markers. In C++ the identity marker is usually address: &a != &n.

All copy constructors must behave this way. If somebody clever comes and says, “oh we’re going to have a semantics where we’re going to have this shared thing”5. Will it work? No. Copy has to construct a different thing.

Assignment operator

T a; a = b;

Axiom: Constructing and assignment are equivalent: T a(b) <=> T a; a = b.

So, in order to for these operations to have correct semantics, they have to have equality defined.

Destructor

~T();

You don’t call destructors, so nothing specifically to say here.

Regular types

The concept Regular extends Semiregular with equality operators which are == and !=.

Equality operator

As we said before, we should define == so that after constructing a copy, the original and the copy are equal.

!= should always behave like: !(a == b). My very strong point is that the semantics of inequality (!=) is absolutely and totally bound to the semantics of equality (==). You should not even be able to have a situation where they have different semantics. But, the standards committee disagrees with me on that. They say that you could have equality be equality and inequality be multiplication operator. I think it’s a very bad idea and a good idea depending on what you want (joking). It will provide you with job security. Because nobody will ever figure out why your code works or doesn’t work.

Many of these operations will be member functions. But what about equality? No, it shouldn’t. Because fundamentally equal is a symmetric function. It compares two things. So even the paradigm of a member is the wrong paradigm. They are symmetric.

Total orderings

The concept TotallyOrdered extends Regular by adding a comparison operator <. This is actually something that Bjarne6 disagrees with me on. I say that <, the total ordering of the type is fundamental, and should be required. He says it’s too much to ask, and Regular types should be less strict. However we will follow his and the standard C++ convention by using a weaker Regular and limiting the requirement of ordering to TotallyOrdered.

Less than operator

< must obey the following mathematical properties:

Axiom 1: Anti-reflexive: !(a < a)

Axiom 2: Transitive: If a < b and b < c then a < c.

Axiom 3: Anti-symmetric: If a < b then !(b < a).

Axiom 4: If a != b then a < b or b > a.

In the same way that the semantics of == is related to !=, we can see the semantics of < must be totally bound to the semantics of equality. But furthermore, < is related to four other operations which must also be defined:

  1. < less than
  2. > greater than
  3. <= less than or equal to
  4. >= greater than or equal to

If you provide them then they have to have natural meaning. For example, !(a < b) should be a >= b otherwise the world perishes.

Later we will talk more about orderings, and define several different kinds.

In the next lesson we will become familiar with these concepts by implementing a class which satisfies each of them.


  1. Dijkstra is also a fantastic programmer to learn from. Instead of focusing on academic publishing the majority of his work was typewritten or handwritten and passed around by mailing list. Consequently, it is very readable, and applicable, free from jargon or obscure technicalities. You can find an archive of these here.
  2. Alex: They’re not Turing complete, but they’re Stepanov complete (joke).
  3. Alex: Some might think regular types are a form of equational reasoning. However, you can have a universe that is strictly functional where you have equational reasoning. Our universe has assignment and state.
  4. Closed is a term from math that has various but similar meanings depending on context. In the context of abstract algebra, from which generic programming is inspired, closure means that applying an operation to an element in a domain D gives you something back in the same domain. For example the integers Z are closed under the operation of multiplication. Multiply two integers, you get an integer. However, it is not closed under division. 1 / 2 is not an integer. Applying this to Alex’s quote, he means that containers should preserve properties of the base types they are constructed from. Containers of Regular types should be Regular.
  5. This is likely a reference to how std::shared_ptr behaves which provides a form of automatic memory management.,
  6. Bjarne Stroustrup is the creator of C++, author of many C++ books, and always has been an active member of the community.
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