Covariance and contravariance - a simple guide
Written by Mike James   
Friday, 26 November 2010
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Covariance and contravariance - a simple guide
Why use co- and contravariance?

Programming, and computer science in particular, has a tendency to use other people's jargon. Often this makes things seem more difficult. You may have heard of  covariance and contravariance and wondered what they were all about. If you want a simple explanation that applies to any computer language here it is.


Programming and computer science in particular has a tendency to use other people's jargon. Often this makes things seem more difficult

Covariance and contravariance are a case in point. They are terms that are used in different  ways in the theory of object-oriented programming and they sound advanced and difficult - but in fact the idea they they encapsulate is very, very simple.

Let's find out.

Functions - the start of co and contra

Covariance and contravariance occur all over mathematics - in vector spaces, differential geometry and so on. They are a very general idea based on observing what happens when you make a change - the result usually goes in the same direction as the change, i.e. it is covariant, but sometimes it goes in the other direction, i.e. it is contravariant.

The most elementary example I can find of the co and contra behavior is the simple mathematical function. A function has an input and an output and these behave differently if you try to change them.

For example, suppose you have the function:


then first lets see what happens to the function if we add a constant to x.

sinA graph of sin(x)

If you draw a graph of the function sin(x+a) you can easily see that the effect of the +a is to move the graph -a units. That is x is translated a units but the function is translated -a units. The function moves in the opposite direction to x and so we say that it is contravariant in x.


contraChanging x to x+a moves the graph in the -a direction


Now consider adding a constant to the function y. You can see that y+a is given by sin(x)+a which moves the graph of the function up by a units. The function moves in the same direction as the constant and so we say it is covariant in y.


coChanging y to y+a moves the graph up a units


In general changes to the inputs of things tend to move the function in the opposite way and are contravariant but changes to the outputs move it in the say way and so they are covariant.

This is where the idea comes from but it turns up with modifications in all sorts of places and it can seem much more sophisticated than this simple example - but in all cases it is contravariant if the change you make results in the opposite change in what you are considering and covariance if it results in the same change.

Object hierarchies

The common use of the terms contravariance and covariance in programming has come to mean something a little more specific - but still often related to the inputs and outputs of functions so lets look at this common usage first.

The whole idea relates to the hierarchy of types.

If type B is derived from type A that is it "inherits" from A or A is the base class for B - then it contains every method and property that type A does and more. In this sense type B is "bigger" than type A. You can see that the type system provides a way of ordering classes.  (Notice that not all classes are on the same branch of the type hierarchy and so not all classes can be compared in this way i.e. we only have a partial ordering.)

The idea that B is in some way "bigger" than A is an important idea. Unfortunately it has long been the case that we use the less than symbol to show that a type is derived from another type which is perhaps the wrong way round.

That is if we write A>B  then A is "higher" in the type hierarchy than B or in other words B is derived from A.

The substitution principle

If B derives from A then A>B and B inherits or contains everything that A has to offer.

As B contains everything that A does you can use a B anywhere that you could have used an A. Of course you can't always use an A in place of a B so the relationship is asymetrical.

This is formally known as the Liskov Substitution principle.

In fact the full "principle" is a little wider than this as it holds that anything that is true of an A should be true of a B and in general this is difficult to achieve - but you get the general idea. In practice it is a more a guiding principle than something that is enforced all of the time. 

Inferred type relationships

Looking at the substitution principle the other way round we can use it to define the hierarchical type relationship between types.

That is if type B can be used anywhere a type A would be ok then you can say that type A is a base type for type B and B<A.

This is a reasonable idea because if B can always be used in place of A then it has everything that A has and perhaps more.

This is a useful idea when you are working with types that are not explicitly defined within a type hierarchy by being explicitly derived from one another - as is the case for delegates say, see later.




Last Updated ( Monday, 29 November 2010 )

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