What mathematical duals are there are in OO programming?

Question

Hello!

If you have watched Going Deep shows of the Channel9 lately, one very frequently mentioned topic is mathematical duality in programming. TomasP has a good blog post about duality in object oriented programming.

This has been since Microsoft Research found that the observer design pattern is actually a mathematical dual of the iterator pattern. Since then they have used the duality concept in various ways.

My question is:

What mathematical dualities are there in programming?

Object oriented programming is a good start. The major GoF design patterns are: Decorator, State, Iterator, Facade, Strategy, Proxy, Factory Method, Adapter, Observer, Template Method, Composite, Singleton, Abstract Factory and Command. Here is a good object-graph-poster.

Answer 1

Not sure if it is entirely what you were looking for, as it's more FP than OO, but there is of course the Curry-Howard Correspondance (a.k.a. Curry-Howard Isomorphism) that "equates" programs with proofs and types with formulae.

Answer 2

I'd say the primary duality in programming is the code-data duality, most clearly exposed in Lisp, but also clear in most contemporary languages which provide introspection functionality.

Answer 3

You could argue that the duality of observer/iterator is (sort of, work with me here :-) ) a manifestation of the more general OO paradigms of inheritence and the alternate paradigm of delegation and aggregation. In the former, more specialized objects use base functionality (point up) to inherit general capabilities and in the latter, more generalized objects use delegation to access more specialized functionality (point down/out) - there is much academic discussion about the fact that oo designs can be expressed in either form, and since the variance between forms is (reasonably) rigourous and defined, I'd say it could be classified as a dual

See Treaty of Orlando 2 for more info

Answer 4

I think of objects and closures/anonymous functions as duals.

An object is a bunch of data, with a set of routines that are "attached" to it (i.e. its methods).

A closure, in the functional-programming sense of the word, is a (callable) reference to a function, with a set of data attached (in the form of its bound free variables).