On Apr 9, 2008, at 10:00 AM, Lawrence Tsang wrote:

> Hi Tibi and Yow Tzu Lim,
>
> Thanks for all the answers. I've got the idea.

Too late! Now you've got me responding too. :-)

It's important to distinguish between "types" and "data". Formally a
type is just a compatibility constraint. If function FOO has a
"return type" of bar, and function BAZ takes one argument which is of
type "bar", then FOO is permitted to fill the argument slot of BAZ.

This doesn't actually have anything to do with data per se; but the
#1 reason why you'd want to state that functions have certain types
is because if FOO sends out an integer and BAZ expects a String, then
yo wouldn't want FOO to be allowed to hook up with BAZ, so you'd
stipulate that they have different types.

Because formal language types are so closely associated with "data
types" -- two functions are typed the same way because they return
the same data or whatever -- people think they're the same thing.
But they're not. There are other reasons you might not want two
functions to hook up. For example, you might want to be guaranteeing
that your tree structure has functions of some form A in its top row,
functions of some form B in its second row, and functions of some
form C in its third row. This kind of thing shows up a lot when the
tree structures aren't "programs" per se, but are used for some other
functionality. To do this, you could say that the root only accepts
functions whose return type is "A", and "A" functions only accept
child arguments of type "B", and so on.

What's nil?

That's easy. By default, most ECJ examples are "untyped", meaning
that there are no constraints on who is able to hook up with whom.
This isn't zero types: it's one type -- everyone shares the same type
so everyone is compatible with everyone. I had to have some name for
that type, so I picked nil. It's just a name. You can change it to
something else like gobbledygook and that'd work fine.

One further note on typing: ECJ's typing is a fairly rudimentary
typing which I call "set typing". Types are sets of objects. Two
things are compatible if their types have a nonempty intersection --
they share an object common in their two sets. You can do a lot with
set typing, not the least of which is generic functions and
replicating the functionality of polymorphism. People actually
typically need an even simpler typing notion, which I call "atomic
typing" -- here types are just symbols, and two functions are
compatible if their types are the same. Atomic typing is really just
a degenerate form of set typing: instead of an atomic type FOO, you
could just have a set type which contains a unique single symbol
inside it, which is special just to that set type: FOO2 = { FOO }.
So you can easily replicate atomic typing with set typing -- I
include it just for convenience. Also, you can mix atomic types and
set types. An atomic type is "compatible" with a set type if the
atomic type is found in the set type's set.

Note that this is strictly NOT as powerful as various "polymorphic
typing" approaches people have taken -- there's been a fair bit of
work on people who have functions whose return types (say) change
based on the argument types of the children that wound up plugging
into them. This is very powerful stuff but it's a nightmare to do
general crossover operators and mutation operators in, and it's
fairly rarely needed. An example of where it'd be nice to have.
Let's say the data structure your functions are passing around is a
matrix. You want to make a MATRIX_MULTIPLY function which takes two
matrices, multiplies them and returns the resulting matrix. In a
sophisticated polymorphic typing mechanism you could say that IF you
have an N x M matrix plugged into your first child, AND you had an M
x P matrix plugged into your second child, then your return type
would be "N x P". Note that this implies that you have a potentially
infinite number of types and so this clearly can't be done with set
typing. Maarten Keijzer (http://www.cs.vu.nl/~mkeijzer/) did work
exactly like this, and wound up building his own extension of EO (I
believe) to handle polymorphic typing. Tina Yu (http://www.cs.mun.ca/
~tinayu/) did polymorphic typing for her thesis I think.

Sean