Ear

Ear is a general-purpose programming language designed on the principles of mathematical equivalence. An Ear program is simply a list of equivalent patterns.

Here is an example of an Ear number implementation:

        num zero     = true
num A | num (succ A) = true
        num X        = false

zero + A     = A + zero
A + zero     = A
A + (succ B) = succ (A + B)

zero * A     = A * zero
A * zero     = zero
A * (succ B) = A + (A * B)

We will be dissecting this example throughout this article.

Elimination

$ ear
Ear Alpha-xxxxxxx
Ready.
[e] succ zero
 =  succ zero
[e] (succ (succ zero)) * (succ (succ zero))
 =  succ (succ (succ (succ zero)))
[e]

First, Ear loads. It may take a little while to load, as Alpha is expected to be slow. You'll see messages like this

Loading 'essential' environment... (5/24)

followed by the Ready. status in the previous example once Ear has finished loading.

The [e] prompt shows that we are in elimination mode, that is, Ear will attempt to find the equivalent form of what you entered which uses the least patterns. for example, (succ zero) * (succ zero) involves 3 patterns (_ * _, succ _ and zero), where an equivalent form of it, succ zero, only involves 2 patterns (succ _ and zero).

So, back to the first entry.

[e] succ zero
 =  succ zero

The form of succ zero with the least number of patterns is succ zero. That is, Ear has found that succ zero already contains the least number of patterns possible (at least, thus far).

[e] (succ (succ zero)) * (succ (succ zero))
 =  succ (succ (succ (succ zero)))

This is the same thing discussed above. What we entered contains 7 entities in 3 patterns, and the result Ear found has 5 entities in 2 patterns. In the elimination strategy, it is the number of patterns that count; that's how it found the answer.

Moving on, consider the following:

[e] one = succ zero
[e] succ zero
 =  one
[e] one
 =  one
[e] one + one
 =  succ one
 =  one + one
 =  succ (succ zero)

Uh-oh. Ear has found that succ one and succ (succ zero) both have the same number of entities, and both are equivalent to one + one. Can something have three right answers? Yes, if the answers are equivalent. one + one = succ one = succ (succ zero). They're just different ways of expressing the same thing.

For convenience, Ear has sorted the three answers by number of entities and complexity of syntax. What happens, now, if we do this?

[e] two = succ one
[e] two + two
 =  two + two
 =  succ (succ two)
 =  succ (succ (succ one))
 =  succ (succ (succ (succ zero)))

Whoa. That's more than we need...

Elimination and Expansion

E mode takes an extra step. After eliminating unneeded patterns, it returns the result(s) with the highest number of entities. This is, in most cases, what you expect in the first place; for example:

[E] two + two
 =  succ (succ (succ (succ zero)))

Warning! There probably are some cases where E mode's output is confusing. In these cases, don't cry, revert to e. It should give you what you're looking for.