Todays tasty treat: a variable-free nearly-pure functional programming language: Unlambda.

Unlambda is based on the SKI combinator calculus. The [SKI calculus][ski] is a way of writing lambda calculus without variables, and without lambda. It’s based on three combinators: S, K, and I:

1. S = λ x y z . x z (y z)

2. K = λ x y . x

3. I = λ x . x

Given only S and K, you can actually write *any* lambda calculus expression – and therefore, you can write any computable function using nothing but S and K. I is often added for convenience; it’s just a shorthand for “SKK”.

You can do recursion in SKI; the Y combinator of lambda calculus is:

Y = S S K (S (K (S S (S (S S K)))) K)

Unlambda is based on SKI; but it extends it with a few additional kinds of sugar to make things easier to write. (Nothing makes Unlambda easier to *read*!) The basic idea is that *everything* in unlambda is a function; and in particular, everything is a function that takes a *single* variable: you don’t need more than one parameter, because you can always using [currying][curry] to express any multiparameter function as a series of single-parameter functions.

(As a refresher, if you have a two-parameter function in lambda calculus, like “λ x y . x + y”; *currying* is translating it into a one parameter function that returns a one parameter function: “λ x .(λ y. x + y)”.)

There are eight basic instructions in Unlambda:

1. ““”` : the application operator. “`’AB`” applies the function `A` to the parameter `B`. You can think of this as being something like the “(” in Lisp; there is no close paren because since every function takes only one parameter, there is no *need* for a close paren. And hey, adding a syntactically unnecessary close paren would only make the code easier to read, and we can’t have that!

2. “`k`” : the K combinator.

3. “`s`” : the S combinator.

4. “`i`” : the I combinator.

5. “`v`” : a “drop” combinator: takes a parameter, discards it, and returns “`v`”

6. “`d`” : the “delay” operator; a way of doing *lazy evaluation* in Unlambda. Unlambda normally evaluates the argument before it

applies a function. `’dA` is the same thing as `A`, except that `A` isn’t evaluated until “`’dA`” is applied to something else.

7. “`c`” : the “call-with-current-continuation” combinator. This is equivalent to [“call/cc”][callcc] in Scheme. What it does is call some function with the current state of the computation captured as a function as its parameter. What this means is that “`’cAB`” calls “`A`”. “`A`” receives the value “`cAB`” as a parameter; during A, it can return a value, *or* it can call the continuation, which will evaluate “`cAB`” again. So this is the function-based equivalent of a loop. “c” turns the “loop body” into a function; so at the end of the loop, you re-invoke the continuation with a new parameter to run the next iteration.

8. “`.`” : The “.” operator is actually a meta-operator. “`.x`” is a function which returns its parameter, *and* prints the character “x”.

9. “`r`” is a shorthand for “`.CR`”, where CR is the carriage return character.

And that is it. The entire language.

Note that there are no numbers! We need to build numbers using church numerals.

* The church numeral “0” is “λ f. (λ x . x)”. In Unlambda, that’s “`’ki`”.

* The church numeral “1” is “λ f . (λ x . f x)”; in unlambda, that’s “`i`”.

* The church numeral “2” is “λ f . (λ x. f f x)”; or “`”s”s’kski`”.

* The church numeral “3” is “λ f . (λ x . f f f x)”; or “`”s”s’ksk”s”s’kski`”.

* And so on.. Just keep appending “`”s”skski`” for each successive number.

So… On to a couple of programs!

Hello world in Unlambda is:

`r“““““`.H.e.l.l.o. .w.o.r.l.di

It’s pretty straightforward. It’s actually sort of easier to understand if you look from right to left. The “`.H.e.l.l.o…`” is basically a series of “`i`” combinators, which print out the characters of hello world. The trailing “`i`” is there because “`.d`” needs a parameter in order to do anything. And the list of “`’`”s at the beginning are there to make the “.” functions be applied. Finally, apply “`’r`” to the whole thing – which prints a newline at the end.

Next, a counter: starts with “N=1″; print “N” asterisks, add one, and then repeat.

“r`ci“s`k`c“s“s`ksk`kr.*

If you look at it, you can see the number pattern in there; “`”s”s’ksk`”. That fragment is an SKI function to add one. You capture a continuation for the main iteration; run “`i”s’k’cN`”; that’s a fragment that adds “1” to N, and then prints N. Since you’ve captured the continuation, you can re-invoke it on 1+n; the rest captures *another* continuation which is the inner loop for printing “*”s; it does that, and then it invokes the out continuation to go back and do it for “N+1″.

One last, which I’ll leave for you to trace through yourself: this one generates the fibonacci sequence; and for each element of it, it prints out the numbers using a line of asterisks containing “N” asterisks for the number “N”:

“`s“s“sii`ki

`k.*“s“s`ks

“s`k`s`ks“s“s`ks“s`k`s`kr“s`k`sikk

`k“s`ksk

I’ll also point out that there are a ton of variants on Unlambda, ranging from the really interesting ([LazyK][lazyk]) to the bizzarely minimal ([Iota][iota]) to the downright goofy ([Jot][jot]).

[callcc]: http://community.schemewiki.org/?call-with-current-continuation

[ski]: http://goodmath.blogspot.com/2006/05/from-lambda-calculus-to-combinator.html

[curry]: http://goodmath.blogspot.com/2006/05/my-favorite-calculus-lambda-part-1.html

[iota]: http://ling.ucsd.edu/~barker/Iota/

[lazyk]: http://homepages.cwi.nl/~tromp/cl/lazy-k.html

[jot]: http://ling.ucsd.edu/~barker/Iota/#Goedel