Programming with Less Than Nothing

By Josh Moody — Published: Oct 19, 2025

Dana. The interviewer. Friendly, efficient. She leans forward.

“So,” she says, “walk me through FizzBuzz.”

You could do this in your sleep. “Is JavaScript okay?”

She nods.

You open your laptop and begin.

let S = (x) => (y) => (z) => x(z)(y(z)); let K = (x) => (y) => x;

“That should do it,” you say to yourself. “Just need to combine these a few times.”

Dana raises an eyebrow. You continue.

let I = S(K)(K); // x => x let B = S(K(S))(K); // x => y => z => x(y(z)) let C = S(B(B)(S))(K(K)); // x => y => z => x(z)(y) let W = S(S)(K(I)); // x => y => x(y)(y) let T = C(I); // x => y => y(x)

Bluebird, cardinal, warbler, thrush. Avian friends you know well.

Dana looks puzzled. “Are we still doing FizzBuzz?”

“Getting there.”

let V = B(C)(T); // x => y => z => z(x)(y) let R = B(B)(T); // x => y => z => y(z)(x) let B1 = B(B)(B); // x => y => z => w => x(y)(z)(w) let E = B(B1); // x => y => z => w => v => x(y)(z(w(v))) let C_STAR = B(C); // x => y => z => w => x(y)(w)(z) let C_STAR_STAR = B(C_STAR); // x => y => z => w => v => x(y)(z)(v)(w)

Dana breaks away from the screen. “Typically I see a for loop by now. Are you sure we’re talking about the same problem?”

“Trust me.”

let TRUE = K; let FALSE = K(I); let NOT = V(FALSE)(TRUE);

She stands. “Okay,” she says finally. “Let me stop you there.”

“Something wrong?”

“You’re not the first candidate to show up here and reinvent programming from first principles using lambda calculus.”

You stare, blankly. “Lambda calculus?”

“Look,” she says. “We’ve all read Programming with Nothing. We’ve all seen What is PLUS times PLUS. You’re not breaking any new ground here.”

You shake your head vigorously. “I would never be caught dead using Lambda calculus. It’s a bloated language.”

Dana’s turn to stare. You move on to numbers.

let ZERO = I; let IS_ZERO = T(TRUE); let SUCC = V(FALSE); // successor let PRED = T(FALSE); // predecessor let DECREMENT = S(W(IS_ZERO))(PRED); // clamped predecessor let ONE = SUCC(ZERO); let TWO = SUCC(ONE); let THREE = SUCC(TWO); let FOUR = SUCC(THREE); let FIVE = SUCC(FOUR); let SIX = SUCC(FIVE); let SEVEN = SUCC(SIX); let EIGHT = SUCC(SEVEN); let NINE = SUCC(EIGHT); let TEN = SUCC(NINE);

Dana leans forward. She seems interested now. “Church numerals?”

“Barendregt. Church is too mainstream.”

You start with a lambda calculus expression.

let ADD = (m) => (n) => IS_ZERO(m)(n)(SUCC(ADD(PRED(m))(n)));

“Dirty,” you mutter. “But it’s a start.”

You scribble in your notepad briefly. Showing all your work would be tedious, but you should at least walk her through the interesting parts.

“First, we add combinators until n is by itself on the right.”

let ADD = (m) => (n) => S(C(B(B(B))(IS_ZERO)(m))(SUCC))(ADD(PRED(m)))(n);

“Ah,” she says. “Now we can remove n.”

She’s warming up.

let ADD = (m) => S(C(B(B(B))(IS_ZERO)(m))(SUCC))(ADD(PRED(m)));

You do it again for m

// let ADD = (m) => // S(B(S)(C_STAR_STAR(B)(C)(B(B(B))(IS_ZERO))(SUCC)))( // B(ADD)(PRED), // )(m); let ADD = S(B(S)(C_STAR_STAR(B)(C)(B(B(B))(IS_ZERO))(SUCC)))( B(ADD)(PRED), );

Point-free. Clean. But not clean enough.

“Now for that pesky self reference…”

let ADD_MAKER = (self) => S(B(S)(C_STAR_STAR(B)(C)(B(B(B))(IS_ZERO))(SUCC)))( B(self)(PRED), ); let Y = W(C)(S(B)(C(W(C)))); // x => (y => x(y(y)))(y => x(y(y)))

“Is that the… Y combinator?” Dana asks.

“Can’t recurse without it.”

“It’s JavaScript. You can literally do recursion without it.”

“It won’t be JavaScript for much longer.”

let ADD = Y(ADD_MAKER); console.log(ADD(ONE)(TWO));

You run the program for the first time.

Uncaught RangeError: Maximum call stack size exceeded

Dana smirks. “Well, yeah. JavaScript is an eager language. Can’t use the Y combinator.”

You saw this coming. You paste your code into Skoobert.

“Skoobert?” Dana asks.

“JavaScript but lazy,” you explain. “And without the bloat.”

You run it again.

[Function]

You sigh. Perfect code tarnished by human consumption.

let extractNumber = Y( B(S(C(IS_ZERO)(0)))(B(B((x) => x + 1))(C(B)(PRED))), ); console.log(extractNumber(ADD(ONE)(TWO))); 3

Back on track. You type faster.

let SUBTRACT = C( Y( B(S(B(S)(C_STAR_STAR(B)(C)(B(B(B))(IS_ZERO))(DECREMENT))))( C(S(K(I))(B))(PRED), ), ), ); let MULTIPLY = Y( B(C)( B(S(B(B)(C(IS_ZERO)(ZERO))))( B(B(S(ADD)))(C(B(B)(C))(PRED)), ), ), ); let LESS_THAN_OR_EQUAL = B1(IS_ZERO)(SUBTRACT); let GREATER_THAN_OR_EQUAL = C(LESS_THAN_OR_EQUAL); let GREATER_THAN = B1(NOT)(LESS_THAN_OR_EQUAL); let LESS_THAN = B1(NOT)(GREATER_THAN_OR_EQUAL); let RECURSE_SUB = B(B(W))(C(B(B)(B))(SUBTRACT)); let MOD = Y(B(S(B(S)(W(C_STAR(LESS_THAN)))))(RECURSE_SUB)); let DIVIDE = Y( B(B(S(C(IS_ZERO)(ZERO))))( B(S(B(S)(C(B(C)(LESS_THAN))(ZERO))))( B(B(B(SUCC)))(RECURSE_SUB), ), ), );

Numbers are done. Lists:

let CONS = V; let FIRST = T(TRUE); let REST = T(FALSE); let EMPTY = K(TRUE); let END = EMPTY; let IS_EMPTY = T(K(K(FALSE))); let NTH = Y( B(S(B(S)(B(C(IS_ZERO))(FIRST))))( C(B(C)(B(B(B))(C(B)(REST))))(PRED), ), ); let CONCAT = B(T(CONS))(FOLD); let RANGE = Y( C( B(C)( B(B(C))( B(S(B(S)(LESS_THAN_OR_EQUAL)))( B(S(B(B)(CONS)))(C(B)(SUCC)), ), ), ), )(EMPTY), ); let FOLD = Y( B(S(B(B)(B(S)(IS_EMPTY))))( B(S(B(S)(C(B(B(B))(C(B)(FIRST))))))(C(B)(REST)), ), ); let MAP = C(B(C)(C(B(B)(FOLD))(B(CONS))))(EMPTY);

More pretty-printing:

let renderList = (fmt) => (sep) => (xs) => IS_EMPTY(xs)("")( fmt(FIRST(xs)) + IS_EMPTY(REST(xs))("")( sep + renderList(fmt)(sep)(REST(xs)), ), ); let showLines = (fmt) => (list) => renderList(fmt)("\n")(list);

“I won’t bother making the formatters point-free, since they’re just output,” you say, as if Dana cares.

Sanity check:

console.log( showLines(extractNumber)(MAP(RANGE(ONE)(THREE))(SUCC)), ); 2 3 4

Dana has given up following along.

let DECIMAL2 = B(ADD)(MULTIPLY(TEN)); // base 10 let CHAR_A = ONE; let CHAR_B = TWO; let CHAR_C = THREE; let CHAR_D = FOUR; let CHAR_E = FIVE; let CHAR_F = SIX; let CHAR_G = SEVEN; let CHAR_H = EIGHT; let CHAR_I = NINE; let CHAR_J = TEN; let CHAR_K = DECIMAL2(ONE)(ONE); let CHAR_L = DECIMAL2(ONE)(TWO); let CHAR_M = DECIMAL2(ONE)(THREE); let CHAR_N = DECIMAL2(ONE)(FOUR); let CHAR_O = DECIMAL2(ONE)(FIVE); let CHAR_P = DECIMAL2(ONE)(SIX); let CHAR_Q = DECIMAL2(ONE)(SEVEN); let CHAR_R = DECIMAL2(ONE)(EIGHT); let CHAR_S = DECIMAL2(ONE)(NINE); let CHAR_T = DECIMAL2(TWO)(ZERO); let CHAR_U = DECIMAL2(TWO)(ONE); let CHAR_V = DECIMAL2(TWO)(TWO); let CHAR_W = DECIMAL2(TWO)(THREE); let CHAR_X = DECIMAL2(TWO)(FOUR); let CHAR_Y = DECIMAL2(TWO)(FIVE); let CHAR_Z = DECIMAL2(TWO)(SIX); let CHAR_0 = DECIMAL2(TWO)(SEVEN); let CHAR_1 = DECIMAL2(TWO)(EIGHT); let CHAR_2 = DECIMAL2(TWO)(NINE); let CHAR_3 = DECIMAL2(THREE)(ZERO); let CHAR_4 = DECIMAL2(THREE)(ONE); let CHAR_5 = DECIMAL2(THREE)(TWO); let CHAR_6 = DECIMAL2(THREE)(THREE); let CHAR_7 = DECIMAL2(THREE)(FOUR); let CHAR_8 = DECIMAL2(THREE)(FIVE); let CHAR_9 = DECIMAL2(THREE)(SIX); let ARRAY = Y( B(S(B(S)(C(B(C(IS_EMPTY)))(END))))( C(B(B)(B))(C(B(B)(B))(CONS)), ), )(I); let FIZZ = ARRAY(CHAR_F)(CHAR_I)(CHAR_Z)(CHAR_Z)(END); let BUZZ = ARRAY(CHAR_B)(CHAR_U)(CHAR_Z)(CHAR_Z)(END); let FIZZBUZZ = CONCAT(FIZZ)(BUZZ);

Printing strings:

let extractString = (lst) => IS_EMPTY(lst)("")( NTH(letters)(PRED(FIRST(lst))) + extractString(REST(lst)), ); console.log(extractString(FIZZBUZZ)); fizzbuzz

Just a few more loose ends. Numbers to digits. Digits to strings. Strings to output.

let array = Y( (h) => (acc) => (x) => x === END ? acc(END) : h((t) => acc(CONS(x)(t))), )(I); let letters = array("a")("b")("c")("d")("e")("f")("g")("h")("i")("j") ("k")("l")("m")("n")("o")("p")("q")("r")("s")("t")("u") ("v")("w")("x")("y")("z") ("0")("1")("2")("3")("4")("5")("6")("7")("8")("9")(END); let DIGITS_NUMERAL = ARRAY(CHAR_0)(CHAR_1)(CHAR_2)(CHAR_3)(CHAR_4) (CHAR_5)(CHAR_6)(CHAR_7)(CHAR_8)(CHAR_9)(END); let NUMBER_TO_DIGIT_LIST = Y( C( B(S)( C(B(B)(B(S(B(S)(IS_ZERO)))(B(CONCAT))))(C(DIVIDE)(TEN)), ), )(C(B(CONS)(C(MOD)(TEN)))(EMPTY)), ); let NUMBER_TO_STRING = S( C(B(FOLD)(NUMBER_TO_DIGIT_LIST))( B(CONS)(NTH(DIGITS_NUMERAL)), ), )(B(T(EMPTY))(K(I)));

You take a deep breath and look Dana in the eye.

“And now… the main event.”

let FIFTEEN = DECIMAL2(ONE)(FIVE); let ONE_HUNDRED = MULTIPLY(TEN)(TEN); let FIZZBUZZ_RESULT = MAP(RANGE(ONE)(ONE_HUNDRED))( S( B(S(C(R(FIZZBUZZ))(FIFTEEN)))( B(S(C(R(FIZZ))(THREE)))( B(C(C(R(BUZZ))(FIVE)))(NUMBER_TO_STRING), ), ), )(B(B(IS_ZERO))(MOD)), ); console.log(showLines(extractString)(FIZZBUZZ_RESULT));

“Are you satisfied?” she cries. But you aren’t. Not yet.

Variables are for wimps. You replace each with its definition.

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It’s time. You press enter.

Several seconds go by.

The tension is invigorating.

Suddenly, the terminal fills with output:

1 2 fizz 4 buzz fizz 7 8 fizz buzz 11 fizz 13 14 fizzbuzz 16 17 fizz 19 buzz fizz 22 23 fizz buzz 26 fizz 28 29 fizzbuzz 31 32 fizz 34 buzz fizz 37 38 fizz buzz 41 fizz 43 44 fizzbuzz 46 47 fizz 49 buzz fizz 52 53 fizz buzz 56 fizz 58 59 fizzbuzz 61 62 fizz 64 buzz fizz 67 68 fizz buzz 71 fizz 73 74 fizzbuzz 76 77 fizz 79 buzz fizz 82 83 fizz buzz 86 fizz 88 89 fizzbuzz 91 92 fizz 94 buzz fizz 97 98 fizz buzz

You lean back, exhausted but triumphant.

Dana is dead.

---

What the heck did I just read?

The culmination of months of effort!

Combinatory logic is a model of computation that is extremely minimal. All you start with are two functions: S and K.

• S is a function that, given three functions x, y, and z, returns x(z)(y(z)).
• K is a function that, given two functions x and y, returns x.

You can compute anything you want just by passing these two functions into each other. For example, the identity function I = (x) => x can be expressed as S(K)(K).

And you can build up arbitrarily complex functions from there. There is no limit.

Combinatory logic is basically lambda calculus without variables. Despite this limitation, combinatory logic is still Turing complete.

It is also extremely difficult to understand.

Further reading

If you found this post interesting (whether or not you understood it), the following resources would be good places start exploring:

• To Mock a Mockingbird, by Raymond Smullyan
• Programming with Nothing, by Tom Stuart
• Kyle Kingsbury’s Technical Interview series

In particular, I strongly recommend reading To Mock a Mockbird if you want to understand combinatory logic. It’s the most fun, accessible way to learn combinatory logic from scratch. It took me about a month’s spare time to work through it. Time well spent!

Skoobert

After reading To Mock a Mockingbird, I wanted to program in combinatory logic but found existing tools lacking. So I spent a weekend building Skoobert, which is intended to be the ideal programming language for experimenting with lambda calculus and combinatory logic. Give it a try and let me know what you think!

• Skoobert on GitHub
• Skoobert playground
• FizzBuzz in Skoobert (with explanatory comments)