This post is heavily inspired by Tom Stuart: Programming with noting, William Byrd on “The Most Beautiful Program Ever Written”, Guowei LV – The Most Beautiful Program Ever Written, John Tromp – Binary lambda calculus.

When I first learned about Lambda calculus, I was amazed by a strong mathematical basis for anonymous functions used in many languages.

Then I wondered: what’s the simplest way to represent lambda functions as data?

Representation

Lambda expressions, such as λa.λb.(a b) have three parts: Variables a , b , Application (a b) and Abstraction (or creating a function) λa.a .

The question of representing code as data would be incomplete without mentioning LISP, where code is data that can be transformed. The Most Beautiful Program Ever Written dives into that topic, and into the question of evaluation.

All of the features in lambda expressions are available in LISP and have a straightforward representation. However, LISP AST has quite complex data structures: symbols, strings, linked lists… we need to go simpler.

First, let’s use De Bruijn indices – instead of variable symbols. Replace variable names with the number of the nested lambda’s argument you want to look up. For example, an implementation of the K combinator λa.λb.a becomes λλ2 , where 2 means “the argument of the second lambda, going up from this position”.

Second, note that the lambda notation has a binary operation – function application, and a single argument syntax for creating a lambda, the argument being lambda body.

Let’s use the following representation:

variable lookup is a positive integer number which represents a De Bruijn index.

creating a function is a tuple of zero and function body. λa.a becomes [0, 1] in JS notation.

becomes in JS notation. function application is a tuple of two values. λa.λb.(a b) becomes [0, [0, [2, 1]]] .

Note: John Tromp has invented Binary lambda calculus, which is a much more efficient representation than this. For the purpose of making a simple interpreter, a binary tree form is used to represent lambda terms. There’s a 1-to-1 mapping between binary tree form and binary representation.

Environment

Since variable are represented as indicies, the most straightforward way to represent environment as linked lists. Or a JS tuple [head, tail] .

Why not a stack or an array? Because multiple scopes could reference the same parent scope, with possibly changing values.

Constructing an environment becomes as simple as constructing a tuple [currentValue, parentEnv] .

Evaluation

It’s easy to construct a simple evaluator once you’ve seen one:

function Eval ( prog , env ) { if ( typeof prog === ' number ' ) { // lookup a variable while ( -- prog ) { env = env [ 1 ]; } return env [ 0 ]; } else if ( prog [ 0 ] === 0 ) { // constructing a new lambda return ( arg ) => Eval ( prog [ 1 ], [ arg , env ]); } else { // function application return Eval ( prog [ 0 ], env )( Eval ( prog [ 1 ], env )); } }

The code mangled to 140 characters or less, for no particular reason:

Eval = function E ( p , e ) { if ( typeof p == ' number ' ){ while ( -- p ){ e = e [ 1 ]} return e [ 0 ]} return p [ 0 ] == 0 ?( a ) => E ( p [ 1 ],[ a , e ]): E ( p [ 0 ], e )( E ( p [ 1 ], e )) }

And of course, translated to Common LISP for completeness:

( defun Eval ( prog env ) ( cond (( numberp prog ) ( nth ( 1- prog ) env )) (( zerop ( car prog )) ( lambda ( arg ) ( Eval ( cdr prog ) ( cons arg env )))) ( 't ( apply ( Eval ( car prog )) ( Eval ( cdr prog ))))))

note: this code is using cons pairs instead of linked lists to represent the binary tree

Does it work?

Let’s test this representation and evaluation with simple combinators:

I combinator or identity: λx.x . Trivially translates to [0, 1] .

var I = Eval ([ 0 , 1 ]); console . assert ( I ( " test " ) === " test " );

K combinator or first of two arguments: λx.λy.x . Translates to [0, [0, 2]] .

var K = Eval ([ 0 , [ 0 , 2 ]]); console . assert ( K ( " first " )( " second " ) === " first " );

S combinator or generalized application: λx.λy.λz.((x z) (y z)) . Translates to [0, [0, [0, [[3, 1], [2, 1]]]]] .

var K = Eval ([ 0 , [ 0 , 2 ]]); var S = Eval ([ 0 , [ 0 , [ 0 , [[ 3 , 1 ], [ 2 , 1 ]]]]]); console . assert ( S ( K )( K )( " test " ) === " test " );

ι combinator, Iota combinator or universal iota combinator:

λf.((f S) K) => [0, [[1, S], K]]

Or, expanded: [0, [[1, [0, [0, [0, [[3, 1], [2, 1]]]]]], [0, [0, 2]]]]

var iota = Eval ([ 0 , [[ 1 , [ 0 , [ 0 , [ 0 , [[ 3 , 1 ], [ 2 , 1 ]]]]]], [ 0 , [ 0 , 2 ]]]]); var iota_I = iota ( iota ); var iota_K = iota ( iota ( iota ( iota ))); var iota_S = iota ( iota ( iota ( iota ( iota )))); console . assert ( iota_I ( " test " ) === " test " ); console . assert ( iota_K ( " first " )( " second " ) === " first " ); console . assert ( iota_S ( iota_K )( iota_K )( " test " ) === " test " );

FizzBuzz

Once the interpreter is working, we can run some more ambitious programs.

Using some trivial transformations, Tom Stuart’s FizzBuzz becomes the following:

var Eval = function E ( p , e ) { if ( typeof p == ' number ' ){ while ( -- p ){ e = e [ 1 ]} return e [ 0 ]} return p [ 0 ] == 0 ?( a ) => E ( p [ 1 ],[ a , e ]): E ( p [ 0 ], e )( E ( p [ 1 ], e )) }; ZERO = [ 0 ,[ 0 , 1 ]]; ONE = [ 0 ,[ 0 ,[ 2 , 1 ]]]; TWO = [ 0 ,[ 0 ,[ 2 ,[ 2 , 1 ]]]]; THREE = [ 0 ,[ 0 ,[ 2 ,[ 2 ,[ 2 , 1 ]]]]]; FOUR = [ 0 ,[ 0 ,[ 2 ,[ 2 ,[ 2 ,[ 2 , 1 ]]]]]]; FIVE = [ 0 ,[ 0 ,[ 2 ,[ 2 ,[ 2 ,[ 2 ,[ 2 , 1 ]]]]]]]; INC = [ 0 ,[ 0 ,[ 0 ,[ 2 ,[[ 3 , 2 ], 1 ]]]]]; DEC = [ 0 ,[ 0 ,[ 0 ,[[[ 3 ,[ 0 ,[ 0 ,[ 1 ,[ 2 , 4 ]]]]],[ 0 , 2 ]],[ 0 , 1 ]]]]]; PLUS = [ 0 ,[ 0 ,[ 0 ,[ 0 ,[[ 4 , 2 ],[[ 3 , 2 ], 1 ]]]]]]; MINUS = [ 0 ,[ 0 ,[[ 1 , DEC ], 2 ]]]; MUL = [ 0 ,[ 0 ,[ 0 ,[ 2 ,[ 3 , 1 ]]]]]; POW = [ 0 ,[ 0 ,[ 2 , 1 ]]]; NINE = [[ PLUS , FIVE ], FOUR ]; TEN = [[ MUL , TWO ], FIVE ]; FIFTEEN = [[ MUL , FIVE ], THREE ]; HUNDRED = [[ MUL ,[[ MUL , FIVE ], FIVE ]], FOUR ]; TRUE = [ 0 ,[ 0 , 2 ]]; FALSE = [ 0 ,[ 0 , 1 ]]; IF = [ 0 ,[ 0 ,[ 0 ,[[ 3 , 2 ], 1 ]]]]; IS_ZERO = [ 0 ,[[ 1 ,[ 0 , FALSE ]], TRUE ]]; LEQ = [ 0 ,[ 0 ,[ IS_ZERO ,[[ MINUS , 2 ], 1 ]]]]; PAIR = [ 0 ,[ 0 ,[ 0 ,[[ 1 , 3 ], 2 ]]]]; LEFT = [ 0 ,[ 1 ,[ 0 ,[ 0 , 2 ]]]]; RIGHT = [ 0 ,[ 1 ,[ 0 ,[ 0 , 1 ]]]]; EMPTY = [[ PAIR , TRUE ], TRUE ]; UNSHIFT = [ 0 ,[ 0 ,[[ PAIR , FALSE ],[[ PAIR , 1 ], 2 ]]]]; IS_EMPTY = LEFT ; FIRST = [ 0 ,[ LEFT ,[ RIGHT , 1 ]]]; REST = [ 0 ,[ RIGHT ,[ RIGHT , 1 ]]]; Z = [ 0 ,[[ 0 ,[ 2 ,[ 0 ,[[ 2 , 2 ], 1 ]]]],[ 0 ,[ 2 ,[ 0 ,[[ 2 , 2 ], 1 ]]]]]]; MOD = [ Z ,[ 0 ,[ 0 ,[ 0 ,[[[[ LEQ , 1 ], 2 ],[ 0 ,[[[ 4 ,[[ MINUS , 3 ], 2 ]], 2 ], 1 ]]], 2 ]]]]]; DIV = [ Z ,[ 0 ,[ 0 ,[ 0 ,[[[[ LEQ , 1 ], 2 ],[ 0 ,[[ INC ,[[ 4 ,[[ MINUS , 3 ], 2 ]], 2 ]], 1 ]]], ZERO ]]]]]; RANGE = [ Z ,[ 0 ,[ 0 ,[ 0 ,[[[[ LEQ , 2 ], 1 ],[ 0 ,[[[ UNSHIFT ,[[ 4 ,[ INC , 3 ]], 2 ]], 3 ], 1 ]]], EMPTY ]]]]]; FOLD = [ Z ,[ 0 ,[ 0 ,[ 0 ,[ 0 ,[[[ IS_EMPTY , 3 ], 2 ],[ 0 ,[[[ 2 ,[[[ 5 ,[ REST , 4 ]], 3 ], 2 ]],[ FIRST , 4 ]], 1 ]]]]]]]]; MAP = [ 0 ,[ 0 ,[[[ FOLD , 2 ], EMPTY ],[ 0 ,[ 0 ,[[ UNSHIFT , 2 ],[ 3 , 1 ]]]]]]]; PUSH = [ 0 ,[ 0 ,[[[ FOLD , 2 ],[[ UNSHIFT , EMPTY ], 1 ]], UNSHIFT ]]]; TO_DIGITS = [ Z ,[ 0 ,[ 0 ,[[ PUSH ,[[[[ LEQ , 1 ], NINE ], EMPTY ],[ 0 ,[[ 3 ,[[ DIV , 2 ], TEN ]], 1 ]]]],[[ MOD , 1 ], TEN ]]]]]; B = TEN ; F = [ INC , B ]; I = [ INC , F ]; U = [ INC , I ]; ZED = [ INC , U ]; FIZZ = [[ UNSHIFT ,[[ UNSHIFT ,[[ UNSHIFT ,[[ UNSHIFT , EMPTY ], ZED ]], ZED ]], I ]], F ]; BUZZ = [[ UNSHIFT ,[[ UNSHIFT ,[[ UNSHIFT ,[[ UNSHIFT , EMPTY ], ZED ]], ZED ]], U ]], B ]; FIZZBUZZ = [[ UNSHIFT ,[[ UNSHIFT ,[[ UNSHIFT ,[[ UNSHIFT , BUZZ ], ZED ]], ZED ]], I ]], F ]; RESULT = [[ 0 ,[[ 0 ,[[ MAP ,[[ RANGE , ONE ], HUNDRED ]],[ 0 , [[[ 3 ,[[ 2 , 1 ], FIFTEEN ]], FIZZBUZZ ], [[[ 3 ,[[ 2 , 1 ], THREE ]], FIZZ ], [[[ 3 ,[[ 2 , 1 ], FIVE ]], BUZZ ], [ TO_DIGITS , 1 ]]]] ]]], MOD ]], IS_ZERO ]; // some helper functions to extract data into a readable form. // lambda bool to JS bool var to_bool = ( fn ) => fn ( true )( false ); // church numeral to JS number var to_int = ( fn ) => fn ( x => x + 1 )( 0 ); // linked list to JS array var to_list = ( fn ) => { var list = []; var LEFT = p => p ( x => y => x ); var RIGHT = p => p ( x => y => y ); var FIRST = l => LEFT ( RIGHT ( l )); var REST = l => RIGHT ( RIGHT ( l )); while ( ! to_bool ( LEFT ( fn ))) { list . push ( FIRST ( fn )); fn = REST ( fn ); } return list ; }; // transform list of numbers to JS string var to_string = ( fn ) => { var alpha = ' 0123456789BFiuz ' ; return to_list ( fn ). map ( c => alpha [ to_int ( c )]). join ( '' ); }; console . log ( ' full program: ' , JSON . stringify ( RESULT )); to_list ( Eval ( RESULT )). forEach ( row => console . log ( to_string ( row )));

Utility

N/A

Downsides

The simplicity of this language and evaluator implies a near-impossibility of providing meaningful error messages.

There’s almost no applications for this in the real world except for learning how lambda calculus works.

Future work

Translate more lambda expressions into data using this notation.

Write a parser that can translate lambda expressions into this form.

Write a parser for Binary Lambda Calculus

Write a lazy evaluator.

Implement an evaluator without using lambda functions provided by the language.

Implement an evaluator in a language that doesn’t support GC.

Make a lambda -> x64 compiler.

Summary

Making a simple evaluator was fun. This is probably the simplest evaluator of a kind.

This evaluator shows once again the simplicity and expressiveness of lambda calculus, and how it maps onto programming languages.