Lambda-calculus is Fun!

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In my last post I talked about a way to get the Abstract Syntax Tree (AST) from a JavaScript function.

So today I decided to make some experiments with AST manipulation. As an exercise I choose to make pure lambda-calculus logical expressions.

First lets define some nice namespaces so that the code doesn’t get confused with normal JS code and generated code:

var lambda = {}
lambda.logic = {}

Next we define the true and false values as lambda abstractions/functions:

lambda.logic.True = function(a,b) {return a;};
lambda.logic.False = function(a,b) {return b;};

And now the all necessary logical connectives:

lambda.logic.nand = function(a,b) {return (a(b, lambda.logic.False))(lambda.logic.False, lambda.logic.True);};

Yep, thats it NAND its all we need 😀 because its functional complete meaning all other logical connectors can be constructed with NAND.

For example a NOT can be defined by:

lambda.logic.not = function(a) {return lambda.logic.nand(a, a);};

This may seems a little confusing but actually is very simple and clever, the True and False functions are the actually boolean values, so the NOT function is called like this:

console.log('not(True) = ' + (lambda.logic.not(lambda.logic.True) === lambda.logic.True?"True":"False"));
console.log('not(False) = ' + (lambda.logic.not(lambda.logic.False) === lambda.logic.True?"True":"False"));

This will print to log console:

not(True) = False 
not(False) = True 

Now lambda-calculus functions are lot of fun for small examples, but not so much when writing an entire program. Therefor I have write a small experimental code to do it for me:

lambda.program = function(prg) {
  var AST = esprima.parse(prg.toString());
  console.log(JSON.stringify(AST, null, 4));

  var str = lambda.gen(AST);
  return str;

lambda.gen = function(prg){
  switch (prg.type) {
    case 'Program':
      var result = '';
      for (var i in prg.body) {
        result += lambda.gen(prg.body[i]) + ';n';
      return result;
    case 'FunctionDeclaration':
      var body = prg.body.body;
      var f = 'function lambda_' + +
        '(' + lambda.genParams(prg.params) + ') {';

      for (i in body) {
        f += lambda.genFuncBody(body[i]);
      f += '}';

      return f;

  return "";

lambda.genParams = function(params) {
  var result, name;
  for (i in params) {
    name = params[i].name;
    result = result? result +','+ name : name;
  return result; 

lambda.genFuncBody = function(body) {
  switch (body.type) {
    case "ReturnStatement":
      return 'return ' + lambda.genFuncBody(body.argument) + ';';
    case "UnaryExpression":
      switch (body.operator) {
        case '!':
          return 'lambda.logic.not('+ lambda.genFuncBody(body.argument)+')';
    case "Identifier":

  return "";

Now I can write the following test functions in JavaScript:

function not(a) {return !a;}
function not_not(a) {return !(!a);}

And call my lambda.program to convert it to lambda functions:


lambd.program(not) will return a string with new function definition and eval will make it executable, the following are the generated functions:

function lambda_not(a) {return lambda.logic.not(a);};
function lambda_not_not(a) {return lambda.logic.not(lambda.logic.not(a));};

To avoid name clashing I used the lambda_ prefix in all generated functions.

And now they can be called like this:

console.log('lambda_not(True) = ' + (lambda_not(lambda.logic.True) === lambda.logic.True?"True":"False"));
console.log('lambda_not(False) = ' + (lambda_not(lambda.logic.False) === lambda.logic.True?"True":"False"));

console.log('lambda_not_not(True) = ' + (lambda_not_not(lambda.logic.True) === lambda.logic.True?"True":"False"));
console.log('lambda_not_not(False) = ' + (lambda_not_not(lambda.logic.False) === lambda.logic.True?"True":"False"));

console output:

lambda_not(True) = False
lambda_not(False) = True
lambda_not_not(True) = True
lambda_not_not(False) = False


Transversing the AST and generating different outputs and functions seems possible and not very difficult but handle all JavaScript possibilities seems a lot of work.
Another thing is that I want to use JS as a proof assistance and simple converting JS functions to pure lambda will not going to help, but generating property or auxiliary functions may prove to be useful.
Other options would probably be creating a tree representing the lambda conversion and apply validation tactics to it.

Well thats it for today, see you soon 😀

What do you think of this post?
  • Awesome (16.7%)
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  • Useful (33.3%)
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