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Programming language: PHP
License: MIT License
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Lambda calculus interpreter in PHP.

Lambda calculus

Lambda calculus is a very minimal programming language that was invented in 1936 by Alonzo Church. It is the functional equivalent of the Turing Machine.

Lambda calculus has only three concepts: Function definitions, lexically scoped variables, function application.

An example term would be the identity function:


The first part 位x defines a function that takes an x, the . signifies that the part that follows is the function body. The body just returns x.

In PHP, you would write the same thing as follows:

function ($x) {
    return $x;

You can nest function definitions. Here is a function returning a function:


And you can also apply a function to an argument, which just means calling the function.

位f.位g.f g

Which is the short hand (left-associative) form of writing

位f.位g.(f g)

Nested calls like:

位f.位g.位h.f g h

Are interpreted as:

位f.位g.位h.((f g) h)

If you want to change the grouping to be right-associative, you need to explicitly group them in parentheses:

位f.位g.位h.(f (g h))

Interestingly, lambda calculus is turing complete. Using just these three concepts you can represent any computation.

Check out the links at the bottom for more details on how to do stuff in lambda calculus.


This project consists of a lambda calculus expression parser using dissect, and an eval-apply interpreter based on Matt Might's implementation in scheme.

The interpreter is call-by-value which means that recursive calls need to be wrapped in a function to prevent them from being evaluated eagerly.

For examples of how to do numbers (church encoding), booleans, arithmetic, boolean logic, looping (recursion), etc. look at example.php.


This project ships with a read-eval-print-loop that you can use to evaluate lambda calculus expressions:

$ php repl.php

By default, it is in int-mode, expecting the result of the expression to be a church-encoded number. Example:

$ php repl.php
i> 位f.位x.f (f (f x))

You can switch to bool-mode by sending the b command:

$ php repl.php
i> b
b> 位x.位y.x

Or r for raw mode:

$ php repl.php
i> r
r> 位x.x


A few things are still a work in progress:

  • Krivine machine: This alternate interpreter would allow call-by-need and indexing into de-bruijn indices, which is needed by...

  • Binary lambda calculus: Allows encoding lambda calculus programs in binary form which produces extremely small programs. This also defines an I/O mechanism.


Thanks to