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Lambda Zero is a minimalist pure lazy functional programming language with:

  • All the standard Lambda Calculus features (first class functions, closures, currying, recursion, nested function definitions, ...)
  • Lazy evaluation (infinite data structures, increased modularity, ...)
  • Algebraic data types (enumerations, tuples, lists, trees, records/structs, maybe/optional types, checked exceptions, monads, ...)
  • Pattern matching (case expressions, tuple destructuring, ...)
  • Automatic garbage collection
  • Semantic whitespace
  • Unicode support (UTF8)
  • User-defined lexically-scoped operator syntax and semantics
  • Uniform function call syntax (z.f(x, y) means f(x, y, z))
  • Hindley-Milner static type inference (implemented in Lambda Zero)
  • A self-interpreter

And the whole language can be interpreted with less than 2500 lines of C code!

Sample Code

Hello World

main(input) ≔ "hello world"


n ! ≔ n ⦊ 0 ↦ 1; ↑ n′ ↦ n ⋅ n′ !


sort ≔ [] ↦ []; n ∷ ns ↦ sort(ns ¦ (≤ n)) ⧺ [n] ⧺ sort(ns ¦ (> n))

Infinite list of natural numbers

iterate(f, x) ≔ x ∷ iterate(f, f(x))
(…) ≔ iterate(↑)
naturals ≔ 0 …

Infinite list of Fibonacci numbers

fibonaccis ≔ f(0, 1) where f(m, n) ≔ m ∷ f(n, m + n)

Infinite list of prime numbers

primes ≔ p(2 …) where p ≔ [] ↦ []; n ∷ ns ↦ n ∷ p(ns ¦ (% n ≠ 0))


The Lambda Calculus is incredibly elegant and can be a very powerful high-level programming language with just a few simple optimizations and syntactic sugars.

Lambda Zero allows you to program in the Lambda Calculus with just the minimal amount of added features to make it practical and readable. Lambda Zero combines the elegance of Haskell with the simplicity of LISP.

Use Cases

The primary use case of Lambda Zero is to bootstrap a future higher level language "Lambda One" i.e. Lambda Zero is a sublanguage of Lambda One and the Lambda One compiler is written entirely in Lambda Zero.

Lambda Zero is also a good way to learn about functional programming since it is much simpler than other languages like Haskell, but operates on the same foundation.

Lambda Zero is not a general purpose language; the interpreter is optimized for simplicity over performance and the language does not allow any I/O besides accepting stdio as a string and returning a string which is sent to stdout by the interpreter.


Core Language

First we will describe the core language without syntactic sugars, which is very small. These are the only things you can do in the core language:

  • Define a one-parameter function with the arrow operator: (x -> body)
  • Call a one-parameter function by putting a space between the function name f and the argument x: (f x)
  • Refer to the parameter of a function by name: x
  • Refer to an integer: 123
  • Refer to a built-in operator: +

For example, we can combine all of these to write a function that returns its argument plus one (just ignore the parentheses around the plus sign; they are to disable a syntactic sugar):

(x -> (((+) x) 1))

Note that you can construct a function that effectively takes two parameters by making a one-parameter function that returns a one-parameter function, so that it takes two function applications to get to the body. This is the concept of currying. For example, to make a function that adds two arguments:

(x -> (y -> (((+) x) y)))

The desugared language can be described by the grammar rules below (technically the desugared language is a sublanguage of expr because this grammar ignores some error cases):

natural = [0-9]+
builtin = '+' | '-' | '*' | '//' | '%' | '=' | '=/=' | '<' | '>' | '<=' | '>=' | 'up' | 'error'
name = ("a token that is not a natural, builtin, delimiter, or arrow")
expr = natural | builtin | name | (name -> expr) | (expr expr)

Syntactic sugars

Let x, y, z be generic expressions, f be a name expression, and * be an operator name expression.

  • Left associative spaces: x y z desugars to ((x y) z)
  • Right associative newlines: x \n y \n z desguars to (z (y z))
  • Redundant Parentheses: (x) desugars to x unless x is an operator (see Operator names sugar below)
  • Definitions (Let expressions): (f := x) y desugars to (f -> y) x
  • Trailing definitions: (x (f := y)) desguars to (x ((f := y) f)) If there is nothing after a definition, it as treated as if the name of the definition follows the definition.
  • Function definitions: f x y := z desugars to f := x -> y -> z
  • Recursive definitions: The right hand sise of a function definition can refer to the function name, in which case the Y combinator is used to convert the definition to a non-recursive one.
  • Curried function application: f(x, y) desugars to f x y
  • Infix operators: x * y desugars to * x y When operators are chained like x * y * z there are precedence and associatvitiy rules that determine how it is parsed.
  • Operator names: (*) disables the infix operator sugar on *
  • Sections: (* y) desguars to x -> (x * y) and (y *) desugars to x -> (y * x)
  • String literals: "abc" desugars to a Church-encoded list of ascii character codes.
  • Character literals: 'a' desugars to the ascii character code for a
  • List literals: [x, y, z] desugars to the Church-encoded list of x, y, and z.
  • Tuple destructuring: (a, b, c) := x desugars to the equivalent of a := first(x), b := second(x), and c := third(x). Tuples can also be used on the left side of -> and in parameters on the left side of function definitions. Destructuring is also recursive e.g. ((a, b), (c, d)) := x.


There is no I/O in the lambda zero language, but there is some basic I/O support in the lambda zero interpreter. In other words, there is no syntax in a lamda zero program that actually performs I/O itself, but there is a special function main that the interpreter treats differently to allow I/O.

When the last function definition of a program defines the symbol main, the interpreter assumes that it is a function and passes it a string corresponding to stdin and then assumes that the return value of main is a string and prints it to stdout. An error will occur if these assumptions aren't true.

A string is a linked list of integers in the range 0-255, where the linked list is constructed according to the standard Church encoding.

The input and output are both evaluated lazily, which allows for non-terminating output and interactive programs. See the samples below for examples of this.


That's the whole language! Take a look at the prelude for more examples.


Make sure a C compiler is installed and run the make script in the bootstrap-interpreter directory:

cd bootstrap-interpreter


From the bootstrap-interpreter directory:


For example, try this sample program that prints out an infinite list of prime numbers:

./run test/samples/

The run script prepends the prelude to the SOURCEFILE and passes the result to main.

Note that when using the run script, line numbers in error messages will be offset by the number of lines in the prelude.


Lambda Zero is in version 0.x, so breaking changes can be made in any commit.

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