lambda_calculus

lambda_calculus is a simple implementation of the untyped lambda calculus in Rust.

The library tries to find a compromise between the spirit of the lambda calculus and Rust's best practices; the lambda Terms implemented by the library are produced by functions (in order to allow arbitrary application), but they are not Copyable and the methods they provide allow memory-friendly disassembly and referencing their internals.

Documentation

Features

• a parser for lambda expressions
• 7 β-reduction strategies with optional display of reduction steps
• standard terms (combinators)
• Church numerals and arithmetic operations
• Church booleans
• Church pairs
• Church lists

The terms are implemented using De Bruijn indices, but are displayed using the classic lambda notation and can be parsed both ways.

The data and operators follow the Church encoding. Library functions utilizing the fixed-point combinator use its call-by-value variant and are built for compatibility with as many β-reduction strategies as possible. The bodies of functions are normalized for maximum performance.

Installation

Include the library by adding the following to your Cargo.toml:

[dependencies]
lambda_calculus = "^1.0"

And the following to your code:

#[macro_use]
extern crate lambda_calculus;

Compilation features:

• backslash_lambda: changes the display of lambdas from λ to \
• no_church: doesn't build the church module; useful if you want to implement it on your own or introduce a different data encoding

To apply a feature setup the dependency in your Cargo.toml like this:

[dependencies.lambda_calculus]
version = "^1.0"
features = ["no_church"]

Usage

Comparing classic and De Bruijn index notation

code:

use lambda_calculus::arithmetic::{succ, pred};

fn main() {
    println!("SUCC := {0} = {0:?}", succ());
    println!("PRED := {0} = {0:?}", pred());
}

stdout:

SUCC := λa.λb.λc.b (a b c) = λλλ2(321)
PRED := λa.λb.λc.a (λd.λe.e (d b)) (λd.c) (λd.d) = λλλ3(λλ1(24))(λ2)(λ1)

Parsing lambda expressions

code:

use lambda_calculus::parser::*;

fn main() {
    assert_eq!(parse(&"λa.λb.λc.b (a b c)", Classic), parse(&"λλλ2(321)", DeBruijn));
}

Showing β-reduction steps

code:

use lambda_calculus::reduction::*;
use lambda_calculus::arithmetic::pred;

fn main() {
    let mut expr = app!(pred(), 1.into());

    expr.beta(NOR, 0, true);
}

stdout:

β-reducing (λa.λb.λc.a (λd.λe.e (d b)) (λd.c) (λd.d)) (λa.λb.a b) [normal order]:

1. (λa.λb.λc.a (λd.λe.e (d b)) (λd.c) (λd.d)) (λa.λb.a b)
=>     λa.λb.(λc.λd.c d) (λc.λd.d (c a)) (λc.b) (λc.c)

2. (λc.λd.c d) (λc.λd.d (c a))
=>     λc.(λd.λe.e (d a)) c

3. (λc.(λd.λe.e (d a)) c) (λc.b)
=>     (λc.λd.d (c a)) (λc.b)

4. (λc.λd.d (c a)) (λc.b)
=>     λc.c ((λd.b) a)

5. (λc.c ((λd.b) a)) (λc.c)
=>     (λc.c) ((λc.b) a)

6. (λc.c) ((λc.b) a)
=>     (λc.b) a

7. (λc.b) a
=>     b

result after 7 reductions: λa.λb.b

Comparing the number of steps of different reduction strategies

code:

use lambda_calculus::reduction::*;
use lambda_calculus::arithmetic::fac;

fn main() {
    let expr = app!(fac(), 4.into());

    compare(&expr, &[NOR, APP, HNO, HAP], false); // compare normalizing strategies
}

stdout:

comparing β-reduction strategies for (λa.a (λb.λc.λd.b (λe.c (d e)) (λe.λf.e (d e f))) (λb.λc.b) (λb.λc.b c) (λb.λc.b c)) (λa.λb.a (a (a (a b)))):

normal:             87
applicative:        65
hybrid normal:      87
hybrid applicative: 40

TODO

• a few more tests
• more independent tests (less integration)
• further optimizations