# Binary lambda calculus

**Binary lambda calculus** (**BLC**) is an extremely small Turing-complete language which can be represented as a series of bits or bytes. Unlike Binary combinatory logic, another binary language with a similar acronym, it is capable of input and output.

## De Bruijn Index

I suggest knowing what a De Bruijn index is before trying to understand this language. In case you don't want to do that, here's an explanation. The explanation also features lambda calculus code which has more parentheses than normal in order to better map onto BLC.

Lambda calculus can use names to denote the term corresponding to each lambda, like the following example.

λxλy.y

This function takes in two inputs x and y and outputs the final one it took in. In this, you know that the term y is referring to the second lambda, the one with y after it. But the term names are arbitrary, which is even more annoying when all you have is binary.

A De Bruijn index can be used to replace the term in lambda calculus without naming the term. The index is a number which counts from the final lambda in the scope until it finds the lambda which the term is referencing.

In BLC, this number starts from 1. So the De Bruijn index in BLC basically indicates how many lambdas the term looks at from the end until it looks at the lambda it's actually referencing, with 1 referring to the last lambda, 2 to the second-last, 3 to the third-last, etc.

The following code is equivalent to the code above, but using a De Bruijn index.

λλ.1

Note that this is *in the scope*. It is possible for there to be lambdas that are in a different scope, and lambdas in a lower scope are skipped in the count.

Here are some examples of lambda calculus with named terms and lambda calculus with De Bruijn indices. I have deliberately put spaces between the numbers so it's easier to understand what's going on.

Command | Description | Notes |
---|---|---|

λxλy.(yx) | λλ.(2 1) | outputs in opposite order |

λxλyλz.((xz)(yz)) | λλλ.((3 1)(2 1)) | S combinator |

λx.(xλy.(yy)) | λ.(1λ.(1 1)) | takes in one input, outputs that input and this other lambda. Note the lambda doesn't need parentheses before it. |

λxλy.((x λz.(zz))y) | λλ.((2 λ.(1 1)) 1) | Note that 1 replaces y and 2 replaces x. They can't be arguments of the inner lambda, so they treat it like it doesn't exist. |

## BLC Syntax

The BLC program is a sequence of bits read left to right. The following commands are defined. Feel free to change how the commands are explained if you think it's too confusing.

`00x`

= Lambda function with body x`x0`

, where x is one or more "1" bits = the number of "1" bits serves as the De Bruijn index- For example
`10`

corresponds to a De Bruijn index of 1,`110`

to 2,`1110`

to 3, etc. - To be clear, this means 10 refers to the final lambda, 110 to the second-last, 1110 to the third-last, etc.

- For example
`01xy`

, where x and y are more code = x is applied to y.- By default, the output of a lambda is one term in this language. Using 01 allows the function to output both x and y, which is the same as applying x to y in lambda calculus because both of the outputs are also lambdas.
- If you want to take in one input and output it twice, you would write 00011010 = 00 01 10 10.
- If you want to take in two inputs and output the first one three times, you would write 00000101110110110 = 00 00 01 01 110 110 110.
- If you want to output code directly starting with 00, it doesn't need to have 01 directly before it. If there is a term before the function, then 01 is needed before the term.

It is possible for the code to have padding at the right end, i.e. code which doesn't affect the result of the command. This fact is especially useful when trying to use bytes to represent this language.

## Basic Program Information

A program is a lambda calculus term that transforms an input to an output. Standard input is represented as a list of boolean values, and standard output has the same format.

A 0 bit in BLC is 0000110 (True), and a 1 bit is 000010 (False), which are the normal lambda calculus representations of these values.

0000110 = 00 00 110 *taking two inputs (with the two lambda functions represented by 00), return the second argument from the inside i.e. the first argument 000010 = 00 00 10 *taking two inputs (with the two lambda functions represented by 00), return the innermost / last argument i.e. the second argument

The empty list, called nil, is 000010 (False).

A list with multiple elements is represented by the pairing or cons function 00010110xy, where x is the head of the list and y is the tail.

00010110xy = 00 01 01 10 x y *taking one input, output that input, the head of the list, and the tail of the list

You might expect programs to consist of multiple bytes, considering all of these have been six bits or over. However, printing out input in this language is done through the code `0010`

, which takes one input (which is the innermost by default) and prints it out. Because padding is ignored and lambdas only output one term by default in this language, a program consisting of just a cat can be represented by any bytes between 32 (00100000) and 47 (00101111) because everything after 0010 is ignored (remember that you have to type 00011010 to output the input twice).

## SKI combinator calculus

The encoding of lambda term S is λxλyλz.((xz)(yz)), which is written as λλλ.((3 1)(2 1)) using De Bruijn indexes instead of names, and as 00 00 00 01 01 1110 10 01 110 10 in BLC.

The K combinator is written as λxλy.x or λλ.2 in a corresponding format, so it would be 00 00 110 in BLC.

The identity function I is the same as the cat: 00 10.

Therefore, you can implement SKI combinator calculus in BLC.

## BLC8

BLC operates on a stream of bits (values of 0 and 1), while BLC8 is the same, but operates on a stream of bytes (values from 0 - 255) with the most significant bit in the smallest value (big-endian). In the following programs, BLC and BLC8 programs are put into different parts.

## Programs (BLC)

### self-interpreter

01010001 10100000 00010101 10000000 00011110 00010111 11100111 10000101 11001111 000000111 10000101101 1011100111110 000111110000101 11101001 11010010 11001110 00011011 00001011 11100001 11110000 11100110 11110111 11001111 01110110 00011001 00011010 00011010

### prime number sieve

000100011001100101000110100 000000101100000100100010101 11110111 101001000 11010000 111001101 000000000010110111001110011 11111011110000000011111001 10111000 00010110 0000110110

### Brainfuck interpreter

0000000 01a15144 02d55584 223070b7 00f032ff 0000020 7f85f9bf 956fe15e c0ee7d7f 006854e5 0000040 fbfd5558 fd5745e0 b6f0fbeb 07d62ff0 0000060 d7736fe1 c0bc14f1 1f2eff0b 17666fa1 0000100 2fef5be8 ff13ffcf 2034cae1 0bd0c80a 0000120 e51fee99 6a5a7fff ff0fff1f d0049d87 0000140 db0500ab 3bb74023 b0c0cc28 10740e6c 0000160

### Universal Turing Machine

0101000110100000000101011000000000011110000101111110011110 0001011100111100000011110000101101101110011111000011111000 0101111010011101001011001110000110110000101111100001111100 0011100110111101111100111101110110000110010001101000011010

## Programs (BLC8)

### self-interpreter

19468 05580 05f00 bfe5f 85f3f 03c2d b9fc3f8 5e9d65e5f 0decb f0fc3 9befe 185f7 0b7fb 00cf6 7bb03 91a1a

There is also a universal Turing machine written in BLC8 which is slightly longer than the one in BLC.

## Computational Class

Lambda calculus is Turing-complete, and because binary lambda calculus is a version of lambda calculus, it is also Turing-complete. It can also implement SKI combinator calculus as seen above, so it is Turing-complete in that way. As a final indication of Turing-completeness, Brainfuck and a Universal Turing Machine have both been represented in this language.