lambda-8cc - An x86 C Compiler Written in Untyped Lambda Calculus

lambda-8cc is an x86 C compiler written as a monolithic closed untyped lambda calculus term.

When printed on letter-sized paper, it becomes 18,506 pages long on a 22 MB PDF without any figures. The PDF can be seen on my GitHub Pages here. The LaTeX source is 448 MB, and the LaTeX compilation log file main.log is 284 MB. I couldn't believe LaTeX was able to do that.

This gigantic lambda calculus term is a C compiler. Here is rot13.c, a program that compiles on GCC with no errors. The same program can be compiled using lambda-8cc producing the x86 executable rot13.bin, runnable on x86/x86-64 Linux:

$ echo 'Hello, world!' | ./rot13.bin
Uryyb, jbeyq!
$ echo 'Uryyb, jbeyq!' | ./rot13.bin
Hello, world!

Despite its massive size, compiling rot13.c finishes in 8 minutes on my machine using a lambda calculus interpreter. You can try it out on your own PC by cloning this repo. Running time stats are summarized in the Running Times and Memory Usage section. Note that although the compilation takes time, the compiled binary runs instantaneously.

Additional Feature: C to Lambda Calculus

As an additional feature, not only can lambda-8cc compile C to x86, but it can also compile C to lambda calculus terms, producing something like rot13.lam. Compiled lambda terms run on the same lambda calculus interpreter used to run lambda-8cc itself.

Using its compilation options, lambda-8cc can compile C to 5 different formats. Here is a full list of its features:

Compile C to:
- x86 executable (a.out)
- Lambda calculus term (runs on the same interpreter as lambda-8cc)
- Binary lambda calculus program
- SKI combinator calculus term (runnable as a Lazy K program)
- ELVM assembly listing (example: rot13.s)
Compile ELVM assembly to:
- x86/lambda calculus/BLC/SKI combinator calculus (all of the above)

Among the list is Lazy K, a minimal purely functional language with only 4 built-in operators, similar to the minimal imperative language BF which only has 8 instructions. I have covered a little bit about it in my blog post as well.

lambda-8cc is based on the following 3 projects: The first one is LambdaVM written by the author of this repo Hikaru Ikuta, a programmable virtual CPU written as an untyped lambda calculus term. This is combined with 8cc by Rui Ueyama, and a modified version of ELVM by Shinichiro Hamaji.

Gallery

The first page of the PDF looks like this. Notice the page count on the top left:

Its grand finale is a round of applause by a page full of right parentheses:

Overview

Lambdas All the Way Down

lambda-8cc is written as a closed untyped lambda calculus term ${\rm lambda8cc} = \lambda x. \cdots$ which takes an input string $x$ representing a C program and outputs an x86 Linux ELF executable expressed as a list of bytes.

Here, even strings are encoded as lambda terms. Characters and bytes are encoded as a list of bits with $0 = \lambda x. \lambda y.x$, $1 = \lambda x. \lambda y.y$, and lists are encoded in the Scott encoding with ${\rm cons} = \lambda x.\lambda y.\lambda f.(f x y)$, ${\rm nil} = \lambda x.\lambda y.y$.

Therefore, everything in the computation process, even including integers, is closed in the world of pure lambda terms, without the need of introducing any non-lambda type object whatsoever. It doesn't use any primitive types other than lambdas. lambda-8cc makes beta reduction the sole requirement for compiling C to x86. Note that the process doesn't depend on the choice of variable names as well. Instead of encoding the character A as a variable with the name $A$, A is encoded as a list of bits of its ASCII encoding 01000001.

The encoding process is a little cumbersome to say at the least to do by hand. This can be solved by using a lambda calculus interpreter. Various lambda calculus interpreters automatically handle this I/O format so that it runs on the terminal - standard input is encoded into lambda terms, and the output lambda term is decoded and shown on the terminal. Using these interpreters, lambda-8cc can be run on the terminal to compile C programs just like GCC.

For further details on how I/O is handled and how programs are written in lambda calculus, please see the implementation details of my other project LambdaLisp, a Lisp interpreter written as an untyped lambda calculus term.

C to Lambda Calculus

In addition to x86, lambda-8cc can compile C to lambda calculus as well. The output program runs on the same lambda calculus interpreter used to run lambda-8cc itself. Compiled lambda terms also run on minimal interpreters such as the 521-byte lambda calculus interpreter SectorLambda written by Justine Tunney, and the IOCCC 2012 "Most functional" interpreter written by John Tromp (its source is in the shape of a λ). This makes lambda-8cc self-contained in the realm of lambda calculus.

It has long been known in computer science that lambda calculus is Turing-complete. lambda-8cc demonstrates this in a rather straightforward way by showing that C programs can directly be compiled into lambda calculus terms.

The nice thing about lambda calculus is that the language specs are extremely simple. With lambda-8cc, we are preserving knowledge about how to compile C in a timeless method. Even if humanity loses knowledge about the x86 instruction set, as long as we remember the rules for lambda calculus and have the lambda term for lambda-8cc, we can still use the entire C language through lambda-8cc and build everything on top of it again.

Basic Usage Example

Here is a program rot13.c that encodes/decodes standard input to/from the ROT13 cipher. It compiles without errors using GCC:

// rot13.c: Encodes/decodes standard input to/from the ROT13 cipher

#define EOF -1

int putchar(int c);
char getchar(void);

char c;
int offset;

int main (void) {
    for (;;) {
        c = getchar();
        if (c == EOF) {
            break;
        }

        offset = 0;
        if (('a' <= c && c < 'n') || ('A' <= c && c < 'N')) {
            offset = 13;
        } else if (('n' <= c && c <= 'z') || ('N' <= c && c <= 'Z')) {
            offset = -13;
        }
        putchar(c + offset);
    }
    return 0;
}

The same program can be compiled by lambda-8cc out of the box as follows.

Build the Tools

First build the tools and prepare lambda-8cc:

$ make tools  # Build the interpreter uni++ and the tools lam2bin, asc2bin
$ unzip bin/lambda-8cc.lam.zip
$ cat lambda-8cc.lam | bin/lam2bin | bin/asc2bin > lambda-8cc.Blc  # Prepare format for uni++

The requirements are:

clang++ for building uni++
gcc or cc for building lam2bin and asc2bin

The tools built here are:

uni++: A very fast lambda calculus interpreter written by Melvin Zhang.
lam2bin: A utility written by Justine Tunney (available at https://justine.lol/lambda/), that converts plaintext lambda calculus notation such as \x.x to binary lambda calculus notation, the format accepted by uni++.
asc2bin: A utility that packs the 0/1 ASCII bitstream to bytes.

The tools are build via the lambda calculus development kit.

The conversion from lambda-8cc.lam to lambda-8cc.Blc is simply a transformation of notation for a format that's accepted by the interpreter uni++. Details are described in details.md.

Compile the Program Using lambda-8cc

Then rot13.c can be compiled as:

$ cat lambda-8cc.Blc examples/rot13.c | bin/uni++ -o > a.out
$ chmod 755 a.out

$ echo 'Hello, world!' | ./a.out
Uryyb, jbeyq!
$ echo 'Uryyb, jbeyq!' | ./a.out
Hello, world!

This runs in about 8 minutes on my machine. But be careful - it takes 145 GB of memory to run it! If you have free storage space or a USB drive, you can use a swap file with mkswap and swapon to extend the swap without configuring the partition settings. Also, by compiling the assembly and x86 executable separately, you can halve the RAM usage to 65 GB, as shown in the Detailed Usage section. Small programs such as putchar.c only take about 40 GB of memory. I suspect that the RAM usage can be decreased by introducing a mark-and-sweep GC to the interpreter, although I haven't confirmed it yet.

More running time stats are available in the Running Times and Memory Usage section. More example C programs compilable by lambda-8cc can be found under ./examples.

Other compilation options are described in the Detailed Usage section.

Detailed Usage

Compilation Options

Being written in lambda calculus, naturally, lambda-8cc's compilation options are expressed as lambda calculus terms as well. These options can be used to unlock the full features of lambda-8cc.

Compilation options are used by applying an optional term as (lambda-8cc option) beforehand of the input. This changes the behavior of the lambda term lambda-8cc so that it accepts/produces a different input/output format.

Here are all of lambda-8cc's compilation options:

Input	Output	Compilation Option
C	x86 executable	$\lambda f. (f ~ (\lambda x. \lambda y. x) ~ (\lambda x.\lambda y.\lambda z.\lambda a.\lambda b.x) ~ (\lambda x.x))$
C	Plaintext lambda calculus term	$\lambda f. (f ~ (\lambda x. \lambda y. x) ~ (\lambda x.\lambda y.\lambda z.\lambda a.\lambda b.y) ~ (\lambda x.x))$
C	Binary lambda calculus notation (BLC program)	$\lambda f. (f ~ (\lambda x. \lambda y. x) ~ (\lambda x.\lambda y.\lambda z.\lambda a.\lambda b.z) ~ (\lambda x.x))$
C	SKI combinator calculus (Lazy K program)	$\lambda f. (f ~ (\lambda x. \lambda y. x) ~ (\lambda x.\lambda y.\lambda z.\lambda a.\lambda b.a) ~ (\lambda x.x))$
C	ELVM assembly	$\lambda f. (f ~ (\lambda x. \lambda y. x) ~ (\lambda x.\lambda y.\lambda z.\lambda a.\lambda b.b) ~ (\lambda x.x))$
ELVM assembly	x86 executable	$\lambda f. (f ~ (\lambda x. \lambda y. y) ~ (\lambda x.\lambda y.\lambda z.\lambda a.\lambda b.x) ~ (\lambda x.x))$
ELVM assembly	Plaintext lambda calculus term	$\lambda f. (f ~ (\lambda x. \lambda y. y) ~ (\lambda x.\lambda y.\lambda z.\lambda a.\lambda b.y) ~ (\lambda x.x))$
ELVM assembly	Binary lambda calculus notation (BLC program)	$\lambda f. (f ~ (\lambda x. \lambda y. y) ~ (\lambda x.\lambda y.\lambda z.\lambda a.\lambda b.z) ~ (\lambda x.x))$
ELVM assembly	SKI combinator calculus (Lazy K program)	$\lambda f. (f ~ (\lambda x. \lambda y. y) ~ (\lambda x.\lambda y.\lambda z.\lambda a.\lambda b.a) ~ (\lambda x.x))$

Each option is in the format of a 3-tuple ${\rm cons3} ~ {\rm input} ~ {\rm output} ~ X$ where ${\rm cons 3} = \lambda x. \lambda y. \lambda z. \lambda f. (f x y z)$. The first element ${\rm input}$ is a selector of a 2-tuple specifying the input format. The second element ${\rm output}$ is a selector of a 5-tuple specifying the output format. The third element $X = \lambda x.x$ is a placeholder used to distinguish the data structure from the standard input, also existing for backward portability in the case when more options are added in the future.

Applying Compilation Options

The compilation options shown before can be used in the terminal as follows.

To compile C to an ELVM assembly listing a.s:

( ( cat lambda-8cc.lam; printf '(\\f.(f (\\x.\\y.x) (\\x.\\y.\\z.\\a.\\b.b) (\\x.x)))' ) \
  | bin/lam2bin | bin/asc2bin; cat input.c ) | bin/uni++ -o > a.s

To compile an ELVM assembly listing a.s to x86 executable a.out:

( ( cat lambda-8cc.lam; printf '(\\f.(f (\\x.\\y.y) (\\x.\\y.\\z.\\a.\\b.x) (\\x.x)))' ) \
  | bin/lam2bin | bin/asc2bin; cat a.s ) | bin/uni++ -o > a.out
chmod 755 a.out

As described before, by separately compiling a.s and a.out using these commands, the maximum RAM usage can be cut in half since the memory is freed when each process finishes.

By running lambda-8cc without any input or options, you can see a usage message showing the full set of options:

$ cat lambda-8cc.lam | bin/lam2bin | bin/asc2bin | bin/uni++ -o
lambda-8cc v1.0.0

Usage:
    apply lambda-8cc.lam [input-file]
    apply lambda-8cc.lam [option] [input-file]

Options:
    (\f.(f [input] [output] (\x.x)))
    (\f.(f (\x.\y.x) (\x.\y.\z.\a.\b.x) (\x.x))) : C to x86 (defualt)
    (\f.(f (\x.\y.x) (\x.\y.\z.\a.\b.y) (\x.x))) : C to *.lam (plaintext lambda calculus program)
    (\f.(f (\x.\y.x) (\x.\y.\z.\a.\b.z) (\x.x))) : C to *.blc (binary lambda calculus program)
    (\f.(f (\x.\y.x) (\x.\y.\z.\a.\b.a) (\x.x))) : C to *.lazy (SKI combinator calculus, as a Lazy K program)
    (\f.(f (\x.\y.x) (\x.\y.\z.\a.\b.b) (\x.x))) : C to ELVM assembly
    (\f.(f (\x.\y.y) (\x.\y.\z.\a.\b.x) (\x.x))) : ELVM assembly to x86
    (\f.(f (\x.\y.y) (\x.\y.\z.\a.\b.y) (\x.x))) : ELVM assembly to *.lam
    (\f.(f (\x.\y.y) (\x.\y.\z.\a.\b.z) (\x.x))) : ELVM assembly to *.blc
    (\f.(f (\x.\y.y) (\x.\y.\z.\a.\b.a) (\x.x))) : ELVM assembly to *.lazy

lambda-8cc includes the following projects. All of the following projects
are released under the MIT license. See the LICENSE in each location for details.
    8cc: By Rui Ueyama - https://github.com/rui314/8cc
    ELVM: By Shinichiro Hamaji - https://github.com/shinh/elvm
    LambdaVM: By Hikaru Ikuta - https://github.com/woodrush/lambdavm
    lambda-8cc: By Hikaru Ikuta - https://github.com/woodrush/lambda-8cc

Running Times and Memory Usage

The following table shows the compilation time and memory usage on Melvin Zhang's lambda calculus interpreter.

Program	Compilation Time	Max. RAM Usage at Compilation Time	x86 Binary Size	Description
putchar.c	1.8 min	31 GB	342 bytes	Prints `A`
hello.c	2.4 min	42 GB	802 bytes	Prints `Hello, world!`
echo.c	2.5 min	46 GB	663 bytes	Echoes standard input
rot13.c	7.7 min	84 GB	2,118 bytes	Encodes/decodes stdin to/from ROT13
fizzbuzz.c	49.7 min	240 GB	5,512 bytes	Prints FizzBuzz sequence up to 30
primes.c	53.0 min	241 GB	5,500 bytes	Prints primes up to 100

Now that is a lot of memory! To compile programs that require a huge RAM, you can extend your swap region without changing the partition settings by using a swap file. If you run Linux and have any free storage or a USB drive, you can use that storage to easily and dynamically extend your swap region using mkswap and swapon. The stats on this table are run with an extended swap region this way. Instructions are explained in this askubuntu thread. I suspect that the RAM usage can be decreased by introducing a mark-and-sweep GC to the interpreter, although I haven't confirmed it yet.

Note that these are the compilation times - the running times for the compiled x86 binary are instantaneous. This even holds when compiling C to lambda calculus terms. Compiled lambda terms also run instantaneously and only use a few gigabytes of memory when run on a lambda calculus interpreter.

The compilations for these stats were run on an Ubuntu 22.04.1 machine with 48 GB RAM, 16GB SSD swap (default partition), and 274GB (256GiB) HDD swap (dynamically added with mkswap and swapon). The running time shown here is the wall clock running time including memory operations. For swap-heavy programs, the running time could be decreased by using a device with a faster I/O speed.

The stats were measured by running

cp examples/[program].c ./input.c
make

which compiles a.s and a.out for input.c separately to save the total memory usage. A more detailed table of stats for each pass is shown in details.md.

How is it done?

Please see details.md.

Building From Source

For details on building from source, please see details.md.

Credits

lambda-8cc is a combination of 3 projects, LambdaVM, ELVM, and 8cc. LambdaVM was written by Hikaru Ikuta, the author of this repository (lambda-8cc). The ELVM architecture was written by Shinichiro Hamaji. 8cc was written by Rui Ueyama. The version of 8cc used in lambda-8cc is a modified version of 8cc included as a part of ELVM, modified by Shinichiro Hamaji and others. lambda-8cc also includes elc, a part of ELVM written by Shinichiro Hamaji, modified by Hikaru Ikuta so that it can compile ELVM assembly to lambda calculus. The lambda calculus backend for ELVM was written by Hikaru Ikuta, by integrating LambdaVM into ELVM. The running time and memory usage statistics were measured using a lambda calculus interpreter written by Melvin Zhang. lam2bin was written by Justine Tunney.

chloe0x0/lambda-8cc