John Tromp

john.tromp@gmail.com

http://tromp.github.io/

On a 32-bit machine:

```
make tromp
```

On a 64-bit machine:

```
make tromp64
# And mentally substitute ./tromp64 for ./tromp everywhere below
```

```
cat ascii-prog.blc data | ./tromp -b
cat binary-prog.Blc data | ./tromp
```

```
(cat hilbert.Blc; echo -n 1234) | ./tromp
(cat oddindices.Blc; echo; cat primes.blc | ./tromp -b) | ./tromp
cat primes.blc | ./tromp -b | ./primes.pl
```

The judges dare to say that the data files this entry is processing are more obfuscated than the entry itself.

This program celebrates the close connection between obfuscation and conciseness, by implementing the most concise language known, Binary Lambda Calculus (BLC).

BLC was developed to make Algorithmic Information Theory, the theory of smallest programs, more concrete. It starts with the simplest model of computation, the lambda calculus, and adds the minimum amount of machinery to enable binary input and output.

More specifically, it defines a universal machine, which, from an input stream of bits, parses the binary encoding of a lambda calculus term, applies that to the remainder of input (translated to a lazy list of booleans, which have a standard representation in lambda calculus), and translates the evaluated result back into a stream of bits to be output.

Lambda is encoded as 00, application as 01, and the variable bound by the n'th enclosing lambda (denoted n in so-called De Bruijn notation) as 1^{n}0. That’s all there is to BLC!

For example the encoding of lambda term S = \x \y \z (x z) (y z), with De Bruijn notation \ \ \ (3 1) (2 1), is 00 00 00 01 01 1110 10 01 110 10

In the closely related BLC8 language, IO is byte oriented, translating between a stream of bytes and a list of length-8 lists of booleans.

The submission implements the universal machine in the most concise manner conceivable. It lacks #defines and #includes, and compiles to a (stripped) executable of under 6K in size.

Without arguments, it runs in byte mode, using standard in- and output. With one (arbitrary) argument, it runs in bit mode, using only the least significant bit of input, and using characters ‘0’ and ‘1’ for output.

The program uses the following exit codes: 0: OK; result is a finite list 5: Out of term space 6: Out of memory 1,2,3,4,8,9: result not in list form

The size of the term space is fixed at compile time with -DA

The shortest (closed) lambda calculus term is \x x (\ 1 in De Bruijn notation) which is the identity function. When its encoding 0010 is fed into the universal machine, it will simply copy the input to the output. (well, not that simply, since each byte is smashed to bits and rebuilt from scratch) Voila: a half byte cat:

```
echo " Hello, world" | ./tromp
Hello, world
```

Since the least significant 4 bits of the first byte are just arbitrary padding that is ignored by the program, any character from ASCII 32 (space) through 47 (/) will do, e.g.:

```
echo "*Hello, world" | ./tromp
Hello, world
```

If the input doesn’t start with a valid program, that is, if the interpreter reaches end-of-file during program parsing, it will crash in some way:

```
echo -n "U" | ./tromp
Segmentation fault
```

Furthermore, the interpreter requires the initial encoded lambda term to be closed, that is, variable n can only appear within at least n enclosing lambdas. For instance the term \ 5 is not closed, causing the interpreter to crash when looking into a null-pointer environment:

```
echo ">Hello, world" | ./tromp
Segmentation fault
```

Since these properties can be checked when creating BLC programs, the interpreter doesn’t bother checking for it.

The BLC universal machine may be small at 650 bytes of C (952 bytes including layout), but written as a self interpreter in BLC it is downright minuscule at 232 bits (29 bytes):

```
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
```

The byte oriented BLC8 version weighs in at 43 bytes (shown in hexadecimal).

```
19468
05580
05f00
bfe5f
85f3f
03c2d
b9fc3f8
5e9d65e5f
0decb f0fc3
9befe 185f7
0b7fb 00cf6
7bb03 91a1a
(cat uni8.Blc; echo " Ni hao") | ./tromp
Ni hao
```

Even shorter than the self-interpreter is this prime number sieve in 167 bits (under 21 bytes):

```
000100011001100101000110100
000000101100000100100010101
11110111 101001000
11010000 111001101
000000000010110111001110011
11111011110000000011111001
10111000
00010110
0000110110
```

The n'th bit in the output indicates whether n is prime:

```
cat primes.blc | ./tromp -b | head -c 70
0011010100010100010100010000010100000100010100010000010000010100000100
```

For those who prefer to digest their primes in decimal, there is oddindices.Blc, which will print the indices of all odd characters (with lsb = 1) separated by a given character:

```
(cat oddindices.Blc; echo -n " "; cat primes.blc | ./tromp -b) | ./tromp | head -c 70
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97
```

Program hilbert.Blc, at 143 bytes, is a very twisty “one-liner” (shown in hexadecimal):

```
1818181 8111154 6806041 55ff041
9d f9 de 16 ff fe 5f 3f
ef f615ff9 46 84 058117e 05
cb fe bc bf
ee86cb9 4681600 5c0bfac bfbf71a
85 e0 5c f4
14d5fe0 8180b048d0800e078 016445f
fe 5f
f7 ffffe5fff2fc 02f7ad97f5bf ff
ff bf ff ca af ff
7817ffa df76695 4680601 57f7e16
05 c1
3fe80b2 2c18581 bfe5c10 42ff805
de ec 06 c2 c0 c0
60 8191a00167fb cbcfdf65f7c0 a20
```

It expects n arbitrary characters of input, and draws a space filling Hilbert curve of order n:

```
(cat hilbert.Blc; echo -n "1") | ./tromp
_
| |
(cat hilbert.Blc; echo -n "12") | ./tromp
_ _
| |_| |
|_ _|
_| |_
(cat hilbert.Blc; echo -n "123") | ./tromp
_ _ _ _
| |_| | | |_| |
|_ _| |_ _|
_| |_____| |_
| ___ ___ |
|_| _| |_ |_|
_ |_ _| _
| |___| |___| |
(cat hilbert.Blc; echo -n "1234") | ./tromp
_ _ _ _ _ _ _ _
| |_| | | |_| | | |_| | | |_| |
|_ _| |_ _| |_ _| |_ _|
_| |_____| |_ _| |_____| |_
| ___ ___ | | ___ ___ |
|_| _| |_ |_| |_| _| |_ |_|
_ |_ _| _ _ |_ _| _
| |___| |___| |_| |___| |___| |
|_ ___ ___ ___ ___ _|
_| |_ |_| _| |_ |_| _| |_
| _ | _ |_ _| _ | _ |
|_| |_| | |___| |___| | |_| |_|
_ _ | ___ ___ | _ _
| |_| | |_| _| |_ |_| | |_| |
|_ _| _ |_ _| _ |_ _|
_| |___| |___| |___| |___| |_
```

The smallest known BF interpreter is written in… you guessed it, BLC, coming in at 112 bytes (including 3 bits of padding):

```
od -t x4 bf.Blc
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
```

It expects its input to consist of a Brainfuck program (looking only at bits 0,1,4 to distinguish among ,-.+<>][ ) followed by a ], followed by the input for the Brainfuck program.

```
more hw.bf
++++++++++[>+++++++>++++++++++>+++>+<<<<-]>++.>+.+++++++..+++.>++.<<+++++++++++++++.>.+++.------.--------.>+.>.]
cat bf.Blc hw.bf | ./tromp
Hello World!
```

Curiously, the interpreter bf.Blc is the exact same size as hw.bf.

Writing BLC programs can be made slightly less painful with this parser that translates single-letter-variable lambda calculus into BLC:

```
echo "\f\x f (f (f x))" > three
cat parse.Blc three | ./tromp
000001110011100111010
```

THe program inflate.Blc and its inverse deflate.Blc allow us to translate between BLC and BLC8. If you assemble a byte oriented program, you’ll need to compact it into BLC8:

So we could assemble an input reversing program as

```
echo "\a a ((\b b b) (\b \c \d \e d (b b) (\f f c e))) (\b \c c)" > reverse
cat parse.Blc reverse | ./tromp > reverse.blc
```

and change it to BLC8 with

```
cat deflate.Blc reverse.blc | ./tromp > rev.Blc
wc rev.Blc
0 1 9 rev.Blc
```

and then try it out with:

```
cat rev.Blc - | ./tromp
Hello, world!
^D
!dlrow ,olleH
```

BLC8 program symbolic.Blc shows individual reduction steps on symbolic lambda terms.
Here it is used to show the calculation of 2^{3} in Church numerals:

```
echo "(\f\x f (f (f x))) (\f\x f (f x))" > threetwo
cat parse.Blc threetwo | ./tromp > threetwo.blc
cat symbolic.Blc threetwo.blc | ./tromp
(\a \b a (a (a b))) (\a \b a (a b))
\a (\b \c b (b c)) ((\b \c b (b c)) ((\b \c b (b c)) a))
\a \b (\c \d c (c d)) ((\c \d c (c d)) a) ((\c \d c (c d)) ((\c \d c (c d)) a) b)
\a \b (\c (\d \e d (d e)) a ((\d \e d (d e)) a c)) ((\c \d c (c d)) ((\c \d c (c d)) a) b)
\a \b (\c \d c (c d)) a ((\c \d c (c d)) a ((\c \d c (c d)) ((\c \d c (c d)) a) b))
\a \b (\c a (a c)) ((\c \d c (c d)) a ((\c \d c (c d)) ((\c \d c (c d)) a) b))
\a \b a (a ((\c \d c (c d)) a ((\c \d c (c d)) ((\c \d c (c d)) a) b)))
\a \b a (a ((\c a (a c)) ((\c \d c (c d)) ((\c \d c (c d)) a) b)))
\a \b a (a (a (a ((\c \d c (c d)) ((\c \d c (c d)) a) b))))
\a \b a (a (a (a ((\c (\d \e d (d e)) a ((\d \e d (d e)) a c)) b))))
\a \b a (a (a (a ((\c \d c (c d)) a ((\c \d c (c d)) a b)))))
\a \b a (a (a (a ((\c a (a c)) ((\c \d c (c d)) a b)))))
\a \b a (a (a (a (a (a ((\c \d c (c d)) a b))))))
\a \b a (a (a (a (a (a ((\c a (a c)) b))))))
\a \b a (a (a (a (a (a (a (a b)))))))
```

As expected, the resulting normal form is Church numeral 8.

Taking only the first line of output gives us a sort of BLC disassembler, an exact inverse of the above assembler. The prime number sieve disassembles as follows:

```
cat symbolic.Blc primes.blc | ./tromp | head -1
\a (\b b (b ((\c c c) (\c \d \e e (\f \g g) ((\f c c f ((\g g g) (\g f (g g)))) (\f \g \h \i i g (h (d f))))) (\c \d \e b (e c))))) (\b \c c (\d \e d) b)
```

Hardly any less obfuscated…

The last line of cat symbolic.Blc primes.blc | ./tromp | head -16 starts out as \a \b b (\c \d c) (\c c (\d \e d) (\d d (\e \f f) (\e e (\f \g g) ((\f (\g \h \i

The \a is for ignoring the rest of the input (to which the universal machine applies the decoded lambda term). The \b b (..) (…) is the list constructor, usually called cons, applied to a head (a list element) and a tail (another list). In this case the element is (\c \d c), which represents the boolean true, and which we use to represent a 0 bit. This is the bit that says 0 is not prime. The next list element (following another cons) is (\d \e d). Another 0 bit, this time saying that 1 is not prime. The third list element is (\e \f f), a 1 bit, confirming our suspicion that 2 is prime. As is the next number, according to (\f \g g). We can see that the tail after the first 4 elements is still subject to further reduction. The bit for number 4 will show up for the first time in line 30, as (\g \h g), or 0, as the result of zeroing out all multiples of the first prime, 2. Since my computer reaches swap hell before line 40, we can’t see the next bit arriving, at least not in this symbolic reduction.

Performance is quite decent, and amazingly good for such a tiny implementation, being roughly 50% slower than a Haskell implementation of the universal machine using so-called Higher Order Abstract Syntax which relies on the highly optimized Haskell runtime system for evaluation. Of course individual blc programs running under the interpreter perform much worse than that same program written in Haskell.

Our interpreter copes well with extra levels of interpretation:

```
time cat primes.blc | ./tromp -b | head -c 210 > /dev/null
real 0m0.043s
time cat uni.blc primes.blc | ./tromp -b | head -c 210 > /dev/null
real 0m0.191s
time cat uni.blc uni.blc primes.blc | ./tromp -b | head -c 210 > /dev/null
real 0m1.919s
time cat uni.blc uni.blc uni.blc primes.blc | ./tromp -b | head -c 210 > /dev/null
real 0m23.514s
time cat uni.blc uni.blc uni.blc uni.blc primes.blc | ./tromp -b | head -c 210 > /dev/null
real 4m52.700s
```

Obfuscation is due entirely to conciseness. Some questions to ponder:

Which of the term space codes 0,1,2,3 serves multiple purposes?

Why is the environment pointer pointing into the term space?

What does the test u+m&1? do?

How does the program reach exit code 0?

And how do any of those blc programs work?

The program freely (without casting) converts between int and int*, causing many warnings; note: expected ‘int *’ but argument is of type ‘int’ warning: assignment from incompatible pointer type warning: assignment makes integer from pointer without a cast warning: assignment makes pointer from integer without a cast warning: incompatible implicit declaration of built-in function ‘calloc’ warning: incompatible implicit declaration of built-in function ‘exit’ warning: passing argument 1 of ‘d’ makes pointer from integer without a cast warning: passing argument 1 of ‘p’ makes pointer from integer without a cast warning: pointer/integer type mismatch in conditional expression

Avoiding these would make the program substantially longer, and detract from its single minded focus on conciseness.

It implicitly declares functions read, write, exit and calloc, the latter two incompatibly. 32 bit and 64 bit executables are separate Makefile targets, involving a change from int to long and from a hardcoded sizeof of 4 to 8.

The program has been tested to work correctly on Linux/Solaris/MacOSX both in 32 and 64 bits.

See the file how13.

Christopher Hendrie, Bertram Felgenhauer, Alex Stangl, Seong-hoon Kang, and Yusuke Endoh have contributed ideas and suggestions for improvement.

Binary Lambda Calculus http://en.wikipedia.org/wiki/Binary_lambda_calculus

G J Chaitin, Algorithmic information theory, Volume I, Cambridge Tracts in Theoretical Computer Science, Cambridge University Press, October 1987. http://www.cs.auckland.ac.nz/~chaitin/cup.html

Jean-Louis Krivine. 2007. A call-by-name lambda-calculus machine Higher Order Symbol. Comput. 20, 3 (September 2007), 199-207. http://www.pps.univ-paris-diderot.fr/~krivine/articles/lazymach.pdf

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Leo Broukhis, Simon Cooper, Landon Curt Noll
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