Most functional

John Tromp

john.tromp@gmail.com

http://tromp.github.io/

Judges' comments:

To build:

On a 32-bit machine:

make tromp

On a 64-bit machine:

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

To run:

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

Try:

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

Selected Judges Remarks:

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

Author’s comments:

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

A half byte `cat'

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

Bad programs

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.

A Self Interpreter

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

A prime number sieve

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

A Space filling program

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 _ _ _ _ _ _ _ _ | |_| | | |_| | | |_| | | |_| | |_ _| |_ _| |_ _| |_ _| _| |_____| |_ _| |_____| |_ | ___ ___ | | ___ ___ | |_| _| |_ |_| |_| _| |_ |_| _ |_ _| _ _ |_ _| _ | |___| |___| |_| |___| |___| | |_ ___ ___ ___ ___ _| _| |_ |_| _| |_ |_| _| |_ | _ | _ |_ _| _ | _ | |_| |_| | |___| |___| | |_| |_| _ _ | ___ ___ | _ _ | |_| | |_| _| |_ |_| | |_| | |_ _| _ |_ _| _ |_ _| _| |___| |___| |___| |___| |_

A BrainFuck interpreter

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.

A BLC assembler

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

Converting between bits and bytes

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

Symbolic Lambda Calculus reduction

BLC8 program symbolic.Blc shows individual reduction steps on symbolic lambda terms. Here it is used to show the calculation of 23 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

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

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?

Portability

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.

How the program works

See the file how13.

Acknowledgements

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

References

Binary Lambda Calculus https://tromp.github.io/cl/Binary_lambda_calculus.html

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