## Claude Shannon hates this one weird trick!

There was a question posted to /r/AskComputerScience recently: “does this compression scheme look fishy to you?". The algorithm in question, called “press” by its author, makes some.. bold claims in the README:

By stringing together 1/0s, using 1’s as negative and a 0 as a positive, I’ve sublimely made a perfect compression. Through outputting into hexadecimal. This shortens <=4096 ‘bits’ into a 2-byte hexadecimal.

I know enough to immediately scoff at these claims, and if somebody hadn’t specifically asked about it, I would have left it there and not thought twice. Since somebody asked, however, I thought it would be fun to give this a thorough examination.

## Shannon says “no”

The first few lines of the README, above, seem to imply this might be some kind of arithmetic coding. On the other hand, the author’s fixation with hexadecimal seems to imply there’s some aspect in which they fail to completely understand the nature of base conversions.

For now disregarding what’s actually going on in the code (we’ll get to that shortly), but Shannon’s source coding theorem states that the size of a losslessly-compressed message cannot be less than the Shannon entropy of the plaintext (uncompressed) message. The claims are vague here, so we can’t immediately discard them as invalid.

Up to 4096 bits can indeed be encoded to two bytes. In a simple case, one could encode strings of zero-bits in a 16-bit number, and all of the numbers in the range 0-4096 can be represented in 16 bits (which has a range 0-65535).

So far so plausible. It gets a little more interesting if we take the claim of “perfect compression” literally, meaning that a compressed message will not exceed the ceiling of the Shannon entropy of the uncompressed message in bits1. For later reference, Shannon entropy can be defined as follows:

$$H(X) = - \Sigma_i P(x_i) \mathrm{log}_b P(x_i)$$

Where the message has an alphabet of $$n$$ possible elements $$x_0 \ldots x_n$$, $$P(x_i)$$ is the probability of a given character in the message being $$x_i$$ (that is, the number of times the character appears divided by the message length), and $$b = 2$$ when expressing $$H$$ in bits.

Since I don’t really want to do a formal analysis of this, it’s time to break out the code and do some concrete tests, attempting to disprove the claims.

## Time for code

I don’t really want to go testing the provided implementation of this system, so I’m instead going to write my own white-box implementation that should be easier to run tests on. Since I like working in Rust and there are some particularly useful tools available for this task, I choose to write implementation in Rust.

Let’s first examine the reference C++ implementation, specifically the encoder (with formatting adjusted to be more readable)2:

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21  std::ifstream in(argv[i+1], std::ios::in); std::ofstream out(argv[i+2], std::ios::out | std::ios::binary); int Y = 1; while (!(in.eof())) { char P=in.get(); char ha[4098]= { P }; for (int j=0;j<=4096;j++) { if (ha[j]==0) { out << (char)Y-1 ; Y=1; } else if (ha[j]==1) { Y++; continue; } else { Y=1; break; } } } 

This takes “bits” from an input stream, writing bytes to an output file. The “bits” here are actually individual ‘0’ or ‘1’ characters, but we’ll do better in the reimplementation and use actual bits. The decoder must be the opposite, taking bytes in and emitting bits. Doing a kind of test-driven development, we’ll start by writing the skeleton of the encoder and decoder.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33  #[derive(Clone, Copy, Debug, PartialEq, Eg)] enum Bit { Zero, One } struct Compress where I: Iterator { src: I } impl Iterator for Compress where I: Iterator { type Item = u8; fn next(&mut self) -> Option { unimplemented!(); } } struct Decompress where I: Iterator { src: I } impl Iterator for Decompress where I: Iterator { type Item = Bit; fn next(&mut self) -> Option { unimplemented!(); } } fn main() { unimplemented!(); } 

Rust doesn’t have a single-bit data type, but I chose to emulate it here with an enumeration that has two variants, zero and one. A boolean value could have worked too, but this approach is clearer to understand. I let the compiler automatically implement a number of traits so Bit values can be copied, displayed and compared.

### Invariants

With some type signatures defined, we can write some tests for the system. There are two axioms that we expect will always be true of any lossless compression scheme. Given $$e = C(x)$$ as the compressor and $$x = D(e)$$ as the decompressor:

1. $$x = D(C(x))$$. A compressed message is always decompressed to the original.
2. $$|C(x)| \geq H(x)$$. The size of the compressed message may not be less than the Shannon entropy of the uncompressed input.

Using quickcheck3, we can write simple tests for these axioms and let the computer generate test cases attempting to cause them to fail, indicating either a bug in our implementation or a false axiom.

Testing these two properties, we can both test that the algorithm is correct, and detect if the creator has somehow disproven the source coding theorem. If both tests pass it is a correct algorithm that appears to obey the principles of information theory, and if the second fails but the first passes we appear to have found a counterexample to the theorem.

### Using quickcheck

First, we’ll write a test that decompressing a compressed message yields the same message back.

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19  impl quickcheck::Arbitrary for Bit { fn arbitrary(g: &mut G) -> Self { g.choose(&[Bit::Zero, Bit::One]).unwrap().clone() } } #[test] fn codec_is_lossless() { fn encodes_losslessly(xs: Vec) -> bool { let roundtrip = Decompress { src: Compress { src: xs.iter().cloned() } }; roundtrip.collect::>() == xs } quickcheck(encodes_losslessly as fn(Vec) -> bool); } 

There’s nothing particularly novel here. We implement quickcheck’s Arbitrary trait for Bits so it can generate random lists of them for us, and define a test function (that will be run when we do cargo test) which uses quickcheck to ensure that compressing then decompressing a list of Bits yields an equivalent list back.

If quickcheck can find a case where encodes_losslessly returns false, it will automatically attempt to minimize the test case, simplifying debugging. This is the second half of what makes quickcheck very nice to use- not only can it generate tests, but it can minimize them to remove unnecessary parts of the test case it generates.

Writing a similar test for preservation of entropy is not particularly remarkable either:

 1   

I perhaps went a little bit overboard on the implementation of entropy in making it so generic, but I like to think it could be adapted to some other application in the future.

### Implementing codecs

With tests out of the way, we can (finally) get around to implementing the (de)compressor. Referring back to the C++ compressor code, it’s pretty easy:

• Initialize counter to 1.
• If bit is zero, output counter - 1 and reset to 1.
• Otherwise increment counter.

So, it looks like the idea is that strings of ones will encode to a number of ones, and zeroes cause the counter to be emitted. In short: any number of ones followed by a zero encodes to the number of ones. We’ll throw together a simple implementation:

 1   

Perhaps unsurprisingly, the tests failed. This is shown to not be lossless, and entropy is not preserved (but that’s entirely allowable in a lossy codec).

running 3 tests
test tests::validate_entropy_computation ... ok
test tests::codec_is_lossless ... FAILED
test tests::entropy_is_preserved ... FAILED

failures:

---- tests::codec_is_lossless stdout ----
thread 'tests::codec_is_lossless' panicked at '[quickcheck] TEST FAILED.  Arguments: ([One])', registry/src/github.com-0a35038f75765ae4/quickcheck-0.2.24/src/tester.rs:113

---- tests::entropy_is_preserved stdout ----
thread 'tests::entropy_is_preserved' panicked at '[quickcheck] TEST FAILED. Arguments: ([Zero, One, Zero])', registry/src/github.com-0a35038f75765ae4/quickcheck-0.2.24/src/tester.rs:113


### Fixing it

We see from the failure on codec_is_lossless that it can emit wrong data if the input is a single one. Looking at the implementation, that makes sense- when the end of input is reached in the encoder, any buffered ones are not emitted. This is a relatively simple fix:

 1   

Testing again, we find that empty input is still a problem. Turns out we should only flush the encoder if it’s received data to begin with, in general meaning if the one-counter indicates there are any buffered. Conveniently, this simplifies the code a little bit, by observing that there’s nothing to be flushed if counter is greater than one.

1. We take the ceiling of the entropy because it doesn’t make sense to emit a fractional bit, so a real implementation must round the number of bits up. ↩︎

2. I honestly couldn’t figure out how the decoder was supposed to work, but it seems to vaguely match what we’d expect the encoder to emit. ↩︎

3. The Rust version of quickcheck is a port of the Haskell library of the same name. This kind of formalized approach to software design is fairly common among Haskell programmers I find, and I think most Rust programmers appreciate the value of this approach too. ↩︎