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The patent application claimed that if it was applied recursively, a file could be reduced to almost nothing. With a little thought you should convince yourself that this is not possible, at least if the source messages can contain any bit-sequence. We can see this by a simple counting argument. Lets consider all 1000 bit messages, as an example. There are 2 power 1000 different messages we can send, each which needs to be distinctly identified by the decoder. It should be clear we can’t represent that many different messages by sending 999 or fewer bits for all the messages — 999 bits would only allow us to send 2 power 999 distinct messages. The truth is that if any one message is shortened by an algorithm, then some other message needs to be lengthened. You can verify this in practice by running GZIP on a GIF file. It is, in fact, possible to go further and show that for a set of input messages of fixed length, if one message is compressed, then the average length of the compressed messages over all possible inputs is always going to be longer than the original input messages. Consider, for example, the 8 possible 3 bit messages. If one is compressed to two bits, it is not hard to convince yourself that two messages will have to expand to 4 bits, giving an average of 3 1/8 bits.

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Although it might be misunderstood in one place, the specification says "This recursion process repeats until the output file is at or below its required size 78 or until it is deemed that any further recursion of the data is fruitless." They are not stating any arbitrary degree of compression can be achieved on any arbitrary file. If a claim said "do this compression recursively until any arbitrary degree of losses compression is achieved" that claim would probably be unpatentable. As having "incredible utility". –  George White Jul 3 at 1:04

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If the claims were directed to something that is mathematically impossible, then they would be invalid under the requirement that a patent be useful. However, the claims don't require that the compression process work on all inputs or work repeatedly. They simply discuss a series of steps you can apply to data. What the specification says incorrectly cannot render the claims invalid -- it is sufficient that there exists any situation in which the claimed method would be useful.

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