But without politicizing this, I think it is safe to say the authors' method is insufficient to accomplish their goal. Considering it's a published paper it might be worth publishing a refutation. Their measure is based on "ASC", which seems to me to be no more than a measure of pattern compressibility: the difference between number of bits in a direct representation of a pattern and the number of bits in a compressed form (using an ad hoc language provided in the paper).
The paper provides numbers for a few specific examples, but I'm not sure it gives enough detail to reproduce them. On a second reading, I think it's not that hard and might be worth a shot. One operator they provide is effectively "run the pattern for n generations." I believe that small infinite growth patterns provide a way to generate arbitrarily compressible patterns this way. Just compare the representation of "run P for n generations" with the resulting pattern (a million steps into a glider producing switch engine for instance). Even Methuselahs might work for this, starting small and generating something big.
Stable ash fields are also fairly compressible relative to size, just owing to their sparsity. While it is not hard to tell by eye that such a pattern is not designed but "originating by chance" I do not think it could be distinguished from a designed pattern on compressibility criteria alone.
I would be interested in any experts here at least taking a look at the paper. Whatever you think of the ultimate aim of the authors, I believe it fails as a research result (disagreement welcome of course). If so, it should not be left to stand in the peer-reviewed literature without comment.
Oh, one other funny thing. Literally the last sentence of the paper is incorrect.
At least, if I understand the 35-glider result , a pattern with 175 cells could be anything that's glider constructible (though one glider would need to be extremely far away and it would take exponential time to complete the synthesis). This claim is in no way crucial to the paper, but it suggests a rudimentary understanding in my view. (Update: Rereading I see that they make this claim relative to largest number of generations used in a ⊕ operation. Fair enough, but the number could be made quite large with a small representation, e.g. using repeated squaring.)Consequently there is a finite number of interesting patterns for a given number of living cells, and we can number them.
Taking the paper on its own terms, I should probably address this statement:
Their compression scheme makes an attempt to capture "meaningful functionality" by providing expressions for objects like "the smallest oscillator of period 32." However, in the interest of completeness, it includes other operators that may really be more useful for producing a compressed pattern.We have demonstrated the ability to describe functional Game of Life pattern using a mathematical formulation. This allows us in principle to compress Game of Life patterns which exhibit some functionality. Thus, ASC has the ability to measure meaningful functionality
They in no way demonstrate that the description in terms of "meaningful functionality" is the smallest description. I suspect in most cases it is not. The other problem is that their notation only helps with patterns exhibiting periodicity. It is unclear how it would produce a small description for "binary ripple adder" or "toggle flip flop" (realized quite naturally by the snake bit). Whether by intent or misunderstanding, they are conflating "maximum compressibility" with "maximum compressibility in terms of meaningful function". Whatever the latter entails, it is not the same as the former.