Hello Lee --
Very nice to see your essay in the mix this year!
I really, really like pathway assembly. This may have been something you were telling me about the last time we met in DC--if so, I finally understand it! It is, indeed, computable--and, OK, fine, it's in NP, I think (just because it's a shortest path problem), but the way in which it is computed is quite simple.
If I understand correctly, you also have some AIT-like stories, where you consider also maps between assembly spaces, and how these lead to relative lower bounds. This reminds me of the (uncomputable) material in Kolmogorov Complexity, and translating between machines.
I know you developed these for molecular biology, but I am wondering how they can be equally well applied to human systems. If I think about a sentence, for example, I should be able to find repeated subunits. In those cases, however, the subunits are often with variable placeholders, e.g., ["the". [NOUN]] gets repeated in different places, but then you have to sub-in for the noun. So I can't just immediately stack it on. But there are lots of other places I can, e.g., if I'm looking at the topology of a social network, where I can see subunit motifs being repeated.
Indeed, what happens if you try to apply these methods to a metabolic network, for example? I feel like there's some material on "graph grammars", but I kind of gave up on it because it seemed like the problem was too hard. But perhaps you have deeper insights here.
Do you have a sense of where pathway grammars fall in the Chomsky hierarchy? Meaning, what kinds of processes they can model well. Is it possible, for example, to capture a context-free language like mathematical syntax? Again, I don't have a sense for the class of patterns your methods can capture well.
(That brings up another thought, which is that one can consider pathway assembly across multiple patterns; now you have, I think two things--you want to get the right subunits for a particular pattern, but also reuse of those subunits across different patterns. This would perhaps be equivalent to the difference between the Kolmogorov Complexity of a distribution, and of a sample from that distribution. This could be quite easy to do, if you considered toy examples, like the pathway assembly of a sequence of coin tosses, where the coin follows different Markov processes--you may even discover a nice relationship between the entropy of the process and the pathway complexity.)
In short, it would be lovely to have a (big, long) introduction to these ideas, perhaps in a journal like J Roy Soc Interface, which is often really open to these kinds of things.
Yours,
Simon