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Tuesday, June 6, 2023

Wouldn’t it be attainable to create a instrument to mechanically diagram papers?

This can be a considerably experimental and speculative submit. This week I used to be on the IPAM workshop on machine assisted proof that I used to be one of many organizers of. We had an attention-grabbing and numerous vary of talks, each from laptop scientists presenting the newest accessible instruments to formally confirm proofs or to automate numerous facets of proof writing or proof discovery, in addition to mathematicians who described their experiences utilizing these instruments to resolve their analysis issues. One can discover the movies of those talks on the IPAM youtube channel; I additionally posted in regards to the talks throughout the occasion on my Mathstodon account. I’m in fact not probably the most goal individual to evaluate, however from the suggestions I obtained it appears that evidently the convention was capable of efficiently obtain its goal of bringing collectively the totally different communities on this matter.

Because of the convention I began desirous about what attainable laptop instruments may now be developed that might be of broad use to mathematicians, notably those that would not have prior experience with the finer facets of writing code or putting in software program. One concept that got here to thoughts was a possible instrument to might take, say, an arXiv preprint as enter, and return some form of diagram detailing the logical move of the principle theorems and lemmas within the paper. That is at the moment executed by hand by authors in some, however not all, papers (and might typically even be mechanically generated from formally verified proofs, as seen as an illustration within the graphic accompanying the IPAM workshop, or this diagram generated from Massot’s blueprint software program from a manually inputted set of theorems and dependencies as a precursor to formalization of a proof [thanks to Thomas Bloom for this example]). As an example, here’s a diagram that my co-author Rachel Greenfeld and I drew for a current paper:

This specific diagram integrated quite a few subjective design decisions relating to structure, which ends to be designated vital sufficient to require a devoted field (versus being seen as a mere instrument to get from one field to a different), and tips on how to describe every of those outcomes (and tips on how to colour-code them). That is nonetheless a really human-intensive activity (and my co-author and I went via a number of iterations of this specific diagram with a lot back-and-forth dialogue till we had been each happy). However I might see the potential for creating an automated instrument that might present an preliminary “first approximation” to such a diagram, which a human consumer might then modify as they see match (maybe utilizing some handy GUI interface, as an illustration some variant of the Quiver on-line instrument for drawing commutative diagrams in LaTeX).

As a crude first try at mechanically producing such a diagram, one couuld maybe develop a instrument to scrape a LaTeX file to find all of the situations of the theory surroundings within the textual content (i.e., all of the formally recognized lemmas, corollaries, and so forth), and for every such theorem, find a proof surroundings occasion that appears like it’s related to that theorem (doing this with affordable accuracy could require a small quantity of machine studying, although maybe one might simply hope that proximity of the proof surroundings occasion to the theory surroundings occasion suffices in lots of instances). Then determine all of the references inside that proof surroundings to different theorems to begin constructing the tree of implications, which one might then depict in a diagram such because the above. Such an method would possible miss lots of the implications; as an illustration, as a result of many lemmas won’t be confirmed utilizing a proper proof surroundings, however as a substitute by some extra free-flowing textual content dialogue, or maybe a one line justification resembling “By combining Lemma 3.4 and Proposition 3.6, we conclude”. Additionally, some references to different ends in the paper won’t proceed by direct quotation, however by extra oblique justifications resembling “invoking the earlier lemma, we receive” or “by repeating the arguments in Part 3, we’ve”. Nonetheless, even such a crude diagram may nonetheless be useful, each as a place to begin for authors to make an improved diagram, or for a pupil making an attempt to grasp a prolonged paper to get some preliminary thought of the logical construction.

Extra superior options could be to attempt to use extra of the textual content of the paper to assign some measure of significance to particular person outcomes (after which weight the diagram correspondingly to spotlight the extra vital outcomes), to attempt to give every end result a pure language description, and to in some way seize key statements that aren’t neatly encapsulated in a theorem surroundings occasion, however I might think about that such duties must be deferred till some cruder proof-of-concept prototype may be demonstrated.

Anyway, I might have an interest to listen to opinions about whether or not this concept (or some modification thereof) is (a) really possible with present know-how (or higher but, already exists in some kind), and (b) of curiosity to analysis mathematicians.

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