Wouldn’t it be attainable to create a device to routinely diagram papers?

It is a considerably experimental and speculative put up. 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 fascinating and various vary of talks, each from pc scientists presenting the newest obtainable instruments to formally confirm proofs or to automate varied points 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 concerning the talks in the course of the occasion on my Mathstodon account. I’m after all not essentially the most goal individual to evaluate, however from the suggestions I acquired plainly the convention was capable of efficiently obtain its purpose of bringing collectively the completely different communities on this subject.

On account of the convention I began fascinated about what attainable pc instruments may now be developed that might be of broad use to mathematicians, notably those that don’t have prior experience with the finer points of writing code or putting in software program. One concept that got here to thoughts was a possible device to may take, say, an arXiv preprint as enter, and return some kind of diagram detailing the logical stream of the principle theorems and lemmas within the paper. That is at present carried out by hand by authors in some, however not all, papers (and may usually even be routinely 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]). For example, here’s a diagram that my co-author Rachel Greenfeld and I drew for a current paper:

This explicit diagram included quite a lot of subjective design decisions concerning structure, which ends up to be designated essential sufficient to require a devoted field (versus being considered as a mere device to get from one field to a different), and how you can describe every of those outcomes (and how you can colour-code them). That is nonetheless a really human-intensive job (and my co-author and I went by means of a number of iterations of this explicit diagram with a lot back-and-forth dialogue till we had been each happy). However I may see the potential for creating an automated device that would present an preliminary “first approximation” to such a diagram, which a human person may then modify as they see match (maybe utilizing some handy GUI interface, as an illustration some variant of the Quiver on-line device for drawing commutative diagrams in LaTeX).

As a crude first try at routinely producing such a diagram, one couuld maybe develop a device to scrape a LaTeX file to find all of the cases 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 may 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 start out constructing the tree of implications, which one may 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 may not be confirmed utilizing a proper proof surroundings, however as an alternative by some extra free-flowing textual content dialogue, or maybe a one line justification comparable to “By combining Lemma 3.4 and Proposition 3.6, we conclude”. Additionally, some references to different leads to the paper may not proceed by direct quotation, however by extra oblique justifications comparable to “invoking the earlier lemma, we acquire” or “by repeating the arguments in Part 3, now we have”. 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 scholar making an attempt to know a prolonged paper to get some preliminary thought of the logical construction.

Extra superior options is perhaps 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 essential outcomes), to attempt to give every consequence a pure language description, and to someway seize key statements that aren’t neatly encapsulated in a theorem surroundings occasion, however I might think about that such duties needs to be deferred till some cruder proof-of-concept prototype will be demonstrated.

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