How do you go from insights to formal solutions?

Question

My question pertains to the problem of translating ideas and insights to the solution (of some problem) into formal notations, thus expressing the answer formally. In other words, having a bunch of ideas (pertaining to the solution) and transforming them into a well composed, formal solution is a process which I often find difficult.

It might be because (at times) I am not aware of enough literature in the problem domain and so have the problem of not knowing how to express my ideas. However, I would like to know if this is something that can be improved upon and if so, how?

Answer 1

Transforming intuition into formal argument is difficult, and the degree to which it is viewed as necessary is sociological, and varies by community. In general, arguments tend to be less formal in AI conferences, reasonably formal in the "Theory A" community, and extremely formal (i.e. machine verifiable) in parts of the programming languages "Theory B" community.

Probably the best way to develop this skill is to take rigorous courses that require formal argument on problem sets. You can also just read papers in the community that you would like to publish in, but you have to make sure to actually read the arguments carefully for this to be helpful. i.e. it is easy to read a paper and feel that you understand it, but be unable to reproduce key parts of the argument. A good way to do this is to plan to actually present the paper to a group of friends. If you actually have to present the arguments on a board and answer questions, you will be able to force yourself to really understand.

Do this a few times and you should be able to construct formal arguments of your own.

Answer 2

One purpose of writing down a solution to a problem in the first place is that it needs to be utterly convincing to your readers. If you think you have an argument, try writing it down fully, first in whatever language / notation you wish. Beware of any instances where you tell yourself things like "It is clear that..." or "The following just has to be true." Then, ask yourself: does it have any holes in it? Am I hand-waving/cheating anywhere? Can I fully explain every detail if asked?

If you find that your work cannot withstand such questions, then you still have some more to figure out. For example, I find that I am not completely sure of my proofs until I try writing them down, and sometimes in the process of writing, I discover problems that I wouldn't have imagined. If I leave writing until last minute, I often find myself in danger. On the other hand, if you can address every issue, then you can go back and rewrite your work up formally, using the proper notation and language so that your peers will be willing to read your work.

Unfortunately, there are no shortcuts here either. It takes reading/presenting a lot of papers until you are familiar with the proper style. One advantage of getting comfortable with writing in proper style is that after a while, you'll be writing proofs "properly" from the first attempt and won't have to do the rewriting step.

Answer 3

As with many other skills, translating your ideas and insights into formal notation (a.k.a. "writing") comes more naturally in some people than others. However, as with most skills, you can improve by practicing.

You can start by rewriting proofs that you have already read or definitions you already know. Read what you write and ask yourself if you can understand that. If you don't, try again. Keep your language as simple as you can, and try to go to the point (be direct). Also, ask yourself why you can understand certain books or papers more than others, and try to capture and emulate their style or simplicity. Once you can say "this is readable", check the spelling, grammar, punctuation, repeated words and other details that you can correct or improve to make your text easier for the reader. Then, for a final test, you can ask colleagues/classmates/professors to read some of your writing exercises and to give their opinion on the quality of the writing.

If what you want to write involves many ideas, try to decide which are more general and start with those before going to specifics. An exception might be when a specific notion (some intermediate result, usually technical) is needed before continuing to other results. In books and papers this is sometimes called "lemma" and it comes before a theorem, because you use the lemma in the proof of the theorem (and probably never again).

Finally, I completely disagree with you when you say

It might be because (at times)I am not aware of enough literature in the problem domain and so have the problem of expressing my ideas.

Not knowing enough literature is perfectly normal when you are approaching a problem or a topic that is new for you. Problems with expressing ideas have more to do with lack of (or misuse of) vocabulary (not necessarily technical!) or the way you structure scattered thoughts into a hierarchy. Whatever the case, you can improve by reading a lot and writing a lot.