# Is there a way to improve the readability of equations in research papers?

I read a lot of computer science papers, and have to understand a lot of equations. Most of the time the concept is so easy, that someone could explain it to me in 5 minutes. I one the other hand need up to an hour to get it.

It is difficult for me to search for the variables in the text, and often I print the paper out and write the meanings of each variable down on the side.

My questions:

So why is it not common to make equations and algorithms more readable?

Is there some PDF reader plugin which highlights the variables in text? Any tips on how to improve in reading such papers?

Why do authors not add an additional list to each equation with a list and explanation of each variable. Should I do that?

• To answer your last question (why don't author do that) - because most authors have a page limit for the paper, and a reference table would very often blow it. Also, a variable's meaning may change (e.g., in Section 1, M is a symmetric matrix, where in Section 2, M is an arbitrary matrix) Jan 9 '15 at 8:00
• On a related note: I once tried to convince LaTeX (which is used to write many/most CS papers) to allow be to typeset "bubbles" that will be displayed when the user hovers over a variable (so that I could remind a reader about what the variable defines) - this would somewhat solve the problem without exceeding page limits. However, the LaTeX package for doing that was not well-maintained or had many side-conditions, so I eventually gave up. Also, papers are often read after printing anyway, so there would have been little use for that, not mentioning the extra writing time needed. Jan 9 '15 at 8:56
• When reading a print out of a paper, I circle with pencil all the definitions I see as I find them. Jan 9 '15 at 9:30
• @DCTLib If you feel like trying again, the todonotes package may work better. You can add permanent "bubbles" to the output in the margins. Obviously the intended use is to put in todo-notes for oneself, but it may work for this case as well :-) Jan 9 '15 at 14:16
• @darthbith I use todonotes quite extensively myself, and think it would be terrible for this purpose. The problem is that it's simply not designed to look good in a finished document, and won't. Jan 9 '15 at 14:56

I do not believe that the problem is fundamentally about space, but rather is a cultural problem that stems from three things:

1. Reviewers do not demand clarity in their mathematics
2. There is often a perception that "fancy" or "difficult" math means more important science (related to #1)
3. Really clear presentation of mathematics is difficult and takes a lot of work ("I would have written a shorter letter, but I did not have the time.")

Ultimately, a community tends to get what it rewards. There is no reason that math must be impenetrable. Even if one is faced with a space problem, you can play the same sort of games that you do with figures and data in order to fit. Right now, however, it is typically understood and accepted that you don't have to do that with your mathematics. In fact, some scientific communities will punish a researcher for presenting mathematics more clearly, because it makes the work look "less significant."

Technology cannot help solve this, because ultimately it is a problem of human communication. Anyone who is bothered by mathematical impenetrability can, however, take their own small steps towards changing this culture:

1. Include tables and clear explanations in your papers.
2. Use as few symbols as possible, and choose the symbols to improve clarity, e.g., matching the symbol to the first letter of its description.
3. Buffer your equations with explanatory prose that restates their content in plain English.

When considering other people's work:

1. Call out mathematical impenetrability as a reviewer
2. Ask for tables, clear prose, etc. in the papers that you review
3. View the mathematical impenetrability of a work as a flaw rather than a good thing.

This is a very difficult problem, and unlikely to change any time soon, but it can be made better one paper at a time. Moreover, it is my belief (and experience), that clearer mathematics can make for a higher impact and better cited paper, so it will likely be valuable to you in the short term as well, unless you are in a community that has a toxic relationship to mathematical impenetrability.

• I've noticed this trend, some insecure authors seem to want their paper to seem more complex and hard than it really is. Sometimes the variables are never defined at all. It's not hard to include a line or 2 with "j is... " "K is the ..." and anyone who doesn't is choosing to make their work harder to understand without good reason. It's a type of bad behaviour I see far more in papers written by grad students than by confident senior academics. Jan 9 '15 at 15:19
• @Murphy Never attribute to malice what can be explained by incompetence. I think a lot of this is can also be attributed to people not realizing how impenetrable their math is and not having feedback that helps them fix it. Jan 9 '15 at 15:25
• That would explain it if they simply explained it badly but no, when they fail to do even the most basic level: simply stating what each variable is. 10 year old are told off for failing to include that level of information. It's so far bellow the level of mere incompetence that it doesn't float. If it's simply hard to understand after the point where they've at least attempted to explain what the variables are then I'm willing to attribute it to something other than intent. Jan 9 '15 at 15:38
• @Murphy When you've been working intensely on something for a long time, you can easily forget that the symbols you are using are not obvious to everybody (e.g., doesn't everybody use n for number of nodes and r for unit disc radius?). I don't like to defend such writing, but I can easily understand how it can happen, and have certainly made these mistakes myself, especially when I was a graduate student and a much less experienced writer. Jan 9 '15 at 15:59

So why is it not common to make equations and algorithms more readable?

The simple reason is paper space constraints. You can only add so many reference tables and clearly separated equations into a paper before the page limit runs out. And given that many CS papers do not only consist of maths, but also need to have some space for a good evaluation, comprehensive related work, and various other sections, paper space is often at a premium, even at journals.

(and, given that most authors are really used to writing papers this way, they often also keep up the same style in those journals without an explicit page limit)

Is there some PDF reader plugin which highlights the variables in text? Any tips on how to improve in reading such papers?

I doubt that there is a plugin for that, but doing what you are already doing (keeping notes with the meaning of the more important definitions and assumptions) seems like a pretty good start to me.

Why do authors not add an additional list to each equation with a list and explanation of each variable. Should I do that?

If you have the space in your manuscripts, sure - whatever helps the readability of the paper.

• +1. You could also put such a list of your variables into an online supplement if your conference/journal allows this (I don't know how common this is in CS; in other fields it is pretty standard to put "further info" online at the journal's website). Jan 9 '15 at 11:52
• @StephanKolassa It's surprisingly uncommon I think. I started doing "Online Companion Pages" on my own blog / website for all major papers, where we put things like data sets, additional figures, or whatever other supplementary material seems useful. Jan 9 '15 at 12:41

I love the LaTeX package \hyperref. There you can link every variable to its definition. (If you use \usepackage[hidelinks]{hyperref} it does not change the visual appearance of your text, but makes variables clickable or you can color the links.)

You can link other symbols to their definition too (e.g. see http://docdro.id/HZMP7Uy from https://tex.stackexchange.com/a/360871/128042). See https://www.overleaf.com/read/mbbqrdpwbqfk for a simple code-example.

The advantage of this solution is that there is no additional distracting content that might annoy experienced reader that are familiar with the used notation, but those reader who forgot the meaning of a certain variable can just click on it.

Additionally, I personally think, sometimes there would be even more possibilities to improve readability by using

1. \underbrace to explain certain terms,
2. \overset like $a\overset{\text{Th. 1}}{=}b+c$ to indicate that a=b+c holds true because of Theorem 1.,
3. highlithing/lowlingthing to indicate more, and less important parts of complicated equations,
4. and probably many other ideas ...,

but every time you do something creative/unconventional, you will have opponents. I use many of this tools and often get very positive feedback in return, but some people don't accept unconventional things which definitely makes it harder to get accepted to certain journals.

At least \hyperref is accepted or even recommended by the majority of the scientific community.