One of my first blog posts was about whether complexity theory took a wrong turn in denoting every model of computation by an incomprehensible-looking acronym. I completely (NP-completely?) agree with you that these acronyms present a daunting obstacle to the would-be theoretical computer science popularizer. Maybe the only thing to do is to play up the classes’ aura of mystery and obscurity — though I suspect that works better with the physicists’ terms, like “quark” and “supersymmetry,” which at least sound cool.

But just like you, I’ve tried to think of a better convention (just as a thought experiment — there’s no hope of actually switching at this point), and have failed completely. As Ryan points out, the key problem is that unlike in organic chemistry, we’re not combining some fixed collection of elements in new ways. Instead, we’re constantly inventing new elements (quantum, interactive, multi-prover, etc.) and combining them with previous elements — and even inventing new ways of combining the elements — and then we need a short name for each of the resulting combinations (at least if it turns out to be important, which of course is hard to predict in advance).

“E-X-P, T-I-M-E,
Find out if it’s B-P-P
E-X-P, T-I-M-E,
Take care, it’s tricky!”

With more lyrics, this might supplant “Find the Longest Path” (sung to the tune of “For the Longest Time” by Billy Joel).

Have you ever tried to explain to your grandmother why NP is named NP?

Curiously, my mammaws have never asked. (I’m from the South, and sometimes when you’re from the South, you call them mammaws.)

Does she get it when you say that problems labeled NP-complete are the hardest problems in NP, but NP-hard problems might be harder, and not in NP?

It might be easier to try this translation. Take something of the form X-Y and think of it as “Y for X.” So NP-complete is “complete for NP” and NP-hard is “hard for NP.” This probably doesn’t begin to solve your problem, but the idea is that something that NP-hard means that it’s “at least as hard as NP”, and something that is complete will essentially characterize NP itself.

Do locutions like P#P and NISZK and (NP ∩ coNP)/poly roll trillingly off your tongue?

Not really. “P to the sharp-P” rolls OK for me.

How about EXP, EEXP, NEXP, PEXP and SUBEXP?

Yeah, those roll. They’re eksp, ee-eksp, en-eksp, pee-eksp, suhb-eksp, etc.

And while we’re on the subject of EXP and friends, I’ve been wondering how to pronounce NEXPTIME.

The P isn’t silent, it’s pronounced “en-eksp-time.” Note N is a prefix, like in NP.

Presumably, the argot of complexity theory works well enough for communication among members of the club—those of you, say, who know that “IP” is neither “internet protocol” nor “intellectual property” but “interactive proof.” If you’d ever like to talk to the rest of the world, however, I think there might be room for improvement.

I hear what you’re saying, but I think that the phrase “interactive proof” is very suggestive, one of the best names that we have actually. It would be cumbersome to write out the phrase everytime we want to talk about the class of problems, wouldn’t it? And what better way to abbreviate “interactive proof” than…

Let’s start with the term “complexity” itself. The root meaning of “complex” is “made up of many interconnected parts”; for example, the innards of a mechanical clock are complex. Is this the best word to describe the property of being hard to compute? To me, it seems the difficulty of an exponential algorithm is not the complexity of the task but the mere mind-numbing repetition of the same operations. An exponential heap of manure is no more complex than a polynomial heap of manure, it just takes longer to shovel it.

When a complexity theorist talks about complexity, he/she doesn’t talk about algorithms being complex. Complexity is a property of problems. Computational problems (which require us a great deal of time to determine an answer) are complex, in that the encoding of the problem instance does not readily help us find the answer– somehow, the answer is embedded deep in the problem and to uncover it requires substantial effort, no matter what we try. So in a sense, complexity can be seen as a property of the problem and its presented encoding. For example, there are no NP-complete problems that are written in unary (instead of binary), unless P = NP. This is a theorem of Piotr Berman from the 70’s.

Another gripe: WHAT’S WITH ALL THE CAPITAL LETTERS? IS IT REALLY NECESSARY TO SHOUT ALL THE TIME?

YES

The sad truth is, the naming conventions for furniture at Ikea make for a more consistent language than those of complexity theory.

Funny, but it’s not quite that bad.

How did we get into this fix?

For the examples you gave, it is because taking the first letter of a word is a bad hash function.

I suppose that if I’m going to complain about somebody else’s notation, I ought to propose an alternative of my own that I believe to be better. I can’t do that, but I would like to point out that certain other fields of study offer models that might be emulated. The most obvious model is organic chemistry, where the naming problem is orders of magnitude larger than the collection of 465 items in the complexity zoo. The sesquipedalian names of organic molecules are not exactly pretty, but the notational system has the wonderful property that if you know the name of a thing, you also know its composition and structure. Wouldn’t it be grand to have the same kind of perspicuous scheme for complexity classes, so that just naming a class would tell you where it lies in the inclusion diagram?

Yes; the trouble is that new classes and definitions are still being invented. The situation is not like organic chemistry in that respect, and I’m not sure when it will be. We don’t know what how important all these classes will be–some are obviously important, some we just don’t know about–maybe they’ll all be of vital importance someday, in which case a more systematic approach will be absolutely vital. For example, few people knew what on earth PPAD was, until two-player Nash was shown to be PPAD-complete. Now many researchers are studying PPAD.

There are many ways to define a computational model, and for each one you can define a corresponding complexity class, and maybe throw in a time bound on top, for good measure. I don’t know how we could systematize this defining process in a nice way, and maybe we can’t even do it (by some diagonalization argument). Right now we’re cataloging everything we know so far, and I agree it’s a mess, but Scott et al.’s efforts to better organize our knowledge is a good first step.

Ouch !! But not too off the mark. I guess people can get used to anything :)

A musing on the same topic:

http://weblog.fortnow.com/2006/07/naming-complexity-classes.html