## Monday, November 14, 2022

### Who first thought of the notion of Polynomial Time?

(Updated version of  Computational Intractability: A Guide to Algorithmic Lower Bound by Demaine-Gasarch-Hajiaghayi is here)

Any question like who first though of X is often hard to answer. I blogged about who first came up with the Fib numbers here. I've heard rumors that IBM had search engines way before Google but could not figure out how to make money off of it. There are other examples.

I had learned that Cobham defined P in the paper The intrinsic computational difficulty of functions, in 1965. It was The conference on Logic, Methodology, and Philosophy of Science. The paper is here.  Jack Edmonds had the notion of P in the paper Paths, Trees, and Flowers here in 1965.

While it is true that Cobham defined P in that paper, and he might have been the first one to do so, was the notion somehow around earlier. I first thought the answer was no. Why?  Because if you look at Joe Kruskal's paper on MST  (see here) you don't see anything resembling time complexity. No O(E+Vlog V) or whatnot. So I thought that if the notion of  this algorithm runs in such-and-such time was not in the air, then certainly any notion of P could not have been.

Hence I was surprised when I accidentally (more on that later) came across the following:

In 1910 (really, 1910)  H.C.Pocklington analyzed two algorithms for solving quadratic congruences and noticed that

one took time proportional to a power of the log of the modulus, where as

the other took time proportional to the modulus itself or its square root.

THAT is the distinction between P and NOT-P.

The paper is titled The determination of the exponent to which a number belongs, the practical solution of certain congruences, and the law of quadratic reciprocity. It appeared in 1910, in the Proceedings of the Cambridge Philosophical  Society, Volume 16, pages 1-5. (I could not find it online. If you know of a place online for it, leave a comment.)

ADDED LATER: Here is the article in pieces: Page 1Pages2,3Pages4,5.

How did I come across this? And why had I NOT come across this in my roughly 40 years working in complexity theory?

I came across it while reading a blog of Scotts, The Kolmogorov Option, see here where Pocklington is mentioned in passing. I am surprised how arbitrary the set of things ones knows can be. I have put the Pocklington story in the Demaine-Gasarch-Hajiaghayi book Computational Intractability: A Guide to Algorithmic Lower Bounds so that this knowledge gets to be better known.

ADDED LATER: That Cobham and Edmonds are known for discovering or inventing P is an example of  the well known

Columbus Principle: Things are named after the LAST person to discover them (note that Columbus was the last person to discover America.)

Bonus Question: Most principles where the author is not on it, the author might be unknown. NOT in this case. I KNOW who coined the term `Columbus Principle' Do you? (It was not me.)

1. Evangelos Georgiadis3:54 AM, November 15, 2022

Scott is referring to a slightly different paper, mentioning 1917 as publication date.

Note that Don Knuth in TAOCP Vol 2 has an exercise under
Factorization of Polynomials (section 4.6.2); see Exercise 15
(Square Roots Modulo a Prime): Design an algorithm to calculate the square root of a given integer $u$ modulo a given prime $p$.
He mentions the paper that Scott is referring to
(i.e., H. C. Pocklington, Proc. Camb. Phil. Soc. 19 (1917), p.57--59).

I think that even Don only became aware of the Pocklington phenomenon later on ... in 1996 ... but that needs double checking.

NB: There's another exercise in section 4.5.4, suggested by H.C. Pocklington in 1914 (!), but this is related to Factoring Primes.

2. Didn't Kurt Gödel consider quadratic time to be a sort of upper limit of feasibility in a letter to von Neumann in the 50s? Certainly not all of P directly, but a notion that a constant degree of n qualifies somewhat as a major precursor.

1. Your are referring to Godel's lost letter. A good article about this is Von Neumann, Godel and Complexity Theory, by Urquhart, in Bull of Symbolic Logic, Vol 16, No 4, 516--530. And YES there wer SOME notions of time complexity in the air between Pocklington and Cobham/Edmonds. but not much.

3. Literally no one gives a shit

4. Cool findings! I mention Pocklington paper in my lectures.

Nitpick about "Columbus principle" is a counterexample to itself.
1) America is not named after him 2) "Things are named AFTER the last person to discover them" I highly doubt that he discover that principle at all.

A correction would be "Things are named BY the last person to discover them", but that doesn't take into account rediscoveries.

A better principle would "Things are named by the person who popularizes it the most".

But then, we can take into account that things can have many names and we get "things have many names, the popularity of a name is usually proportional to the popularity of the people using that name".

I propose to rename "Columbus principle" to "Nothing should be named after Columbus except Columbus principle"

1. AH, I should have said

Credit is given to the last person to discover something.

Columbus is given credit for discovering America.

And yes, life is more complicated than that. It would be interesting to study how things get their name.

5. I don't know to what extent this is relevant, but the computational complexity of the Euclidean algorithm for GCD has been studied since the 19th century. See https://en.wikipedia.org/wiki/Euclidean_algorithm#Algorithmic_efficiency

1. Great- Yes- very relevant! If my book was not already over 540 pages I might include that (I might anyway).

More generally, a history of awarness of computational issues and how formal they are could be... another whole book!

6. > I've heard rumors that IBM had search engines way before Google but could not figure out how to make money off of it.

You might be referring to IBM's early Text Analytics platform WebFountain here (Gruhl et al. 2004, IBM Sys. J.), which contained aspects from a search engine, but I wouldn't say it qualifies to be called a Web search engine. Before Google, there notably was AltaVista, though.

7. Hi, for von Neumann the article [0] claims some notions of computational complexity from a paper of his in 1953. I would also add on page 28 of [1] Gregory Chaitin claims that he first heard ideas about computational complexity from von Neumann some time around 1950, so perhaps giving him an email about this might give you a bit of history.

[0]: https://www.redalyc.org/articulo.oa?id=265219625004