Saturday, July 30, 2022

Juris Hartmanis passed away on July 29 at the age of 94

 Juris Hartmanis, one of the founders of Complexity Theory, passed away on July 29 at the age of 94.  The Gödel's Last Letter blog has an obit posted here.  Scott Aaronson has some words and a guest post by Ryan Williams here. When other bloggers post obits I will update this paragraph to point to them. 

Here is one non-blog obit: here.  For an interview with Juris see here.

Hartmanis and Stearns shared the 1993  Turing award for the paper On the Computational Complexity of Algorithms (see here for the paper and see here for his Turing Award Talk). In that paper they define DTIME(T(n)) for Turing Machines. They also proved some theorems about how changes to the model (1-tape, 2-tape, 1-dim, 2-dim others) change the notion of DTIME(T(n))- so they were well aware that the definition was not robust. They also have some theorems about computable numbers. 

We are used to the notion that you can measure how long a computation takes my counting the number of steps on a Turing Machine. Before the Hartmanis-Stearns paper this was not how people thought of things. Knuth (I suspect independently) was looking at what we might now call concrete complexity- how many operations does an algorithm need. Hartmanis-Stearns were beginning what is now called Complexity Theory. 

 Recall that later, the Cook-Levin Theorem used Turing Machines. 

If Hartmanis-Stearns did not start the process of putting complexity on a rigorous mathematical basis how might complexity theory have evolved? It is possible we would not have the Cook-Levin Theorem or the notion of NP. It is possible that we would ASSUME that SAT is hard and use that to get other problems hard, and also do reverse reductions as well to get some problems equivalent to SAT. Indeed- this IS what we do in other parts of theory with assuming the following problems are hard (for various definitions of hard): 3SUM, APSP, Unique Games. And in Crypto Factoring, DL, SVP, and other problems. 

Hartmanis did other things as well. I list some of them that are of interest to me - other people will likely list other things of interest to them. 

0) He had 21 PhD Students, some of them quite prominent. The list of them is here.

1) The Berman-Hartmanis Conjecture: All NP-Complete sets are poly isomorphic. Seems true for all natural NP-complete sets. Still open. This conjecture inspired a lot of work on sparse sets including that if a sparse set S is btt-hard for NP, then P=NP (proven by Ogiwara-Watanabe)

2) The Boolean Hierarchy: we all know what NP is. What about sets that are the difference of two NP sets? What about sets of the form A - (B-C) where A,B,C are all in NP? These form a hierarchy. We of course do not know if the hierarchy is proper, but if it collapse then PH collapses.

3) He helped introduce time-bounded Kolmogorov complexity into complexity theory, see here.

4) He was Lance Fortnow's undergraduate advisor. 

5) There is more but I will stop here.

Sunday, July 24, 2022

100 Best Number Theory books of all Time---except many are not on Number Theory

I was recently emailed this link:

That sounds like a good list to have!  But then I looked at it. 

The issue IS NOT that the books on it are not good. I suspect they all are.

The issue IS that many of the books on the list are not on Number Theory.


A Mathematicians Apology by Hardy

The Universe Speaks in Numbers by Farmelo (looks like Physics)

Category theory in Context by Riehl

A First Course in Mathematical logic and set theory by O'Leary

Astronomical Algorithms by Meeus (Algorithms for Astronomy)

Pocket Book of Integrals and Math Formulas by Tallardia

Entropy and Diversity by Leinster


Too many to name, so I will name categories (Not the type Riehl talks about)

Logic books. Here Number Theory  seems to mean Peano Arithmetic and they are looking at what you can and can't prove in it. 

Combinatorics books:  Indeed, sometimes it is hard to draw the line between Combinatorics and Number Theory, but I still would not put a book on Analytic Combinatorics on a list of top books in Number Theory. 

Discrete Math textbooks: Yes, most discrete math textbooks have some elementary number theory in them, but that does not make them number theory books.

Abstract Algebra, Discrete Harmonic Analysis, other hard math books: These have theorems in Number Theory as an Application.  But they are not books on number theory. 


Lists like this often have several problems

1) The actual object of study is not well defined.

2) The criteria for being good is not well defined.

3) The list is just one person's opinion. If I think the best math-novelty song of all time is William-Rowan-Hamilton (see  here) and the worse one is the Bolzano--Weierstrass rap (see here) that's just my opinion. Even if I was the leading authority on Math Novelty songs and had the largest collection in the world, its still just my opinion. (Another contender for worst math song of all time is here.)

4) Who is the audience for such lists? For the Number Theory Books is the audience ugrad math majors? grad math majors? Number Theorists? This needs to be well defined.

5) The list may tell more about the people making the list then the intrinsic qualify of the objects. This is more common in, say, the ranking of presidents. My favorite is Jimmy Carter since he is the only one with the guts to be sworn in by his nickname Jimmy, unlike  Bill Clinton (sworn in as William Jefferson Clinton- a name only used by his mother when she was mad at him) or Joe Biden (sworn in as Joseph Robinette Biden which I doubt even his mother ever used). My opinion may seem silly, but it reflects my bias towards nicknames, just as the people who rank presidents use their bias. 

Wednesday, July 20, 2022

What is known about that sequence

 In my last post I wrote:


Consider the recurrence


for all n\ge 2, a_n = a_{n-1} + a_{n/2}.

For which M does this recurrence have infinitely many n such that a_n \equiv  0 mod M?

I have written an open problems column on this for SIGACT News which also says
what is known (or at least what I know is known).  It will appear in the next issue.

I will post that open problems column here on my next post.

Until then  I would like you to work on it, untainted by what I know. 

I will now say what is known and point to the open problems column, co-authored with Emily Kaplitz and Erik Metz. 

If  M=2 or M=3 or M=5 or M=7 then there are infinitely many n such that a_n \equiv 0 mod M

If M\equiv 0 mod 4 then there are no n such that a_n \equiv 0 mod M

Empirical evidence suggests that

If M \not\equiv 0 mod 4 then there are infinitely many n such that a_n\equiv 0 mod M

That is our conjecture. Any progress would be good- for example proving it for M=9. M=11 might be easier since 11 is prime. 

The article that I submitted is HERE

Monday, July 18, 2022

An open question about a sequence mod M.

In this post n/2 means floor{n/2}

Consider the recurrence


for all n\ge 2, a_n = a_{n-1} + a_{n/2}.

For which M does this recurrence have infinitely many n such that a_n \equiv  0 mod M?

I have written an open problems column on this for SIGACT News which also says
what is known (or at least what I know is known).  It will appear in the next issue.

I will post that open problems column here on my next post.

Until then  I would like you to work on it, untainted by what I know. 

ADDED LATER: I have now posted the sequel which includes a pointer to the open problems column. To save you time, I post it here as well.

Monday, July 11, 2022

Review of The Engines of Cognition: Essays From the LessWrong Forum/Meta question about posts

 A while back I reviewed A Map that Reflects the Territory which is a collection of essays posted on the lesswrong forum. My review is here. I posted it to both this blog and to the lesswrong forum. In both cases I posted a link to it. My post to lesswrong is here

On the lesswrong post many of the comments, plus some private emails, told me NO BILL- don't post a link, post it directly as text. It was not clear how to do that, but I got it done with help.

On complexity blog nobody commented that this was a problem. Then again, nobody commented at all, so its not clear what to make of that. 


Meta Question: Is posting a link worse than posting direct text? Note that the book review was 12 pages long and looked great in LaTeX. 

Meta Question: Why did lesswrong care about the format but complexityblog did not (Probably answer: omplexityblog readers did not care at all, whereas Lesswrong cared about what I though about Lesswrong)

Another Question, not Meta. One of the comments was (I paraphrase)

When I open a pdf file I expected to see something in the style of an academic paper. This is written in very much chatty, free-flowing blog post style with jokes like calling neologisms ``newords'', so the whole think felt more off-kilter than was intended. The style of writing would prob work better as an HTML blog post (which could then be posted directly as a Lesswrong post here instead of hosted elsewhere and linked.)

I think its interesting that the format of an article telegraphs (in this case incorrectly) what type of article it will be. Is this a common problem?  I have had the experience of reading a real academic paper and being surprised that some joke or cultural-reference is in it, though I do not object to this. 

Another comment and question

I was surprised the post only had 11 karma when I saw it (William had send me an advance copy and I'd really liked reading it) but when I saw that it was a link post, I understood why.

I find this hilarious- they have some way the posts are rated!  For one, Lance told me very early on to never worry about comments, and I don't. Second, it reminds me of the Black Mirror episode Nosedive.

ANYWAY, I have reviewed another collection of essays for less wrong, this one called The Engines of Cognition. I am posting it here as a link: here  and I will post it on lesswrong as full text (with help) in a few days. 

I am posting it so I can get comments before I submit it to the SIGACT News book review column. But this is odd since I think this blog has more readers than SIGACT news has subscribers, so perhaps THIS is its real debut, not that. And of course the lesswrong forum is a place where more will read it since its about them. 

So- I appreciate comments to make it a better review!

Wednesday, July 06, 2022

The Highland Park Shooting

This week I should be celebrating Mark Braverman's Abacus Medal and the Fields Medalists. Instead my mind has been focused 25 miles north of Chicago.

Mass shootings in the United States have become far too commonplace, but the shooting at a fourth of July parade in Highland Park, Illinois hit home. Literally Highland Park was home for me, from 2003-2012. We've been in downtown Highland Park hundreds of times. We've attended their fourth of July parade in the past. My daughter participated in it as part of the high school marching band. 

We were members of North Shore Congregation Israel. My wife, who had a party planning business back then, worked closely with NSCI events coordinator Jacki Sundhein, tragically killed in the attack.

We lived close to Bob's Deli and Pantry and we'd often walk over there for sandwiches or snacks, sometimes served by Bob Crimo himself. The alleged shooter, Bobby Crimo, was his son.

We spent the fourth with friends who came down from Glencoe, the town just south of Highland Park. We spent much of the day just searching for updates on our phones.

I wish we could find ways to reduce the shootings in Highland Park and those like it, the violence that plagues Chicago and other major cities and the highly polarized world we live in which both hampers real gun reforms and creates online groups that help enable these awful events. But right now I just mourn for the lives lost in the town that was my home, a town that will never fully recover from this tragedy.