Complexity in Michigan

Invited Speaker Nutan Limaye, Conference Chair Valentine Kabanets,
2024 PC Chair Rahul Santhanam, myself, 2025 PC Chair Srikanth Srinivasan
and 2025 Local Arrangements chair Swastik Kopparty enjoy some tapas.

I have a long history with the Computational Complexity conference. I attended the first 26 meetings (1996-2011) and 30 of the first 31. I chaired the conference committee from 2000-2006. According to DBLP I still have the most papers appearing in the conference (32). I even donated the domain name for the conference (with the caveat that I could keep the subdomain for this blog).

Only Eric Allender had a longer streak having attended the first 37 conferences through 2022 in Philadelphia (if you count the two online during the pandemic) before he retired.

But I haven't been back to Complexity since that 31st conference in Tokyo in 2016. Due to my administrative roles, various conflicts and changes in the field you just start missing conferences. But with the conference at the University of Michigan in Ann Arbor within driving distance of Chicago it was time to go home for the 39th meeting. And it's a good thing I drove as more than one person had flight delays due to the Crowdstrike bug.

The complexity conference remains relatively stable at about 75-100 registrants, the majority students and young researchers. I've moved from wise-old sage to who is that guy. But I'm having great fun talking to old acquaintances and new. I'm impressed with the newer generations of complexity theorists--the field is in good hands.

Best paper goes to Michael Forbes Low-Depth Algebraic Circuit Lower Bounds over Any Field. the work of Limaye, Srinivasan and Tavenas I talked about last month gave an explicit polynomials with superpolynomial-size over constant depth algebraic circuits but it required polynomials over large fields. Forbes extended the lower bounds to all field sizes.

Best student paper goes to Ted Pyne from MIT for Derandomizing Logspace with a Small Shared Hard Drive for showing how to reduce space for randomized log-space algorithms on catalytic machines.

Check out all the papers in the online proceedings.

From a relatively quiet business meeting: 36 papers accepted out of 104 submissions, a bit up from previous years. 75 attendees including 42 students, similar to recent years. 2025 conference at the Fields Institute in Toronto August 5-8. 2026 in Lisbon or Auckland.

The loss of Luca Trevisan, PC Chair 2005 and local arrangements chair in 2013 in San Jose, loomed large in the business meeting and at the conference.

FLT solution annouement had its 31's anniv was about a month ago. Some poems about FLT NOT from ChatGPT

On June 21, 1993, at the Issac Newton Institute for Mathematical Science, Andrew Wiles announced that he had proven Fermat's Last Theorem. That wasn't quite right- there was a hole in the proof that was later patched up with the help of Richard Taylor (his former grad student). A correct proof was submitted in 1994 and appeared in 1995. Wiles is the sole author. 

June 21, 2024 was the  31st anniversary of the announcement. (So today is the 31-years and 1-month anniversary).  I COULD have had ChatGPT write some poems about it. But there is no need. There are already some very nice poems about it written by humans. Will humans eventually lose the ability to write such things? Would that be a bad thing? Either ponder those questions or just enjoy the poems. (My spellcheck still thinks ChatGPT is not a word. It needs to get with the times.)

1) A link to a set of poems about FLT: here.

2) Here is a poem that is not in that set but is excellent.

A challenge for many long ages
Had baffled the savants and sages
Yet at last came the light
Seems that Fermat was right
To the margins add 200 pages 

(I don't know who wrote this or even where I read it. If you know anything about where it was published or who wrote it, please let me know.)

3)  Here is a poem by Jonathan Harvey that mentions the gap in the original proof.

A mathematician named Wiles
Had papers stacked in large piles
Since he saw a clue
He could show Fermat true
Mixing many mathematical styles

He labored in search of the light
To find the crucial insight
Young Andrew, it seems
Had childhood dreams
To prove Mr. Fermat was right

He studied for seven long years
Expending much blood, sweat, and tears
After showing the proof
A skeptic said “Poof!
There’s a hole here”, raising deep fears.

This shattered Mr. Wiles’s belief
His ship was wrecked on a reef
Then a quick switcheroo
Came out of the blue
Providing his mind much relief.

Mr. Wiles had been under the gun
But the obstacle blocking Proof One
Fixed a much older way
From an earlier day
And now Wiles has his place in the sun

4) Here is a poem by John Fitzgerald that mentions other unsolved problems including P vs NP

Fermat’s theorem has been solved,
What will now make math evolve?

There are many problems still,
None of which can cause that thrill.

Years and years of history,
Gave romance to Fermat-spree,

Amateurs and top men too,
Tried to push this theorem through.

Some have thought they reached the goal,
But were shipwrecked on the shoal,

So the quest grew stronger still;
Who would pay for Fermat’s bill?

So what is now the pearl to probe,
The snark to hunt, the pot of gold,

The fish to catch, the rainbows end,
The distant call towards which to tend?

One such goal’s the number brick,
where integers to all lengths stick:

To sides, diagonals, everyone,
Does it exist or are there none?

Then there are those famous pearls,
That have stymied kins and earls:

Goldbach, Twin Primes, Riemann Zeta;
No solutions, plenty data.

Find a perfect number odd;
Through 3n + 1 go plod;

Will the P = N P ?
Send a code unbreakably.

Are independence proofs amiss;
Continuum Hypothesis;

Find a proof which has some texture
of the Poincaré conjecture.

And so, you see, onward we sail,
there still are mountains we must scale;

But now there’s something gone from math,
At Fermat’s end we weep and laugh.

The Story of Shor's Algorithm

The quantum factoring algorithm of Peter Shor (FOCS 1994, SIAM Review 1999) turns thirty this year. Before his algorithm, quantum computing lacked the killer app, something practical that quantum could do that seems hard for classical computers. Back in 1994, I said Shor's algorithm bought quantum computing another twenty years. How I misjudged the longevity of quantum hype. 

Peter got the idea for his algorithm from a paper by Daniel Simon solving a theoretical complexity problem. The quantum factoring algorithm is a great example of how a complexity result can open doors to new algorithmic ideas.

Simon came up with a beautifully simple example of a problem that required exponential-time on a probabilistic machine but polynomial-time on a quantum computer. Let's define addition over the \(n\)-bit strings, for \(x\) and \(y\) in \(\{0,1\}^n\), \(x+y\) is the bitwise parity of \(x\) and \(y\). For example if \(x\) is 0110 and \(y\) is 1100, \(x+y = 1010\).

Suppose we have a Boolean function \(f:\{0,1\}^n\rightarrow\{0,1\}^n\) (maps \(n\) bits to \(n\) bits) with the property that \(f(x)=f(y)\) iff \(x=y+z\) for some fixed \(z\). The problem is given \(f\) as an oracle or a circuit, find the \(z\). A classical machine would need exponential steps in to find \(z\) in the worst case.

Simon gave a simple quantum algorithm that would with a single query output a random w such that \(w\cdot z=0\). With \(n = \log N\) linearly independent \(w\), you can solve for \(z\).

Shor's asked what if we could do the same for regular integer addition instead of bitwise parity. Suppose you have a function \(f(x)=f(y)\) iff \(x-y\) is a multiple of \(z\) for a fixed \(z\). (In Simon's case over bits the only multiples are zero and one.) That means \(f\) is periodic and \(z\) is the period. Shor knew that by an algorithm by Miller, finding a period leads to factoring.

Let m be an odd number with multiple prime factors. Consider \(f(x)=a^x\bmod m\) for a randomly chosen \(a\) relatively prime to \(m\). If this function has a period \(z\), then \(a^z\bmod m=a\), \(a^{z-1}\bmod m=1\) and with probability at least one-half, the gcd of \(a^{\frac{z-1}{2}}\) and \(m\) will be a nontrivial factor of m. 

Getting all this to work on a quantum computer requires a number of addition tricks beyond what Simon did but once Shor had the inspiration the rest followed. 

Peter Shor really understood the landscape of theory from complexity to cryptography, a curiosity for quantum computing and the vision to see how it all connected together to get the quantum algorithm that almost single-handedly brought billions of dollars to the field. 

Peter just received the Shannon Award for his work on quantum error correction that would help enable quantum computers to run his algorithm. Still the largest number present day quantum computers can factor with the algorithm is 21. If (and its a big if) that number gets up past the RSA challenge numbers, Peter will have far larger prizes in his future.

The Term Quantum Being Misused ... Again

In a post from 2015 I noted that the word quantum is often misused (see here). Have things gotten better since then? I think you know the answer. But two uses of the word quantum caught my attention

1) The episode Subspace Rhapsody of  Star Trek- Strange New Worlds is described on IMDB as follows:

An accident with an experimental quantum probability field causes everyone on the Enterprise to break uncontrollably into song, but the real danger is that the field is expanding and beginning to impact other ships--- allies and enemies alike.

 (I mentioned this episode and pointed to my website of all the songs in it  here.)

SO- is this an incorrect use of of the word quantum? Since ST-SNW is fictional, its a silly question. However, it seems like a lazy Sci-Fi convention to just use the word quantum for random technobabble. 

2) The Economist is a serious British weekly newspaper. Or so I thought until I read this passage in the June 15-21, 2024 issue, the article featured on the cover The rise of Chinese Science

Thanks to Chinese agronomists, farmers everywhere could reap more bountiful harvests. Its perovskite-based solar panels will work  just as well  in Gabon as in the Gobi desert. But a more innovative China may also thrive in fields with military uses, such as quantum computing or hypersonic weapons.

So The Economist is saying that Quantum Computing has military uses. I am skeptical of this except for the (in my opinion unlikely) possibility that QC can factor and break RSA which, if it will happen, won't be for a while. 

It also makes me wonder if the rest of the paragraph, which is on fields I don't know anything about, is also incorrect or deeply flawed. (See Gell-Man Amnesia which I've also heard called The Gell-Man Affect.) 

I am not surprised that ST:SNW uses quantum incorrectly (Or did it?  Maybe an experimental quantum probability field would cause people to sing.) but I am surprised that The Economist  misused it. I thought they were more reliable. Oh well. 

 

Favorite Theorems: Extracting Ramsey Graphs

June Edition

Two decades ago, I named the recently departed Luca Trevisan's paper connecting extractors to psuedorandom generators as one of my favorite theorems from 1995-2004. I'm dedicating this month's favorite theorem to him.

Suppose we have two independent sources with just a little bit of entropy each. Can I pull out a single random bit? This month's favorite theorem shows us how, with a nice application to constructing Ramsey graphs.

Explicit Two-Source Extractors and Resilient Functions

Eshan Chattopadhyay and David Zuckerman

More formally (feel free to skip this part) suppose we had two independent distributions U and V each of poly log min-entropy, which means for every string x of length n, the probability of choosing x from U and the probability of choosing x from V is at most \(2^{-(\log n)^c}\) for some c. There is a deterministic polytime function (which doesn't depend on U and V) such that f(x,y) with x and y chosen independently from U and V will output 1 with probability \(1/2\pm\epsilon\) for \(\epsilon\) smaller than any polynomial.

Previous work required a linear amount of min-entropy for U and V. 

As a corollary, we can use f to deterministically generate a Ramsey graph on n vertices with no cliques or independent sets of size \(2^{(\log\log n)^c}\) for a sufficiently large c. This is also an exponential improvement from previous constructions. Gil Cohen gave an independent construction that doesn't go through extractors.

There have been several papers improving the bounds of Chattopadhyay and Zuckerman. In FOCS 2023 Xin Li gave a construction of extractors with \(O(\log n)\) min-entropy, the current state-of-the-art for extracting a single random bit with constant error, and Ramsey graphs with no cliques or independent sets of size \(\log^c n\) for some constant c.

The combinatorics of picking a Vice President

 Trump is pondering who to pick for his vice president. For a recent podcast about it go here. Spoiler alert: Doug B or Marco R or J.D. Vance. 

In 2008 I did a blog post titled  I would bet on INTRADE that INTRADE will do badly picking VP nominations where I showed that about half the time the VP candidate is not on anyone's short list, and hence would do badly in betting markets. At the time INTRADE was synonymous with betting markets. I would  not have bet that INTRADE would go out of business. 

What criteria does a prez nominee  use when picking a vice president? How many combinations are there? 

1) Someone who will help with a block of voters. 

Trump-Pence 2016: Mike Pence was thought to help Trump with Evangelicals and establishment Republicans.

Biden-Harris-2020: Kamala Harris was thought to help Biden with women and African-Americans.

JFK-LBJ-1960-LBJ was thought to help JFK in the South. 

Kerry-Edwards-2004: Edwards was thought to help win  North Carolina (Edwards state). It didn't work. 

Dukakis-Bentson-1988- Mike Dukakis (liberal) picked Lloyd Benson (moderate) as the VP. The ticket lost though its possible that Benson brought in some votes, just not enough.

There are other examples. Even for the cases where the candidate won its not clear if the VP mattered.  The podcast says that Trump thinks that this kind of thing (e.g., picking a women or an African American) won't help him get their votes. He might be right. But (my speculation) a women on the ticket might help some men be more comfortable voting for him. That is, they could think Trump is not a misogynist, see- he picked a women for VP.  Similar for an African-American.

Caveat: Perhaps a candidate who would help in Swing States.

2) Someone who will help him if he wins. 

Obama-Biden-2008:  Biden helped new-comer Obama since Biden had Congressional experience, having been a senator for X years for some large value of X.

Bush-Cheney-2000:  Dick Cheney knew Washington DC and hence could help George W Bush (who had been a governor but had no FEDERAL experience).

3) Someone who the voters can see taking over the presidency in case that is needed.

Clinton-Gore-1992: I've heard that Clinton chose Gore for that reason. I'm NOT an insider so it may not be true. 

FDR-Truman-1944: The party chose Harry Truman as VP knowing that FDR would likely pass away and we'd have President Truman. (I've read this and believe it is true on some level.) 

4) Party Unity- Pick someone who you fought in the primary to show that the party is united. Bonus: the VP nominee has been vetted and is some-known to the public. 

Biden-Harris-2020 may have had had some of this.

This mentality is rarer now since people tend to NOT pick people they ran against in the primary lately.

JFK-LBJ-1960 was in this category.  

Biden did run for the nomination in 2008 but didn't run much (I think he dropped out either right before or right after the Iowa Caucus) so that one doesn't really count.

5) DO NO HARM. Counterexamples:

Some people voted against McCain since they didn't wan to see Sarah Palin one heartbeat away from the presidency. This was especially important since McCain was old. And hence this may be important for Trump in 2024.

Biden may have the same problem with  Harris. Note that the issue is NOT if Harris would BE a bad prez, its if people THINK she would be a bad Prez.

Krisiti Noem- Trump doesn't want to answer questions about why his VP shot a dog and a goat. (Note- if Trump himself had shot a dog and a goat the party and FOX news would be defending that action.) 

6) Someone who the Prez candidate gets along with personally. I've heard that Clinton-Gore and Obama-Biden got along. JFK and LBJ did not.

7) Someone who won't outshine the president.

Dukakis-Benson=1988  might have had this problem. 

 8) All of the above might matter less than usual since there are so few undecided people in swing states. And that's NOT just because the country is polarized. Ponder the following:

In most elections its either two people NEITHER of whom has been president, so you don't quite know what they will do, OR one has been prez and the other has not, so you don't know what the newcomer will do.

But in this election BOTH have been president. We KNOW what they will do. So there is less room for doubt. 

History: This only happened once before: 

1884: Cleveland beats Blaine

1888: Harrison beats Cleveland

1892: Cleveland vs Harrison and Cleveland wins

Even though I say its hard to predict, and it could be someone NOT on the short list, here are my thoughts.

a) Marco R. The electors in the electoral college cannot vote for a Prez and vice-Prez who are residents of the same state. (Note1- This is an idiotic rule which dates from either the late 1700's or the early 1800's. Note2- Dick Cheney changes his residency from Texas to Wyoming so he could be Bush's VP in 2000.  I have NO problem with that whatsoever.) So one of Marco R or Trump would have to change residencies. Trump won't bother. Marco R is a SENATOR from Florida so I doubt he would either. Also, Marco said nasty things about Trump when he ran against him for the nomination in 2016. I am surprised Marco is on anyone's short list. NOT going to be VP nominee.

b) Doug B. Who? He doesn't outshine Trump, and he gets along with Trump. Won't bring in any voters, but Trump says he doesn't care about that. How would American's view him as a possible prez? I doubt Trump cares. QUITE POSSIBLE to be VP nominee.

c) JD Vance. Author of a thoughtful book, Hillbilly Elegy, which indirectly explains why poor white rural voters are attracted to Trump. He then became a Senator and is now all-in on Trump. This is NOT hypocritical, but its odd. In 2016 he was anti-Trump but now he is pro-Trump. Even that is NOT hypocritical using the usual standards for politicians. He has praised Trump and there may be people who think he would be a good president. He is young (39) and handsome, so I wonder if Trump worries that Vance might outshine him. Even so QUITE POSSIBLE to be VP nominee.

d) I am surprised that Tim Scott and Elise Stefanik seem to have fallen out of Trump's Short list, though they were at one time on it, so would not be to big a surprise if either becomes the VP nominee.  IF one thinks that Tim Scott will help with the African-American vote, or that Elise Stefanik will help with the women-vote (OR as noted above, would help white men feel more comfortable voting for Trump) then either would be a politically good choice. However, Trump does not think this is true, and he may be right.  I've also heard that Trump doesn't want people saying something like Tim Scott helped Trump win the African-American Vote since Trump wants to think that HE won the election without help. I would think neither will be VP but YOU NEVER KNOW.

e) Someone NOT on the horizon. This brings us back to my 2008 post- IT REALLY COULD BE SOMEONE THAT NOBODY IS TALKING ABOUT. So Who? I DON"T KNOW SINCE NOBODY IS TALKING ABOUT THEM. Maybe Lance.

Why is called the Turing Award (revisited)?

Avi Wigderson gave his ACM Turing Award lecture last week, and instead of telling his own story, he focused on Alan Turing and his influence on complexity. If you didn't see it live, you can watch on YouTube or below.

I want to revisit a post I wrote for the Turing centenary in 2012 asking why the prize is named after Turing. Since that post, Turing has become even more popular, especially through the 2014 movie starring Benedict Cumberbatch. I caught this picture of Turing as a legend in the Chicago Pride Parade last Sunday.

But Turing was not always a legend. The first Turing award was awarded to Alan Perlis in 1966. Turing's work during World War II remained classified until the 1970s and wasn't widely known until the 80's. Alan Turing's homosexuality would have granted him no favors in the mid-60s. 

When I gave a talk celebrating Juris Hartmanis, I posited that Juris not only received the Turing award but may have been the reason the award has its name. The Hartmanis-Stearns paper, as Avi also noted, established the Turing machine as the right model for studying computational complexity, as the model easily capture time and space (memory). That paper was published in 1965, fresh in the minds of those at the ACM creating the award. Perhaps combined with the early days of AI and Turing's intelligence paper, may have been enough to decide to name the award for Turing. 

Today there is no question that ACM made the right move in naming the award for Turing. Just watch Avi show that Turing's influence on complexity justifies the award's name on its own.

Technology: 1966, 2006, 2023.

 In 2013 I wrote a blog to celebrate Lance's 50th birthday by contrasting what things were like when Lance was 10 to when he was 50. That post is here.

But life has even changed from 2006 to 2023. I will tell three stories, one from 1966, one from 2006, one from 2023. They all have to do with my hobby of collection novelty songs; however, I am sure there are similar stories in other realms

1) On Sept 21, 1966 there was an episode of Batman with special guest villain The Minstrel. He sang several songs in the episode that I thought were funny. My favorite was when Batman and Robin are tied up over a rotisserie, the Minstrel sings, to the tune of Rock-a-bye baby. 

Batman and Robin Rotate and Resolve

As the heat grows, your bodies Dissolve

When its still hotter, then you will Melt

Nothing left but your Utility Belt. 

I LIKED the song and WANTED it. So I found out when the episode would re-run and set up my tape recorder to record it. I still have the tape, though I don't have a tape player (see my blog post here) however it doesn't matter because a compilation of the songs in  that episode (actually two episodes) is on YouTube here.

2) On March 6, 2006 there was an Episode of Monk Mr. Monk goes to the dentist which has in it The Randy Disher Project singing Don't need a badge. This was great and I wanted that song. At the time I was buying the DVDs of Monk. When the DVD of that season came out I assumed the song  would be included as an extra. It was not :-(.  By that time I was busier than in 1966 so I  didn't have the time, patience, or tape recorder to track it down. But that does not matter since 8 years later it was on  YouTube here. But I had to wait 8 years.

3) On Aug 23, 2023 there was an episode of ST-SNW entitled Subspace Rhapsody that had NINE songs in it, sung by the crew (actually sung by the actors!)  I don't have streaming so I didn't watch it but I heard about it (people know I am interested in novelty songs so they tell me about stuff like that). I spend about 30 minutes on YouTube finding ALL NINE and putting them in my file of novelty song links, see here. And it was worth the effort- three of the songs are GREAT and the rest are pretty good (in my opinion).

Points

1) Also easier to find now then it was in 2006 and certainly in 1966: Everything. Okay, lets list some examples: Music (not just novelty), TV shows, Movies, Journal articles, Conference articles, books. But see next point. 

2) Big Caveat: For a recording from 1890 to have survived it would have to be on wax cylinder, then vinyl, then CD, maybe back to vinyl (Vinyl is having a comeback), and perhaps mp3, streaming, You Tube, or Spotify. Some music will be lost. I would like to think that the lost music is not the good stuff, but I know of cases that is incorrect (my blog post here gives an example). For journal articles there is also the issue of language. Some articles were never translated.  And some are written in a style we no longer understand. And some you really can't find. And there may be some songs where the only copy is in my collection.

3) Corollary to the Big Caveat: Some things are on YouTube one day and gone the next. There is an SNL short video Conspiracy Theory Rock which seems to come and go and come and go. I don't think its on YouTube, but I found it here. Lets hope it stays. I have that one on VHS tape but I don't have a VHS tape player. And modern e-journals might vanish. See my post on that issue here.

4) Some of my fellow collectors think they miss the days when only they had access to (say) Weird Al's Patterns which he sang on Square One Television (a math-for-kids show on PBS which I discovered and liked when I was 45). The song is on YouTube here. I find this point of view idiotic. The PRO of the modern world is I can find lots of stuff I like and listen to it (and its free!). The CON is a loss of bragging rights for people like me. Really? Seems like a very minor CON. I do not miss the days of hunting in used record shops for an old Alan Sherman record (ask your grandmother what a used record shop is and what an Alan Sherman is).

5) When I played the song Combinatorics (see here) in my discrete math class the students liked it (for some reason the TA hated it, oh well) and the students asked 

Is that a real song

I asked them to clarify the question. They couldn't. To ask if it ever came out on a physical medium is a silly question- it didn't, but that doesn't matter. Did it make money? Unlikely, but that would be a rather crass criteria. There are lots of VERY GOOD songs on You Tube (whether Combinatorics is one of them is a question I leave to the reader) so the question Is that a Real Song is either ill-defined or crass. All that matters is do you like it. 

E versus EXP

Why do we have two complexity classes for exponential time, E and EXP?

First the definitions:

E is the set of problems computable in time \(2^{O(n)}\).

EXP is the set of problems computable in time \(2^{\mathrm{poly}(n)}\).

The nondeterministic variants NE and NEXP have similar definitions and properties.

By the time hierarchy theorem, E is strictly contained in  EXP. But they have basically the same complexity:

• There are polynomial-time many-one complete sets for EXP in E.
• EXP is the closure of E under polynomial-time many-one reductions.
• E is in NP if and only if NP = EXP. You can replace NP by PSPACE, BPP, BQP or any other class closed under poly-time many-one reductions.

Quiz: Show that PSPACE \(\neq\) E. Hint: The proof doesn't tell you which class might be larger.

EXP is the natural class for exponential time since it is closed under polynomial-time reductions and is known to contain PSPACE and all those other classes above. You have results like MIP = NEXP but not MIP = NE since MIP (interactive proofs with multiple provers) is closed under polynomial-time reductions. 

E = NE implies EXP = NEXP but not necessarily the other way around. P = NP implies both equalities but again not the other way around. You get P = NP implies E = NE because poly(\(2^n)\) = \(2^{O(n)}\). That equality plays a role in other theorems related to E and NE:

Impagliazzo-Widgerson: If E is not computed by subexponential-size (\(2^{o(n)}\))-sized circuits then P = BPP. A similar assumption for EXP would only put BPP in quasipolynomial time. 

Hartmanis-Immerson-Sewelson: show that there are sparse (polynomial-sized) sets in NP-P if and only if E \(\ne\) NE. Their paper leads to endless confusion because they state the result as EXPTIME \(\ne\) NEXPTIME without defining the terms before the terminology was set.

In fact I just fixed the Wikipedia article on EXPTIME which had the incorrect statement. Aargh!

Soliciting open problems in honore of Luca T for my Open Problems Column

As you all know Luca Trevisan, a Giant in our field, passed away at the too-young age of 52. See Lance's post on Luca HERE. 

As the editor of the SIGACT News Open Problems Column I am putting together an open problems column in his memory.  (I did the same for Juris Hartmanis, see here, so you will have an idea of what I want.) 

If you want to submit an open problem, email me (gasarch@umd.edu) either 

a) Your IDEA for an open problem to see if its in scope, or 

b) If you are sure it's in scope,  Just Do It and send me the LaTeX code.  Page limit \le  2 page.

The problems should be either BY Luca or INSPIRED by Luca. 

I am thinking of open problems about derandomization and extractors; however, if Luca did some work in some other area that I am less familiar with (this is likely), that's fine; however,  cite that work. 

Luca Trevisan (1971-2024)

Complexity theorist Luca Trevisan lost his battle to cancer yesterday in Milan at the age of 52. A terrible loss for our community and our hearts go out to his family.  

The community will honor Trevisan's life and legacy 12:30 PM Pacific Time Monday at the TCS4All talk that he was scheduled to give at the STOC conference in Vancouver. Register to watch the talk online.

Luca was one of the great minds of our field, an expert on randomness and pseudorandomness. He's the first computer science member of Italy's National Academy of Science. He has taught at Columbia, Berkeley and Stanford until 2019 when he moved back to his home country to join Bocconi University in Milan. 

My favorite result from Trevisan is his connections between extractors and pseudorandom generators, especially as the first works on arbitrary distributions and the latter fools computationally randomized algorithms. This paper laid the framework for better bounds for both extractors and generators. I had one paper with Trevisan, where, with Rahul Santhanam, we show time hierarchies for almost all natural semantic classes with a small amount of advice.

Trevisan had his own blog In Theory full of technical course notes and wonderful stories. Bill has two guest posts on the polynomial van der Waerden theorem in Luca's blog following up on Luca's posts on Szemeredi’s theorem. 

A few years ago Trevisan started the BEATCS theory blogs column to highlight theory blogs and bloggers. Bill and I were both highlighted in this column. 

Trevisan is one of the first theoretical computer scientists to come out as openly gay and many followed. We've come a long way from Turing.

More remembrances from Boaz and Scott.

In 2014 Luca Trevisan returned to Berkeley and joined the Simons Institute as its first permanent senior scientist. Christos Papadimitriou interviewed Luca for the occasion. 

Rethinking Heuristica

I've argued that more and more we seem to live in an Optiland, a computational utopia where through recent developments in optimization and learning we can solve the NP-problems that come up in practice and yet cryptography remains unscathed. We seem to simultaneously live in Heuristica ( and Cryptomania of Russell Impagliazzo's Five Worlds.

But we don't. Impagliazzo defined Heuristica as the world where P \(\ne\) NP but we can solve NP-complete problems easily on average. Since cryptography requires problems that are hard on average, if we are in Cryptomania we can't be in Heuristica. 

That definition made sense in 1995 but it didn't envision a world where we can solve many NP-problems in practice but not easily on average. Despite its name, Heuristica as defined does not capture solving real-world problems. To be fair, Impagliazzo entitled his paper "A Personal View of Average-Case Complexity," not "A Treatise on Solving Real World Problems". 

So we need to rethink Heuristica or create a new world (Practica?) that better captures real-world problems. How would we do so? 

When I talked with the SAT Solving researchers at Simons last spring, they had suggested that problems designed to be hard are the ones that are hard. But how can we mathematically capture that? Maybe it's connected to learning theory and TC⁰ (low depth circuits with threshold gates). Maybe it's connected to constraint-satisfaction problems. Maybe it's connected to time-bounded Kolmogorov complexity. 

As complexity theorists this is something we should think about. As we study the mathematics of efficient computation, we should develop and continue to revise models that attempt to capture what kinds of problems we can solve in practice.

But for now I don't have the answers, I don't even know the right questions.

Should Prover and Verifier have been Pat and Vanna?

LANCE: I had my first Quanta Article published! I explore computation, complexity, randomness and learning and feeling the machine.

BILL: Feels to me like a mashup of old blog posts. Changing topics, I told Darling that you used Pat for Prover and Vanna for Verifier in a 1987 conference talk but those terms did not catch on. She was shocked!

LANCE: I'm shocked you two are married 32 years.

BILL: We hope to get to 64. However, she thought those were really good names for the concept (she has a masters degree in Computer Science so she knows stuff) and wondered why wouldn't those have caught on.

LANCE: I think that its frowned upon to use a cultural icon to tied to one country. There are Europeans who have no idea who Pat and Vanna are. For that matter, there are some Americans, particularly academics, who have no idea who Pat and Vanna are. And who would remember either of them once they stopped hosting the show? And who thought that would be 2024?

BILL: Who do papers on Interactive Proof Systems use?  Of course Author-Merlin games. Is the legend of King Author so well known (or at least it's well know that there IS a legend) that its okay to use those names? I think yes. 

LANCE: Did you really think his name is Author? I command thee to see Excalibur and learn the legend for yourself. Excalibur also being the name of a Computer Othello program I wrote in the 80's.

BILL: All right, Arthur. For one thing, we, or at least everyone but me, still knows who they are many years later, whereas Pat and Vanna will be lost to history. Hey Arthur and Merlin even got a science cartoon for their role in interactive proofs.

LANCE: Did Arthur and Merlin ever host a game show? I used Victor and Pulu in my thesis. I've also written papers where we use Prover and Verifier.

BILL: Pulu? Anyway, Prover and Verifier are boring!

LANCE: Sometimes boring works. We need to only use cultural icons that spans many cultures and won't be forgotten in 200 years. Just to be on the safe side, use cultural icons that are over 200 years old. 

BILL: Can you think of any cultural icon that has been used in Math or Computer Science and the name did catch on?

LANCE: The Monty Hall Problem.

BILL: I suspect there are many people who know who Monty Hall is only because of the paradox. And that is a paradox. Here is a name that didn't catch on: Sheldon's Conjecture was named after Sheldon Cooper from The Big Bang Theory. However, since it was solved, the name won't catch on, which is probably just as well. 

LANCE: How does the Chicken McNugget Theorem fit into this?

BILL: I don't know but it's making me hungry. Let's eat!

Favorite Theorems: Algebraic Circuits

May Edition

Most of my favorite theorems tell us something new about the world of complexity. But let's not forget the greatest technical challenges in our area: proving separations that are "obviously" true. Here's the most exciting such result from the past decade.  

Superpolynomial Lower Bounds Against Low-Depth Algebraic Circuits
Nutan Limaye, Srikanth Srinivasan and Sébastien Tavenas

In this model, the inputs are variables and constants, and the goal is to create a specific formal polynomial using the gate operations of plus and times. Limaye, Srinivasan and Tavenas find an explicit polynomial such that any polynomial-size constant-depth algebraic circuit will compute it. 

How explicit? Here it is: Take d nxn matrices, multiply them together and output the top left element of the product. The \(N=dn^2\) variables are the entries of the matrices. The top left element is a polynomial of the inputs that can be computed by a simple polynomial-size circuit that just computes the iterated multiplication, just not in constant depth. The paper shows that for an unbounded d that is \(o(\log n)\), there is no constant-depth polynomial-size algebraic circuit.

The authors first prove a lower bound for set multilinear circuits and then extend to more general algebraic circuits.

CFG-Kolm-complexity is singleton sets with Lance and Bill

For this post all Context Free Grammars (henceforth CFGs) are assumed to be in Chomsky Normal Form. The size of a CFG \(G\)  is the number of rules. We denote this by \(|G|\).

BILL: In my automata theory class I want to do some lower bounds on the size of CFGs. It is easy to show that if   \(w=0^n\) then there is a CFG G such that \(L(G)=\{w\}\) and \(|G|=O(\log n)\). I showed that if \(w\) is a Kolmogorov random string of length \(n\), and G is a CFG such that \(L(G)=\{w\}\), then \( |G|=\Omega(n/\log n\)), though this is surely known. So here is my question: Is there a natural such \(w\)? I will blog about that and make an open problems column out of it.

LANCE: Kolmogorov strings are natural!

BILL: Oh yeah. If that was true then spell check would not flag Kolmogorov as being misspelled.  So there!

LANCE: Can you ask a more rigorous question?

BILL: Okay. We can view the Kolm-result as saying that there is a function \(f\) from \(1^*\) to \(\{0,1\}^*\) such that  \(f(1^n)\) is a string of length \(n\) such that any CFG for \( \{w\}\) is large. But the function f is not computable!

LANCE: That shouldn't bother you. You wrote an entire book about how many queries to HALT and other incomputable sets are needed to solve certain problems (see here).  Also now that you know you there are such strings, you can simply search for a w and test all small CFGs. So Computable!

BILL: Still not natural. And what is the complexity? Exponential? Poly?

WE DROP THE TOPIC AND PICK IT UP AGAIN A FEW TIMES. WE (meaning mostly Lance) HAVE SOME BRILLIANT INSIGHTS THAT LEAD TO THE FOLLOWING RESULTS:  

1) For every \(w \in \{0,1\}^n\) there is a CFG G with \(L(G)=\{w\}\) and \( |G|=O(n/\log n)\)

2) If  \(w\) is a de Bruijn sequence of length \(n\) and order \(k=\log n\) (we assume n is a power of 2). Then every CFG G with \(L(G)=\{w\}\) has \( |G|=\Omega(n/\log n)\).  There is a known algorithm that will, given \(1^n\), produce a de Bruijn sequence or length n and order \(k=\log n\), in time quasilinear in \(n\). 

BILL: That bums me out for two contradictory reasons

a) The problem is NOT solved since de Bruijn is flagged by spellcheck, so the sequences are not natural.

b) The problems IS solved, so I can't use it for an open problems column. 

LANCE: Do not despair!

a) De Bruijn sequences have a Wikipedia page and therefore are natural. 

b) We can post on ArXiv. 

WE DID and a day later Markus Lohrey emailed us that, aside from the De Bruijn result, the results are already known using a different terminology, word chains.  See his survey here. Then the next day, Giovanni Pighizzini emails us that he had previously published lower bounds for De Bruijn sequences. We have since withdrawn the paper. We revised it by putting in references and history but will not put it on arxiv. The revised paper is here.

LANCE: Bill, are you bummed out? Why did we even write the paper anyway?

BILL: Not at all!  My original goal was pedagogical, and the paper we have can still be taught in automata theory next spring. PLUS, we got invited to submit to Advanced in AI and ML with a 10% discount on publication fees (see here.) Since we are used to getting 100% discount on publication fees we won't be submitting, but it was nice to be asked. 

LANCE: Yeah, nice to be asked to be parted from my money. At least I learned about word chains.

The Godzilla Moment

On the plane earlier this week I got around to watching the Academy Award winning movie Godzilla Minus One, one of the best monster movies I've seen set in Japan during the aftermath of World War II, with a pretty emotional substory about a man dealing with his demons from the war. I had to hide my tears from the nearby passengers.

It wasn't the story that earned the movie an Oscar. Godzilla Minus One won the awards for Best Visual Effects. I found nothing wrong with the effects, but they didn't excel beyond what you see in any typical movie of the genre.

In 2008, I lamented that special effects in movies had improved so much that we had lost the amazement we felt in the 70s. Perhaps I spoke too soon, as James Cameron's Avatar came out the following year and did amaze. However, special effects have since become a commodity, something filmmakers must include because audiences expect it but rarely do you go to a movie for the effects. In the not-too-distant future, special effects will be automated with AI, becoming just another plugin for Final Cut Pro. 

It's time to retire the visual effects award, especially with new awards coming to the Oscars.

I wrote that 2008 column to mirror the lack of enthusiasm about computing at the time which also felt like a commodity. Now we're at an exciting time in computing particularly with the advances in artificial intelligence. But we should be wary, once (if?) AI gets consistently good it may feel like a commodity again and once again we become victims of our own success. 

FOCS 2024 Test of Time Award. Call for nominations and my opinion

 The call for nominations for the Test of Time Award at FOCS 2024 has been posted here.

Eligibility and past winners are here.

Points

1) It is good to have an award that waits until the dust settles and we can see what was really important.

2) The winners are all excellent papers that really have passed the test of time. 

3) And of course it is really important that they appeared in FOCS. NO IT ISN"T! See next point

4) I would prefer a test-of-time award that is independent of WHERE the paper first appeared. Tying it to FOCS or STOCS or FOCS-or-STOC seems bad. I would opt for appearing in ANY journal or conference. Appearing in a journal of low quality is not a problem since this award should be  for papers that are judged on their merit and influence, and not on their pedigree.

5) My proposal to allow any journal or conference may be impractical because some organization has to give it out, and if that organization is IEEE or ACM they will restrict to their own publications. 

6) STOC also has a test of time award, see here 76) I tried to find out of the SODA conference has a test of time award but mostly got hits about the Baking Soda Test for determining if a pregnant women is going to have a boy or  girl. It actually worlds 50% of the time! See here

7) I was not able to find any other test-of-time award for Comp Sci THEORY. 

8) I DID find test of time awards for

SIGCSE- Comp Sci Education, here. Must be for a paper published in a conference co-sponsored by SIGCSE or in an ACM journal.  So an excellent paper published elsewhere wouldn't count. 

SC2- High Performanc Computing, see here. Paper must have been published in the SC conference. 

ACM CCS - Security, Audit(?) and Control, see here I think these must appear in the CCS conference. 

Double Digit Delights

It started with a post from Fermat's Library.

132 is the sum of all the 2-digit numbers made from its digits. It is the smallest such number. pic.twitter.com/hrByAXbr51

— Fermat's Library (@fermatslibrary) May 26, 2024

My immediate reaction was why not list them all? Giving the smallest such number suggests there are an infinite number of them. But the value of a d-digit number grows exponentially in d, while the 2-digit sum grows quadratically so there must only be a finite number. 

Let's be a little more formal. Let's restrict ourselves to positive integers with no leading zeros. The 2-digit sum of x is the sum of all 2-digit numbers formed by concatenating the ith digit of x and the jth digit of x for all i,j with i\(\neq\)j. The 2-digit sum of 132 is 13+12+31+32+21+23 = 132. The 2-digit sum of 121 is 12+11+21+21+11+12 = 88. A number x if 2-idempotent if the 2-digit sum of x is x.

Let's look at the possible lengths of 2-idempotent numbers.

For 1-digit numbers the 2-digit sum is zero.

For 2-digit numbers the 2-digit sum is that number plus another positive number so never equal.

For 5-digit numbers, the 2-digit sum is bounded by 20*99 = 1980 < 10000. So there are no 2-idempotent numbers with 5-digits. More than 5 digits can be discarded similarly. 

For 4-digit numbers, the two digit sum is at most 12*99 = 1188. So a 2-idempotent number must begin with a one. Which now bounds it by 19*3+91*3+99*6=924. So there are no 2-idempotent numbers of 4 digits.

So every 2-idempotent must have 3 digits. I wrote up a quick Python program and the only three 2-idempotents are 132, 264 and 396. Note that 264 is 2*132 and 396 is 3*132. That makes sense, if you double every digit and don't generate carries, every two-digit part of the sum also doubles.

Biscuit asks if there is some mathematical argument that avoids a computer or manual search. You can cut down the search space. Every length 3 2-idempotent is bounded by 6*99=594 and must be even since every digit appears in the one's position twice. But I don't know how to avoid the search completely.

Two more Python searches: 35964 is the only 3-idempotent number. If you allow leading zeros then 0594 is 2-idempotent. There may (or may not) be infinitely many such numbers.

National BBQ day vs World Quantum Day

 After my post on different holiDAYS, here, such as Talk like a Pirate Day, and Raegan Revor day, two other Days were brought to my attention

1) Lance emailed me about National BBQ day, which is May 16. See here

2) While at a Quantum Computing Prelim I saw a poster for World Quantum Day, which is April 14. See here.

The obvious question: Which of these days is better known? I Googled them again but this time note the number of hits. 

I found out that Google seems to have removed that feature!

When using Google on both Firefox and Chrome, I did not get number of hits. 

Some points about this

1) Is there a way to turn the number-of-hits feature on?

2) Bing DOES give number of hits.

World Quantum Day: 899,000 hits

National BBQ Day: 418,000 hits

To get a baseline I binged Pi Day. This did not reveal the number of hits. An unscientific set of Bing searches seems to indicate that if the number of hits is large then they are not shown.

Is hits-on-Bing a good measure of popularity? I do not know.

3) Duck Duck Go does not give number of hits. This might be part of their privacy policy.

4) I also noticed a while back that You Tube no longer allows DISLIKES, just likes. That may explain why my Muffin Math song on You Tube (see here), with Lance on the Piano,  has 0 dislikes. It does not explain why it got  19 likes.

5) Google said that the number-of-hits is really an approximation and one should not take it too seriously. 

YouTube said that (not in these words) the haters caused dislikes to be far more than they should be.

On the one hand, I want to know those numbers. On the other hand I think Google and YouTube are right about about the numbers not being that accurate. And more so for Bing which is used less so (I assume) has less data to work from.

6) Back to my question: What is better known National BBQ day or World Quantum Day? The nation and the world may never know. 

7) All of the above is speculation.

Peer Review

Daniel Lemire wrote a blog post Peer Review is Not the Gold Standard in Science. I wonder who was claiming it was. There is whole section of an online Responsible Conduct in Research we are required to take on peer review which discussing its challenges: "honesty, objectivity, quality control, confidentiality and security, fairness, bias, conflicts of interest, editorial independence, and professionalism". With apologies to Winston Churchill, Peer Review is the worst form of measuring academic quality, except for all of the others.

Peer review requires answering two questions.

1. Has the research been done properly?
2. What is the value of the research?

For theoretical research, the first comes down to checking the proofs, which sounds like an objective check. Here we have a "gold standard", formalizing the proof so it can be verified in a proof system like Lean. That's a heavy burden so we generally only require authors to give enough details so it's clear that we could formalize the proof given enough time. That becomes subjective and reviewers, especially for conferences, may not have the time or inclination to check the details of a 40-page proof. Maybe one day AI can take a well-written informal proof and formalize it for a proof system.

But the second question is almost entirely subjective. How does the work advance previous research? What value does it give to a field and how does it set up future research? Different researchers will give different opinions. And then there are the people who consciously or unconsciously cheat, helping their friends get papers accepted to citations rings. As we focus on metrics to judge researchers, too many people will game the system to pump up those metrics.

In 2013, NeurIPS had over 13,000 submission for 3500 slots. Even with the best or reviewer's intentions, it's impossible to maintain any sense of consistency for these large volume conferences.

Despite the problems with peer review, you'd hate to us a different system, say delegating the reviewing to some AI process, even if it could lead to more consistency. I suspect many reviews are being delegated anyway.

Peer review grew in importance as journals and conferences had to make choices to fill a limited proceedings. These days we have the capacity to distribute every papers. So perhaps the best form of measuring academic quality is no review at all.

I don't do that well when the Jeopardy category is Math

Bill and Darling are watching Jeopardy.

DARLING: Bill, one of the categories is MATH TALK. You will kick butt!

BILL: Not clear. I doubt they will have the least number n such that R(n) is not known. They will ask things easy enough so that my math knowledge won't help.

DARLING: But you can answer faster.

BILL: Not clear. 

--------------------------------------------

Recall that in Jeopardy they give the answers and you come up with the question.

Like Sheldon Cooper I prefer my questions in the form of a question. 

Even so, I will present the answers that were given on the show (that sounds funny), then 

I will provide the questions (that sounds funny), what happened, and what I would have gotten right. 

$400

ANSWER: Its a demonstrably true mathematical statement; Calculus has a ``Fundamental'' one.

QUESTION: What is a Theorem?

WHAT HAPPENED: Someone buzzed in and said AXIOM. This one I knew the answer and would have won!

$800

ANSWER: Fire up the engines of your mind and name this solid figure with equal and parallel circles at either end. 

QUESTION: What is a Cylinder?

WHAT HAPPENED: Someone buzzed in with the correct answer. I had a hard time parsing this one and only got it right in hindsight. This one I would have lost on. Note that the phrase Fire up your engines is supposed to make you think of Fire on all cylinders. This did not help me.

$1200

ANSWER: Multiply the numerator of one fraction by the denominator of another (and vice versa) to get the ``cross'' this. 

QUESTION: What is a Product?

WHAT HAPPENED: I got this one very fast. So did the contestant on the real show. Not clear what would happened if I was there.

$1600

ANSWER: See if you can pick off this term for the point at which a line or curve crosses an axis. 

QUESTION: What is an Intercept?

WHAT HAPPENED: Someone buzzed in with the correct answer. I really didn't know what they were getting at. Even in hindsight the answer does not seem right, though I am sure that it is. The phrase pick off this term is  supposed to remind me of something, but it didn't. Lance happened to read a draft of this post and did the obvious thing: asked ChatGPT about it. ChatGPT said that in football a pick off is an interception. To see the ChatGPT transcript see here.

$2000

ANSWER: In 19-5=14 19 is the minuend; 5 is this other ``end''

QUESTION: What is a  Subtrahend?

WHAT HAPPENED: Someone buzzed in with the correct answer. The answer was news to me. It is correct; however, I am not embarrassed to say I never heard these terms. Spellcheck thinks that minuend and subtrahend words. This is similar to when I was not smarter than a fifth grader (see blog post here). 

----------------------------------------------------------------

So the final tally:

The $400 question I would have gotten right

The $1200 question I might have gotten right if I was fast on the buzzer

But that's it. Why did I do so badly? 

1) Two of the ones I got wrong were phrased in funny ways. I thought so anyway. And note that they did not use advanced math knowledge, so my math knowledge didn't help. (This is not a complaint- it would be bad if they used advanced math knowledge. Like when a crossword puzzle my wife was working on wanted  Log-Man and it began with N and I knew Napier. Why was that in a crossword puzzle for laypeople? Because  Napier has a lot of vowels in it.)

2) One of them I really did not know the math knowledge. Is it arrogant to say that if there is a math question on Jeopardy where I don't know the answer then its a bad question? I leave that as an exercise for the reader. 

On questions about  presidents, vice presidents, or American history, I do well.

On questions about novelty songs  (sometimes comes up) I do very well. (One question was about this song here. The question: here.)

But math... not so much. 

For computer science questions I also do not do that well, but I've learned some common abbreviations that I did not know: 

BIT: Binary Integer (A reader named Anonymous, who makes many comments, pointed out that BIT is actually Binary Digit. I have a possibly false memory of Jeopardy telling me Binary Integer. Either my memory is wrong or Jeopardy is wrong. But Anonymous is right- its Binary Digit.) 

HTTP: Hypertext Transfer Protocol

HTML: Hyper Text Markup Language

FORTRAN: Formula Translation

Those were more interesting than learning about minuend and subtrahend, terms I had never heard before and won't hear again unless I catch a rerun of Jeopardy (at which time I will get it right).

Jim Simons (1938-2024)

Jim Simons passed away Friday at the age of 86. In short he was a math professor who quit to use math to make money before it was fashionable and used part of his immense wealth to start the Simons Foundation to advance research in mathematics and the basic sciences.

While his academic research focused on manifolds, Simons and his foundation had theoretical computer science as one of its priorities and helped fund and promote our field on several fronts.

Foremost of course is the Simons Institute, a center for collaborative research in theoretical computer science. Announced as a competition in 2010 (I was on team Chicago) with the foundation eventually landing on UC Berkeley's campus. At the time, I wrote "this will be a game changer for CS theory" if anything proven to be an understatement over the last dozen years.

Beyond the institute, the Simons Foundation has funded a number of theorists through their investigator and other programs.

Let's not forget Quanta Magazine, an online science publication funded by the foundation without subscriptions or paywalls while science journalism has been seeing cuts elsewhere. Quanta has been particularly friendly to the computational complexity community such as this recent article on Russell and his worlds.

The Simons Foundation will continue strong even without its founder. But as we see challenges in government funding, how much can or should we count on wealthy patrons to support our field?

Read more on Jim Simons from Scott, Dick, the foundation and the institute.

What is Closed Form? The Horse Numbers are an illustration

In the book Those Fascinating Numbers by Jean-Marie De Konick they find interesting (or `interesting') things to say about many numbers. I reviewed the book in a SIGACT News book review column here. The entry for 13 is odd: 13 is the third Horse Number.  The nth Horse number is the number of ways n horses can finish a race. You might think: OH, that's just n!. AH- horses can tie. So it's the number of ways to order n objects allowing ties. 

Is there a closed form for H(n)? We will come back to that later. 

0) The Wikipedia Entry on horse races that ended in a dead  heat is here. They list 78 dead heats (two horses tie for first place) and 10 triple dead heats (three horses tie for first place). For the horse numbers we care if (say) two horses tie for 4th place. In reality nobody cares about that. 

1) I have found nowhere else where these numbers are called The Horse Numbers. 

2) They are called the Ordered Bell Numbers. The Wikipedia entry here has some applications.

3) They are also called the Fubini Numbers according to the Ordered Bell Number Wikipedia page.

4) I had not thought about the Horse Numbers for a long time  when they came up while I was making slides for the proof that  (Q,<) is decidable (the slides are here).

5) There is an OEIS page for the Horse Numbers, though they are called the Ordered Bell Numbers and the Fubini Numbers. It is here. That page says H(n) is asymptotically \(\frac{1}{2}n!(\log_2(e))^{n+1}\) which is approx \(\frac{1}{2}n!(1.44)^{n+1}\).

6) There is a recurrence for the Horse Numbers:

H(0)=1

H(1)=1

H(2)=3

For all  \(n\ge 3\) we split H(n) into what happens  if i horses are tied for last place (choose i out of n) and if the rest are ordered H(n-i) ways. Hence

\( H(n) = \binom{n}{1}H(n-1) + \binom{n}{2}H(n-2) +  \cdots  + \binom{n}{n}H(0) \)

Using \(\binom{n}{i} = \binom{n}{n-i}\) we get

\( H(n) = \binom{n}{0}H(0) + \binom{n}{1}H(1) +  \cdots  + \binom{n}{n-1}H(n-1) \)

STUDENT: Is there a closed form for H(n)?

BILL: Yes. Its H(n).

STUDENT: That's not closed form.

BILL: Is there a closed form for the number of ways to choose i items out of n?

STUDENT: Yes, \(\binom{n}{i}\) or \( \frac{n!}{i!(n-i)!}\) 

BILL: Does that let you compute it easily? No. The way you compute \(\binom{n}{i}\) is with a recurrence. The way you compute H(n) is with a recurrence. Just having a nice notation for something does not mean you have a closed form for it. 

STUDENT: I disagree! We know what n! is!

BILL: Do not be seduced by the familiarity of  the notation. 

Favorite Theorems: Dichotomy

April Edition

A constraint satisfaction problem has a group of constraints applied to a set of variables and we want to know if there is a setting of the variables that make all the constraints true. In CNF-Satisfiability the variables are Boolean and the constraints are ORs of variables and their negations. In graph coloring, the variables are the colors of the nodes and the constraints, corresponding to edges, are two variables must be different. These problems lie in NP, just guess the values of the variables and check the constraints. They are often NP-complete. They are sometimes in P, like 2-coloring graphs. But they are never in between--all such problems are either in P or NP-complete.

A Dichotomy Theorem for Nonuniform CSPs by Andrei Bulatov

A Proof of the CSP Dichotomy Conjecture by Dmitriy Zhuk

Ladner's Theorem states that if P \(\neq\) NP then there exists a set in NP that is not in P and not NP-complete. Ladner's proof works by blowing holes in Satisfiability, an unsatisfying construction as it gives us a set that is NP-complete on some input lengths and easy on others. One could hope that some version of a constraint satisfaction problem could lead to a more natural intermediate set but dichotomy theorems tell us we need to look elsewhere.

In 1978, Thomas Schaefer gave a dichotomy theorem for satisfiability problems, basically CSP problems over Boolean variables. In 1990, Pavol Hell and Jaroslav Nešetřil showed a dichotomy result for homomorphisms of undirected graphs as described in my 2017 blog post. In 1998 Tomás Feder and Moshe Vardi formalized the constraint satisfaction dichotomy conjecture and expressed it as homomorphisms of directed graphs. The blog post described a claimed but later retracted solution to the dichotomy conjecture. Bulatov and Zhuk announced independent and different correct proofs later that year. In 2020 Zhuk received the Presburger Award for his paper (Bulatov was too senior for the award). 

May the fourth be with you. Too many -days?

(This post was inspired by Rachel F, a prior REU-CAAR student, emailing me wishing me a happy Star Wars Day.) 

 I am writing this on May 4 which is Star Wars day. Off the top of my head I know of the following special days (I exclude official holidays, though the term official has no official meaning.)

Jan 25: Opposite Day Wikipedia Link

Feb 2: Groundhog Day Wikipedia Link

Feb 12: Darwin Day Wikipedia Link

March 14: Pi Day Wikipedia link

May 4: Star Wars Day Wikipedia Link

April 22: Earth Day Wikipedia link

April 25: Take your Child to Work Day Wikipedia Link

Sep 21: National Cleanup Day Wikipedia Link

Sept 22: Hobbit Day Wikipedia Link

Oct 1: International Coffee Day Wikipedia Link

Oct 8: Ada Lovelace Day Wikipedia Link

Oct 16: Boss's Day  Wikipedia Link

Oct 23: Mole Day Wikipedia Link

Nov 13: Sadie Hawkins Day Wikipedia Link

Sept 19: Talk like a Pirate Day Wikipedia Link

A few notes

1a) Oct 23 is also Weird Al's birthday.

1b) May 4 is also Edward Nelson's birthday (he invented the problem of  finding the chromatic number of the plan). See my post (actually a guest post by Alexander Soifer) on the problem here for more information on that.

1c) I left off St. Patrick's Day (March 17) and International LGBT + Pride day (June 28) and many others.  Those left off are so well known that they are official where as I was looking for unofficial holidays. But see next point.

2) The Wikipedia entry for Talk Like a Pirate Day says it's a parodic holiday. The entries on the others holidays use terms like unofficial. I prefer unofficial since ALL holidays are made up, so the only real question is which ones are recognized. But even that is problematic since one can ask recognized by who? Also, despite collecting parody music and videos for the last 50 years, I have never heard the term parodic. Therefore it is not a word. Spellcheck agrees!

3) Darwin Day should be Darwin-Lincoln day since they were both born on Feb 12. In fact,they were both born in 1809. Most famous same-birthday-and-year pair ever. Second place is Lenny Bruce and Margaret Thatcher (Oct 13, 1925). 

4) The page on Pi Day mentions Tau Day, but Tau day has no page of its own. Tau is \(2\pi\) which some say comes up more often then \(\pi\) and hence should be THE constant. Some say that \(2\pi i\) comes up so often that it should be THE constant. However, there can't really be a day to celebrate it.(I blogged about is-tau-better-than-pi here.)

5) In the future every day will be some kind of day. The Future Is NOW: Website of Fun Holidays

Are the holidays on the list real? Depends what you mean by real. Because of the web anyone can post a list of anything and its just one person's opinion. I do not know who controls that website but even if I did, it would be hard to say YES THOSE ARE REAL or NO THOSE ARE NOT. 

One could say that to be a real DAY, it has to be on Wikipedia. But there are two problems with this:

a) Goodhart's law. When a measure becomes a target it stops being a measure. If I want Jan 15 to be Bagel and Lox Day, I'll make a page for it.

b) I'm still waiting for Raegan Revord, who has played Missy on Young Sheldon for 7 years, to get a Wikipedia Page. So what hope does Polar Bear Plunge day (Jan 1) have for getting a Wikipedia Page? 

Our Obsession with Proofs

Bullinger's post on this blog last week focused on Vijay Vazirani's public obsession of finding a proof for the 1980 Micali-Vazirani matching algorithm. But why does Vijay, and theoretical computer science in general, obsess over proofs? 

You can't submit a paper to a theory conference without a claimed complete proof, often contained in an appendix dozens of pages long. Often we judge papers more on the complexity of the proof than the statement of the theorem itself, even though for a given theorem a simpler proof is always better.

A proof does not make a theorem true; it was always true. The Micali-Vazirani algorithm is no faster with the new proof. Would we have been better off if the algorithm didn't get published before there was a full proof?

We're theoretical computer scientists--doesn't that mean we need proofs? Theoretical economists and physicists don't put such an emphasis on proofs, they focus on models and theorems to justify them.

Once a senior CS theorist told economists that his group had given the first full proof of a famous economics theorem and wondered why the economists didn't care. The economists said they already knew the theorem was true, so the proof added little to their knowledge base.

More than one journalist has asked me about the importance of a proof that P \(\ne\) NP. A proof that P = NP would be both surprising and hopefully give an algorithm. While a proof that P \(\ne\) NP would be incredibly interesting and solve a major mathematical challenge, it wouldn't do much more than confirm what we already believe.

I'm not anti-proof, it is useful to be absolutely sure that a theorem is true. But does focusing on the proofs hold our field back from giving intuitively correct algorithms and theorems? Is working out the gory details of a lengthy proof, which no one will ever read, the best use of anyone's time? 

As computing enters a phase of machine learning and optimization where we have little formal proof of why these models and algorithms work as well as they do, does our continued focus on proofs make our field even less relevant to the computing world today?

Math Thoughts Inspired by the TV show Succession

I watched Succession one-episode-a-day on treadmill for 39 days. I'm glad I did this in 2023 since Season 2 aired its last show on Oct 19, 2019, and Season 3 had its first show on Oct 17, 2021, so I would have been in suspense (or at least as much suspense as corporate board meetings can generate) for about 2 years. 

The show inspired the following mathematical thoughts.

1) There was a scene which I paraphrase as follows:

Alice: I'll give you two billion dollars for your shares in the company.

Bob: Don't insult me. It's worth at least 2.5 billion. 

My thought: I would take the two billion. 

My Math Thought: Let's say the shares really were worth 2.5 billion. Is it worth haggling? I think not, since I can't imagine I will ever spend that much money in my life. (See the blog post here  on the St. Petersburg paradox which is about  how much money is enough.) To be fair, if I wanted to buy Tesla (note that I mean buying the company, not buying the car) or buy the Comedy Cable Station (I would run it so that it only airs funny commercials) then I might need the extra half a billion to even begin trying (Tesla is worth around 740 billion).  SO the question of is it worth the haggling depends on your situation. 

So let's assume you don't need the money for some large purchase. Would you haggle? One reason to haggle is that you don't want the person who is ripping you off by only offering 2 billion to get away with it and/or think you are a chump. Whether one cares about this might depend on the relationship you have with that person. Even so, I would just take the 2 billion. Perhaps that's why I am in academia instead of business.

MATH QUESTION: Can we quantify what amount of money its not worth haggling over? 

2) Alice wants to buy Bob's company. After months of negotiations they agree to x dollars (and there are many side issues as well). The next day 

Alice thinks: OH, if he was willing to sell for x, he would be willing to sell for x-1. 

Bob thinks:OH, if she was willing to buy for x, he would be willing to buy for x+1.

(In the show this scenario happened many times, but usually with only one party wanting to re-negotiate and its not just  +1 and -1, its things like seats-on-the-board, who-will-be-CEO.) 

MATH QUESTION:  If Alice and Bob behave as above is there any mechanism to make them actually come to an agreement? Might involve assuming they can't factor fast or can't solve NPC problems fast.

Is Persistence an Anachronism?

Guest post by Martin Bullinger

Very recently, Vijay Vazirani's paper A Theory of Alternating Paths and Blossoms, from the Perspective of Minimum Length got accepted to Mathematics of Operations Research. For the first time, it gives a complete and correct proof that the Micali-Vazirani algorithm finds a maximum cardinality matching in time \(\mathcal O\left(m\sqrt{n}\right)\). I would like to give an account of the extraordinary story of this proof and how Vazirani's contribution inspires persistence.

My fascination for matching already started during my undergrad when I gave a talk on Edmonds' blossom algorithm. It was at this time that I first heard about the Micali-Vazirani (MV) algorithm. Naturally, I was quite excited when I got to know Vazirani personally years later. When I talked to him about the MV algorithm I was, however, shocked: Vazirani admitted that even to that day, there did not exist a complete proof of its correctness. How can a theoretical result be accepted to FOCS without a proof?

Now, 44 years after publication of the algorithm, a proof exists and has been peer-reviewed in great depth. But why did it take so long? Apparently, some results just need time. Sometimes a lot of time. Think of Fermat's Last Theorem, whose proof took 358 years! So what is the story behind the MV algorithm? It can without a doubt be seen as a lifework. Together with his fellow PhD student Silvio Micali, Vazirani discovered it in the first year of his PhD in 1979-80. Without even attempting a proof, it was published in the proceedings of FOCS 1980. The first proof attempt by Vazirani was published in 1994 in Combinatorica. Unfortunately, this proof turned out to be flawed. It took another 30 years until his current paper.

What kept Vazirani going for so long? In the acknowledgements of his paper, he thanks matching theory for its gloriously elegant structure. Vazirani was driven by his passion for the subject matter---but passion by itself can only go so far. Even more important was his belief in the correctness of the algorithm and the theory, which he had broadly outlined in his 1994 paper. Similar to Andrew Wiles' story, his perseverance led him to the idea which clinched the proof. In Vazirani's case, this was to use the new algorithmic idea of double depth-first search, which forms the core of the MV algorithm, and now, its proof as well. But Vazirani's result is also the story of an excellent research environment. Finding deep results requires colleagues or friends to discuss ideas with. Vazirani had these in the form of strong postdocs and PhD students. About ten years ago, he had been discussing ideas towards his proof with his former postdoc Ruta Mehta, and in the last three years, he discussed the final touches of his proof with his current PhD student Rohith Gangam. Needless to say, both of them gained a lot from these discussions.

So why should we care for the MV algorithm? I have several reasons. First, without doubt, it is a historic result within combinatorial optimization. Matching is one of the most fundamental objects in discrete mathematics and we keep finding new applications for it, for example, in health, labor markets, and modern day matching markets on the Internet, basically in every part of our lives. But there is more. Once again, one can look at Vazirani's paper where he describes the impact of matching to the development of the theory of algorithms: Matching theory has led to foundational concepts like the definition of the complexity classes \(\mathcal P\) (Edmonds, 1965a) and \(\# \mathcal P\) (Valiant, 1979), the primal-dual paradigm (Kuhn, 1955), and polyhedral combinatorics (Edmonds, 1965b). The impact of matching on complexity theory was an earlier topic of this blog.

Despite being around for decades, the MV algorithm is still the fastest known algorithm for computing a maximum cardinality matching. This is surprising, to put it mildly. Similar to many other fundamental problems in combinatorial optimization, I would have expected the discovery of better algorithms in the last four decades. Why has this not happened? Vazirani appears to have gotten to the essence of the problem: a profound theory that interleaves algorithmic invariants and graph-theoretic concepts. It seems to be the kind of theory which would play an active role in the field of combinatorial optimization.

However, Vazirani's result proves something else, possibly even more important: the massive gains to be made by single-minded persistence. In a world in which departments and promotion procedures focus on publishing large numbers of papers, it seems impossible to work on one result for more than a year, let alone for decades. Vazirani managed to achieve both: pursue his passion and get the unfinished job done, but not let it come in the way of the rest of his otherwise-active research career. As a young researcher, this inspires me! In the end, it is through such persistence that science will take big steps forward.

This blog post evolved from many enjoyable discussions, which I had with Vijay Vazirani during a research stay at UC Irvine in spring 2024. I am grateful to Ruta Mehta for feedback on the initial version of this post. Vazirani recently presented his paper in a mini series of two talks available online.