An investment puzzle and speculation as to why some think its hard

 This is a Guest Post by David Marcus. He gives a puzzle and its solution, which is interesting, and then speculates as to why some people get it wrong. 

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THE PROBLEM:

Investing Puzzle or Arithmetic Can Be Useful

The following is an example of investment results that I saw in an

investment newsletter. There are two portfolios that use different

strategies. Both portfolios start with $1 million twenty years ago and

withdraw 5% each year. The idea is that you are retired and withdrawing

money to spend. Not all years are shown in the tables.

Portfolio A

Year   Return  Withdrawal    Balance

2000   15.31%      57,655  1,095,445

2005    1.81%      59,962  1,139,273

2008  -12.65%      51,000    969,004

2009   34.26%      65,049  1,235,936

2010   11.94%      69,175  1,314,331

2015   -2.48%      64,935  1,233,764

2020   10.27%      66,935  1,271,765

Total Withdrawal: 1,685,190

Change in Balance: 27.18%

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Portfolio B

Year   Return  Withdrawal    Balance

2000   -0.95%      49,524    940,956

2005    3.80%      44,534    846,154

2008  -20.11%      35,922    682,523

2009   18.27%      40,360    766,833

2010   11.57%      42,777    812,764

2015    0.99%      50,767    964,567

2020   13.35%      65,602  1,246,433

Total Withdrawal: 1,425,573

Change in Balance: 24.64%

Portfolio A has a final balance that is 25,000 more than Portfolio B's and

had about 260,000 more in withdrawals. Does the example lend credence to

the Portfolio A strategy being better than the Portfolio B strategy?

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THE ANSWER:

Investing Puzzle or Arithmetic Can Be Useful: Analysis

Summary: The two portfolios have about the same performance over the 20

years. The difference is mainly due to Portfolio A having a good year or

years near the beginning before much money was withdrawn. The example

merely shows that it is better to withdraw money after a gain rather than

before.

Detailed Analysis:

The scenario is: Start with X = $1 million. Withdraw 5% a year.

Define "gain factor" to be 1 plus the percentage return. For example, if a

portfolio returns 5%, then the gain factor is 1.05.

Let A_j, j = 1, ..., 20, be the gain factors each year for portfolio A.

Let B_j, j = 1, ..., 20 be the gain factors each year for portfolio B.

The final amount in portfolio A is

   F = X * A_1 * 0.95 * A_2 * 0.95 * ... * A_20 * 0.95 .

The final amount in portfolio B is

   G = X * B_1 * 0.95 * B_2 * 0.95 * ... * B_20 * 0.95 .

From the "Change in Balance" values or the balances for year 2020, we see

that F and G are almost the same:

   F = 1.271865 * X,

   G = 1.246433 * X.

But, as we learned in elementary school, multiplication is commutative, so

   F = X * 0.95^20 * \prod_{j=1}^20 A_j,

   G = X * 0.95^20 * \prod_{j=1}^20 B_j.

Since F and G are almost the same, the total gains (product of the gain

factors) for the two portfolios are almost the same, i.e.,

   \prod_{j=1}^20 A_j \approx \prod_{j=1}^20 B_j.

Then what accounts for the big difference in the amounts withdrawn?

Portfolio A must have had some good years near the beginning. (We see in

the tables that Portfolio A did better in 2000 than Portfolio B.) So, all

the example shows is that it is better to withdraw your money after your

gains rather than before.

To take an extreme example, suppose an investment is going to go up 100%

this year. It is better to take your money out at the end of the year

(after the gain) than at the beginning of the year (before the gain). This

is a triviality.

The example tells us nothing useful about the two strategies.

Note: The total gains aren't exactly the same, but the timing of the yearly

gains is what is driving the results. We have (rounding off)

   ( F - G ) / 0.95^20 = 70942.81 .

So, if there had been no withdrawals, the difference in the portfolio

balances would have been about $71,000, much less than the $260,000 +

$25,000 difference with the withdrawals.

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WHY IS THIS HARD FOR PEOPLE?

Many people have trouble with this puzzle. The difficulty may be that such

people don't make a mental model (or a written model) of the process that

is producing the balance. If you write down (or have in your head) a

formula for the balance, then you see that the gain factors are independent

of the withdrawal factors. That is, we could withdraw more or less money,

or even deposit money, without affecting the gain factors we would use in

the model. This then leads us to consider the gain factors on their own,

and to recognize that the gain factors are the true measures of how the

strategies perform.

Ordering Beauty

First, congratulations to fellow complexity theorist and blogger Scott Aaronson for receiving the 2020 ACM Prize in Computing for "groundbreaking contributions to quantum computing". The prize is ACM's highest honor for mid-career researchers. Well deserved! 

Now back to our regularly scheduled post...

Every freshman at Cornell back in 1981 had to take two seminar courses, basically one-shot courses in an usually humanities area which required no prerequisites but lots of writing. I took my first course in philosophy. The instructor, a PhD student, at one point described his research, a philosophical argument that there is an intrinsic total ordering of beauty, say that Beethoven would always sit above the Beatles, no matter the beholder. I didn't believe him then and still don't today. A few months ago the Washington Post ran a story with the same theme entitled Maradona was great, and maybe the greatest. Can we make similar claims about artists?

Somehow we have this belief when it comes to conference submissions, that there is some perfect ordering of the submissions and a good PC can suss it out. That's not really how it works. Let's say a conference has an accept rate of 30%. Typically 10% of the submissions are strong and will be accepted by any committee. About half the submissions are just okay or worse and would be rejected. The other 40% of the submissions will be chosen seemingly randomly based on the tastes of the specific members of the program committee. Experiments in the NeurIPS and ESA conferences have bourn this out. 

Why not make the randomness explicit instead of implicit? Have the PC divide the papers into three piles, definite accepts, definite rejects and the middle. Take the third group and randomly choose which ones to accept. It will create a more interesting program. Also randomness removes biases, randomness doesn't care about gender, race and nationality or whether the authors are at senior professors at MIT or first year grad students at Southern North Dakota. 

We put far too much weight on getting accepted into a conference given the implicit randomness of a PC. If we make the randomness explicit that would devalue that weight. We would have to judge researchers on the quality of their research instead of their luck in conferences.

Given that conferences, especially the virtual ones, have no real limits on the number of papers and talks, you might say why not just accept all the papers in the middle. Works for me.

Is the following reaction to getting the first COVID shot logical?

 Alice works at a charity that puts together bag and box lunches for children.

They all wear masks and they are 12 feet apart and very careful, and nobody there has gotten COVID.

Then Alice gets here first COVID shot and says:

I am not going to work for that charity until I have had my second shot and waited  4 weeks so I am immune. 

She is really scared of getting COVID NOW that  she is on the verge of being immune. 

Is that logical? She was not scared before. So does it make sense to be scared now? I see where she is coming from emotionally, but is there a logical argument for her viewpoint? I ask nonrhetorically.

bill g. 

Quantum Stories

Scott Aaronson wrote last month about the hype over quantum computing. I'd thought I'd drop a few stories.

I was once asked to review a grant proposal (outside the US) that claimed it would find a quantum algorithm for NP-hard problems. I wrote a scathing review but the grant was funded because I failed to prove that it was impossible. I replied that they should fund my research to teleport people from Chicago to Paris because they couldn't prove I couldn't do it. I never got a response.

I was at an NSF sponsored meeting on quantum computing. I suggested, as a complexity theorist, that we need to explore the limits of quantum computing. A senior researcher said we shouldn't mention that in the report or it might hurt our chances of funding the field if they think quantum computing might not be a complete success.

I went to a Microsoft Faculty Research Summit which had a big focus on quantum computing. I complained of the quantum computing hype. My friends in the field denied the hype. Later at the summit a research head said that Microsoft will solve world hunger with quantum computing.

I was meeting with a congressional staffer who had worked on the National Quantum Initiative which coincidentally was being announced that day. I said something about high risk, high reward. He looked shocked--nobody had told him before that quantum computing is a speculative technology.

Quantum computing has generated a large number of beautiful and challenging scientific questions. Thinking about quantum has helped generate classical complexity and algorithmic results. But quantum computing having a real-world impact in the near or mid-term is unlikely. Most scientists I know working directly in quantum research are honest about the limitations and challenges in quantum computing. But somehow that message is not often getting to the next layers up, the policy makers, the research managers, the university administrators, the media and the venture capitalists. 

But who knows, maybe some quantum heuristic that doesn't need much entanglement will change the world tomorrow. I can't prove it's impossible.

Do any NP-reductions use deep mathematics? Non-rhetically

BILL: Lets say we didn't know that FACTORING NPC --> NP=coNP.

then what direction would we think Factoring in P or NPC? 

STUDENT: In P. After all, Number Theory is a deep subject and I can imagine some deep Theorem in it leading to FACTORING in P. 

BILL: That cuts both ways. I can imagine some deep theorem in NT being the key to showing 

FACTORING poly-reduces to  SAT

STUDENT: 

Deep mathematics HAS been used for algorithms. Factoring algs is one example, The graph minor theorem yielding MANY problems in P is another. Can you give me an example where deep mathematics has been used for an NPC reduction?

BILL: Oh. Gee. I do not know of any. 

STUDENT: If only you had some mechanism to ask the theory community. Maybe you could call it a web log, or weblog.

BILL: If only...

QUESTIONS

1) Are there any NPC reductions that use deep math? (I realize that the phrase `deep math' is not well defined,)

2) Are there other reductions that use deep math?

3) If P NE NP then: 

For all epsilon there is no approx-alg for MAX3SAT which yields  \ge  (7/8 + epsilon)OPT

For all delta  there is no approx-alg for CLIQ which yields > n^{-delta} OPT

No approx-alg for SET COVER which yields \ge (ln n + o(1))OPT. 

All of these proofs use  the PCP machinery or something similar. My impression is that the original PCP theorem, while long, hard, and impressive, did not use deep math. I have a vague memory of some paper or PhD thesis stating that the ONLY theorem needed was that a poly of degree d over a finite field has \le d roots. 

However to get the optimal lower bounds seemed to use some deep math. But I am more ASKING than telling. 

Want to Buy a Theorem?

This is embarrassing to admit but after a few badly timed trades on GameStop options I find myself a bit tight on money. To raise some cash, I reluctantly decided to sell one of my prized possessions, one of my theorems.

For Sale: Boolean formula satisfiability cannot be solved in both logarithmic space and quasilinear time. For a more formal and slightly more general statement and a proof, see this paper.

Bidding starts at 12 BTC (about $705,000). 

The winning bid, upon verified payment, will receive:

• The ability to give the theorem the name of your choice such as your own name, your best friend's mother's name or "Teddy McTheoremface".
• A non-fungible token (NFT) attesting ownership of the theorem and the name you have chosen for it.
• Anyone citing this result will be required to note that you own it and use the name you chose above. You cannot, however, limit the use of the theorem or receive compensation for its use. 
• By virtue of owning this theorem you will a Fortnow number of zero. This immediately gives you an Erdős number of 2. If you have previously written a paper with Paul Erdős then both of us will now have an Erdős number of 1.

Frequently Asked Questions

Q: Why this theorem?

A: The theorem is in one of my few solely authored papers and I can't afford to share the proceeds of the sale. 

Q: Doesn't Ryan Williams and others have stronger theorems?

A: The results are incomparable. Ryan gives bounds on a single algorithm with low time and space. My theorem allows different machines for the time and space bounds.

Also, to the best of my knowledge, Ryan's theorem is not for sale.

Q: Doesn't the proof of the theorem rely on other people's theorems such as Nepomnjascii? Shouldn't he get some of the value from this sale?

A: I'm not selling the proof of the theorem, just the theorem itself.

Q: If I purchase this theorem will I get to write next year's April Fools post?

A: No.

Slicing the Hypercube

Here's a neat result I heard about at virtual Dagstuhl last week, a new lower bound on the number of hyperplanes that cuts all the edges of a hypercube.

A n-dimensional hypercube has 2ⁿ vertices corresponding to the binary strings of length n. Edges are between two vertices that differ in exactly one bit, for a total of (n/2) 2ⁿ edges. Hypercubes played a large role in Hao Huang's recent proof of the sensitivity conjecture. 

A hyperplane is just a linear inequality in x₁,…,x_n the bits of the string corresponding to a vertex. An edge is cut if the inequality is true for one vertex and false for the other.

With n hyperplanes you can cut all the edges in two very different ways. 

• The hyperplanes x_i > 0 for each i. These are n orthogonal hyperplanes.
• The hyperplanes x₁+…+x_n > i for each i from 0 to n-1. These are n parallel hyperplanes.

Do you need n hyperplanes to cut all the edges? Mike Paterson found 5 explicit hyperplanes that cuts all the edges of a 6-dimensional hypercube (see survey by Mike Saks). Scaling that up means you need 5n/6 hyperplanes to cut an n-dimensional hypercube. That remains the best known upper bound.

For the lower bound, in 1971 Patrick O'Neil showed that any hyperplane can cut at most O(n^0.5) fraction of the edges (sharp by the hyperplane x₁+…+x_n > n/2). Thus you need at least O(n^0.5) hyperplanes which for 50 years was the best known bound.

Gil Yehuda and Amir Yehudayoff have given a new lower bound of O(n^0.57). The paper gives a O(n^0.51) bound but Yehudayoff said in a talk last week the bound has been improved.

Yehudayoff didn't go into much details in his 20 minute talk but the proofs uses geometry, probability and antichains. 

The result has some applications to complexity, namely you need at least n^1.57 wires in a depth-two threshold circuit for parity. But the main fun is the question itself, that we finally made progress and there is still a long way to go.

The key to my Taylor series problem: Buddy can you spare a penny, nickel, dime, or quarter

 In my last blog post I posed a question about finding the coeff of x^100 in a particular Taylor Series. The question and answer are given  here:

The key to the problem was to recognize that it was asking how many ways you can make change of a dollar using pennies, nickels, dimes, and quarters. This can be done by hand (its 242). 

1) Someone who I gave the problem to solved it by available software, but when he saw the answer was 242 he realized how to do it via making change.

2) How hard would this problem be to do completely by hand- say as a Putnam Problem? Its hard for me to say since I started with the trick and then found the problem. 

3) Is this a well known trick? 

A Taylor Series Problem

 I post a problem today and its solution on Thursday.

Comments are fine, though if you don't want to get a hint, don't read them. 

Find the coefficient of x¹⁰⁰ in the Taylor series for the rational function which has 

numerator 1 

and denominator

x⁴¹ - x⁴⁰ - x³⁶ + x³⁵ -x³¹ + x³⁰ + x²⁶ - x²⁵- x¹⁶ + x¹⁵ + x¹¹ - x¹⁰ + x⁶ - x⁵ -x+1

For better readability see my pdf file with the problem in it here

Is there a clever way to do the problem?  If the way to do it was to actually do the Taylor series then 

1) I wouldn't post it

2) I probably could not do it (or it would take too long to bother)  though maybe there are freely available programs that could. 

So yes, there is a clever solution. At least I think it's clever. 

I read the news today oh boy

I'm absolutely elated that Lázló Lovász and Avi Wigderson won the Abel Prize. More from the New York Times, Quanta Magazine and Gil Kalai. Another example of how complexity theory and theoretical computer science has reached the upper echelons of mathematics.

I'm horrified of the spa shootings in Atlanta. I've driven by those spas many times when I lived there. 

I'm saddened by the closing of Mills College in Oakland, California. I lived in a dorm Berkeley rented at Mills College during my crazy year there.

I've got mixed emotions about the death of James Levine. What he did musically with the Metropolitan opera and orchestra was nothing short of incredible. What he did to the young people he abused is inexcusable. I remember watching a Met taped videocast of Die Walküre with my daughter, enjoying the production but telling her "Do you realize we are watching an opera composed by an anti-Semite and conducted by a child molester?"  Can you separate art from artist?

I didn't understand The Art Market before the NFT sale, and I understand it less now

 (This post is a sequel to my post from Feb 13, 2007 which you can find here. While the gap from 2007 until 2021 seems large, its not as long as the gap between Knuth Vol 3 and Knuth Vol 4, nor as long as the gap between Stan Freberg Vinyl Record History of America Part I and his CD History of America Part 2, both novelty records, and quite good.)

My 2017 post was about people posting a clip on youtube and calling it `rare footage of...' 

My point was: How can something be rare if its on you tube?

I also pondered: if someone can make a REALLY REALLY GOOD copy of the Mona Lisa, that is at talent that should be respected and such a person should be able to sell it for about the same price as the original (not sure if I want the copy to be worth A LOT or the original to be worth LESS than it is.)

IF Art is to-look-at-cause-its-pretty then it should not matter who the artist is. 

IF Art is an investment then that could be risky since it does not have intrinsic value. 

IF Art is neither to-look-at or an investment then... What is it? We'll see below that one answer might be Bragging Rights. 

This is NOT a RANT, this is an admitance of my lack of understanding. (Spell check things admitance is not a word. Maybe its admitence. No, Hmmm.) 

And now there is a new wrinkle in all of this: 

69 million for a NFT (Non-Fungable Token) of an art work:story here

1) What is the buyer getting? Bragging rights?

2) Can anyone SEE the art but doesn't OWN it? I don't know..

3) If someone hacked in and got a perfect copy of the art and posted it on a website, would that be theft? Nothing was taken. 

4) In the normal art world does it happen that prices go DOWN because people wake up and say WHAT? Why did I EVER think that White on White was worth 10 million dollars? 

5) Might this happen here also? 

6) Is My Feb 13 blog, which is in a diff format (or I didn't know how to use the blogger interface then) going to one day be worth something?

Cake Cutting in Overtime

There's a new proposal out of Baltimore for a new way to handle overtime in National Football League games. This post is about American Football, soccer has its own overtime issues we won't talk about here.

Instead of randomly choosing who gets the ball, which gives an advantage to the team on offense, the Raven's coach John Harbaugh suggests a "spot and choose" rule based on cake cutting. One team picks a starting position and the other team decides whether to be on offense or defense.

Sounds intriguing but there's a problem. In cake cutting, if you cut off a crumb, everyone would definitely choose the rest of the cake. But in football suppose the spotting team chose the offensive's team one-yard line (99 yards needed for a touchdown). For spot and choose to work the one-yard line would have to be an obvious choice for defense. But many teams might still choose starting at the one-year line on offense. There's a chance you can march down the field and if not you can always punt it away. So the team that gets to choose whether to be on offense could get an advantage no matter what the spotting team did.

I still like the idea of "spot and choose". Maybe you let the first team choose not only the yard line but what down to start. Because no one would start at their one yard line at 4th down.

Are there variations for spot and choose in other sports? I like using game theory to figure out how to play actual games. 

When do I need to warn about Spoilers?

In a recent post here I mentioned in passing a plot point from the last season of The Big Bang Theory. Note that the last season was in 2019.  WARNING- do not read that post if you are watching The Big Bang Theory and do not want a plot point revealed. 

Someone who perhaps thinks Lance and I are the same person (are we? See here) left Lance a tweet complaining about the spoiler. At least I think they are complaining. The tweet is in Spanish and its here.

Either

1) Some country is two years behind America on showing The Big Bang Theory. 

2) The person who tweeted has them on DVR (or something like that) and is watching them a few years after they air (I watched Firefly on a DVD I borrowed from a friend 10 years after it went off he air. Ask your grandparents what a DVD used to be.) 

3) They are kidding us and making fun of the notion of spoilers.

This raises the question: When is it okay to post spoilers without warning? A few random thoughts:

1) ``Don't tell me who won the superb owl! I have it on tape and want to watch it without knowing who won!''  This always seemed odd to me.  Routing for events to happen that have already happened seems weird to me. When I was 10 years old I was in New York listening to a Knicks-Celtics Basketball game on the radio and during halftime I accidentally found a Boston radio station that had the game 30 minutes later (I did not realize that the channel I was on originally was 30 minutes behind). So I heard how the game ended, then switched back listening to a game knowing how it would end. I didn't route for my team (the Knicks, who lost) but it just felt very weird listening to it. If I had thought of it I might have noticed how the different broadcasts differ and got a paper out of the data, but as a 10 year old I was not thinking about how to pad my resume quite yet. 

2) I like seeing a mystery twice- first time I don't know who did it, second time I do but can look for clues I missed the first time.

3) I would have thought 2 years after a show is off the air its fine to spoil. But... maybe not.

4) It also matters how important the plot point is. I didn't think the plot point I revealed was that important. 

5) Many TV shows are predictable so I am not sure what `spoiler' even means. If I said to Darling:

 The bad guy is an unimportant character we meet in the first 10 minutes.

that does not show I've seen it before. It shows that I am a master of TV-logic.

6) With Arc TV shows this is more of a problem. While it was possible to spoil an episode (Captain Kir will survive but Ensign Red Shirt will bite the dust) it was impossible to spoil a long-term arc. TV has gotten to complicated. And I say that without having watched Game of Thrones. 

Using number-of-PhD's as a measure of smartness is stupid.

In Thor:Ragnorak Bruce Banner mentions that he has 7 PhDs. Gee, I wonder how he managed to slip that into a conversation casually.  Later in the movie:

Bruce: I don't know how to fly one of those (it an Alien Spacecraft)

Thor: You're a scientist. Use one of your PhD's 

Bruce: None of them are for flying alien spaceships.

On the episode Double Date of Archer (Season 11, Episode 6) Gabrielle notes that she has 2 PhD's whereas Lana only has 1 PhD. 

I am sure there are other examples of a work of fiction using number of PhDs as a way to say that someone is smart. In reality the number of PhD's one has is... not really a thing. 

In reality if a scientist wants to do work in another field they... do work in that field.

Godel did research in Physics in the 1950's, but it would have been silly to go back and get a PhD in it.

Fortnow did research in Economics, but it would have been silly to go back and get a PhD in it. 

Amy Farrah Fowler worked in neurobiology and then in Physics. Her Nobel prize in physics (with Sheldon Cooper) is impressive, getting a PhD in Physics would be ... odd. Imagine someone looking at here resume: She has a Nobel Prize in Physics, but does she have a PhD? Did she pass her qualifying exams?  This is the flip side of what I mentioned in a prior post about PhD's: Not only does Dr. Doom want to take over the world, but his PhD is from The University of Latveria, which is not accredited. 

There are other examples.

There ARE some people who get two PhDs for reasons of job market or other such things. That's absolutely fine of course. However, I wonder if in the real world they brag about it. I doubt it. 

Is there anyone who has 3 PhDs? I would assume yes, but again, I wonder if they brag about it. Or should. 

WHY do TV and movies use number-of-PhDs as a sign of genius? I do not know- especially since there are BETTER ways say someone is a genius in a way the audience can understand:  number-of-Nobel-prizes, number-of-times-mentioned-in-complexityblog,  number of Dundie's (see here), etc. 

Complexity is the Enemy of Speed

The title of this post came from an opinion piece in the Wall Street Journal yesterday on vaccine distribution. Many attempts to get the vaccines to the right groups first have slowed down distribution and sometime even caused vaccines to go to waste. Rules to help spread vaccines across minority groups often backfire. Often when some rules lead to inequity, we try to fix it with more rules when we need less much less. Attempts to distribute vaccines to multiple medical and pharmacy sites have made it difficult to get appointments even if you are eligible.

Randomness is the simplest way to fairness. The movie Contagion got it right, just choose birthdays by picking balls from a bin to distribute the vaccine. Then people can just show up at a few chosen sites with proof of birthday. No need to sign up.

You could argue to add back conditions like age, medical conditions, jobs but that just leads you down the same problematic path. The fastest way to get past this pandemic is to get vaccines into arms. Trust the randomness.

Good Names and Bad Names of Game Shows and theorems

 In my post on Alex Trebek, see here, I noted that Jeopardy! is not a good name for the game show since it doesn't tell you much about the show. Perhaps Answers and Questions is a better name.

The following game shows have names that tell you something about the game and hence have better names: 

Wheel of Fortune, The Price is Right, Lets make a Deal, Beautiful women have suitcases full of money (the original name for Deal-No Deal), Win Ben Stein's Money, Beat the Geeks. 

In Math we often name a concept  after a person. While this may be a good way to honor someone, the name does not tell us much about the concept and it leads to statements like:

A Calabi-Yau manifold is a compact complex Kahler manifold with a trivial first Chern class. 

A Kahler manifold is a Hermitian manifold for which the Hermitian form is closed.

A Hermitian manifold is the complex analog of the Riemann manifold. 

(These examples are from an article I will point to later---I do not understand any of these terms, though I once knew what a Riemann manifold was. I heard the term Kahler Manifold in the song Bohemian Gravity.  It's at about the 4 minute 30 second place.) 

While I am amused by the name Victoria Delfino Problems (probably the only realtor who has problems in math named after her, see my post here) it's not a descriptive way to name open problems in descriptive set theory. 

Sometimes  a name becomes SO connected to a concept that it IS descriptive, e.g.:

The first proof of VDW's theorem yields ACKERMAN-LIKE bounds. 

but you cannot count on that happening AND it is only descriptive to people already somewhat in the field. 

What to do? This article makes the  ballian point that we should   STOP DOING THIS and that the person who first proves the theorem should name it in a way that tells you something about the concept. I would agree. But this can still be hard to really do.

In my book on Muffin Mathematics (see here) I have a sequence of methods called

Floor Ceiling, Half, Mid, Interval, Easy-Buddy-Match, Hard-Buddy-Match, Gap, Train. 

There was one more method that I didn't quite name, but I used the phrase `Scott Muffin Problem' to honors Scott Huddleton who came up with the method, in my description of it. 

All but the last concept were given ballian names.  Even so, you would need to read the book to see why the names make sense. Still, that would be easier than trying to figure out what a Calabi-Yau manifold is. 

Two examples of Journalists being... Wrong. One BIG one small

 Journalists sometimes get things wrong.

This is not news, but it is interesting when you KNOW they are wrong. 

1) Scott Aaronson has a GREAT example regarding an IMPORTANT story. I recommend you to read his blog post here. Most of the comments are good also, though they go off on some tangents (e.g., is the Universal Basic Income a progressive idea?)

2) I have my own example. It is far less important than the one Scott discusses; however, inspired by Scott, I will discuss it. My example also involves Scott, but that's a coincidence. 

Quanta Magazine emailed me that they wanted to talk to me about an upcoming article on The Busy Beaver Problem. Why me? Because Scott's (same Scott as above!) survey/open problems column appeared in the SIGACT News Open Problem Column that I edit. 

This sounded fine (Spoiler Alert: It was fine, the errors they made were odd, not harmful).

Here is the Quanta Article (though I do not know if it is behind paywalls- I can never tell if I am getting access because I have a UMCP account of or anyone can have access or if I am breaking copyright laws by posting the link):    here

Here is Scotts article: here

The interviewer asked me 

a) Why did I ask Scott to write the article?

ANSWER: He had a blog post on it, and I was skeptical of why these numbers are interesting, so I asked a question in the comments. He gave a great answer, so I asked him if he wanted to write a column for my open problems column. Actually I asked him if either he or perhaps a grad student would do it- I assumed he would be too busy since his `day job' is quantum  computing. However, much to my surprise and delight he said YES he would do it.

b) Is the Busy Beaver Function important?

ANSWER: In my opinion the actual numbers are not that important but its really neat that (a) we know some of them, and (b) they are far smaller than I would have thought. Also these numbers are interesting for the following reason:  there is some  n so that proving 

BB(n)=whatever it equals

 is Ind of Peano Arithmetic. When I hear that I think the number must be really large. Its not. Its 27. NEAT! And stronger theories are related to bigger numbers. This is a way to order theories. For  ZF they have something in the 700's- MUCH SMALLER than I would have thought. Scott and others can even relate BB to open problems in Math! 

There were some other questions also, but I don't recall what they were. 

SO when the article came they mentioned me once, and its... not quite wrong but odd:

William Gasarch, a computer science professor at the University of Maryland College Park,

said he's less intrigued by the prospect of pinning down the Busy Beaver numbers than by 

``the general concept that its actually uncomputable.'' He and other mathematicians are mainly interested in using the yardstick for gauging the difficulty of important open problems in mathematics--or for figuring out what is mathematically knowable at all. 

The oddest thing about the paragraph is they do not mention my connection to Scott and the article he wrote! I reread the article looking for something like `Scotts article appeared in the SIGACT News Open Problems column edited by William Gasarch' Nothing of that sort appears. 

Without that its not clear why they are soliciting my opinion. My colleague Clyde says this is GOOD:  people will ASSUME I am some sort of expert. Am I an expert? I proofread Scott's paper so... there is that...

Also I come off as more down on BB than I really am. 

Did I claim that Mathematicians are more interested in using it as a yardstick. Actually I may have said something like that. I don't know if its true. That's my bad- I should have said that I am interested in that.

After the article came out I asked the interviewer why my role was not in the article. He said it was cut by the editor. 

NOW- NONE of this is important, but even on small and easily correctable things, they get it wrong. So imagine what happens on hard issues that are harder to get right. 

MISC: One comment on Scott's blog was about the Gell-Mann amnesia effect, see this article on it:

here

The Victoria Delfino Problems: an example of math problems named after a non-mathematician

 If you Google Victoria Delfino you will find that she is a real estate agent in LA (well, one of the Victoria Delfino's you find is such).  After this blog is posted you may well get this post on the first Google page. 

If you Google Victoria Delfino Problems you will find a paper:

The fourteen Victoria Delfino Problems and their Status in the year 2015

(ADDED LATER: a comment pointed me to an updated version, so  you can see that- I got to a pay wall.) 

How did a real estate agent get honored by having 14 problems in descriptive set theory named after her?

Possibilities before I tell you which one.

1) Real estate is her day job. Her hobby is Descriptive Set Theory. Recall that Fermat was a lawyer (or something like that- see his Wikipedia page) so perhaps she is similar. Doubtful- I think math is too hard for that now.  Or at least descriptive  set theory is too hard for that now. 

2) She just happened to remark one day, Gee, I wonder if

 ZFC + SEP(Sigma_3^1) + #   implies DET(Delta_2^1). 

Its just the kind of thing someone might just say. That was problem 4 of the 14. 

3) There are two Victoria Delfino's- one is a realtor, one is a mathematician. While plausible, that would not be worth blogging about. 

4) And now the truth: Victoria was the realtor who helped Moschovakis (a descriptive set theorist who I will henceforth describe as M) buy his house. When Tony Martin (another Desc. Set Theorist) moved to UCLA, M referred him to Victoria and she did indeed help Tony find a house. Victoria gave M a large commission which he tried to turn down. She did not want it returned, so M used the money to fund five problems. Later problems were added, but for no money. The article The Fourteen... linked to above has the full story. It also has the curious line: 

Contrary to popular belief, no monetary prize is attached to further problems. 

I didn't think any of this was so well known as to have popular believes. 

ANYWAY, this is an example of a math problem named after a non-math person. Are there others? Will the name stick? Probably not- already 12 of the 14 are solved. I have noted in a prior blog (here) once a conjecture gets proven, the one who made the conjecture gets forgotten. Or in this case the person who the conjectures is named after. 

So are there other open problems in math named after non-math people? How about Theorems?

Near Misses: 

Pythagoras: Not clear what he had to do with the theorem that bears his name. 

L'hopital's Rule: the story could be a blog in itself, and in fact it is! Not mind, but someone else: here. However L'hopital was a mathematician. 

Sheldon's conjecture (see here) was named after a FICTIONAL physicist. Note that Sheldon inspired the conjecture but did not make it. It has been solved. 

The Governor's  Theorem (see here) was named because Jeb Bush was asked for the angles of a 3-4-5 right triangle (not a fair question). 

The Monty Hall Paradox.

SO- are there Open Problems, Theorems, Lemmas, any math concepts, named after non-math people? I really mean non-STEM people. If a Physicist or an Engineer or a Chemist or Biologist or...  has their name on something, that would not really be what I want.

(ADDED LATER - someone emailed me two oddly-named math things:

Belphegor's prime, see here

Morrie's law- odd since Morrie is the FIRST name of who the name is honoring, see here 

)

Are there any other open problems in descriptive set theory  named after realtors?

A Blood Donation Puzzle

In the US you can donate whole blood every eight weeks. Suppose Elvira does exactly that. Will she hit every date of the year? For example, if Elvira gave blood today, will she in some future year give blood on the 4th of July? Can we figure it out without having to rely on a computer simulation or even a calculator?

Let's make the assumptions that the blood center is open every day and that Elvira gives blood exactly every 56 days for eternity. 

A year has 365 days which is relatively prime to 56=2³*7 since 365 mod 2 =1 and 365 mod 7 = 1. By the Chinese remainder theorem her next 365 blood donations will be on 365 distinct dates. If Elvira started giving blood at age 17, she will have hit every date at age 73.

That was easy but wrong. We have to account for those pesky leap years.

In a four year span, there will be one leap day. The total days in four years (using modular arithmetic) will still be odd and 5 mod 7, so still relatively prime to 56. So Elvira will donate on every day on the calendar exactly four times, except February 29th which she will hit once, over a period 56*4=224 years. 

Alas not quite. Years ending 00 are not leap years, unless the year is divisible by 400. 2000 was a leap year but 2100 won't be. Any stretch of 224 years will hit at least one of those 00 non-leap years.

In 400 years, there will be 97 leap years. Since a regular year is 1 mod 7 days and a leap year is 2 mod 7 days, 400 years will be 497 mod 7 days. Since 497=71*7, 400 years has a multiple of 7 days. Every 400 years we have exactly the same calendar. February 3, 2421 is also a Wednesday.

The cycle of blood donations will repeat every 3200 years, the number of years in the least common multiple of 56 and the odd multiple of seven number of days in 400 years. But we can no longer directly apply the Chinese remainder theorem and argue that every day of the year will be hit. In those 3200 years Elvira will have over 20,000 blood donations. If the dates were chosen randomly the expected number to hit all dates would be 2372 by coupon collector. So one would expect Elvira would hit every day, but that's not a proof.

So I had to dust off my Python skills and do the computer simulation after all. No matter what day Elvira starts donating she will eventually hit every date of the year. If Elvira starts donating today, she would give blood on the 4th of July for the first time in 2035 and hit all dates on January 8, 2087 after 431 donations. The longest sequence is 3235 donations starting April 25, 2140 and hitting all dates on February 29, 2636.

Grading policies during Covid-No easy answers

 Because of COVID  (my spellecheck says covid and Covid are not words, but COVID is) various schools have done various things to make school less traumatic. Students already have problems, either getting COVID or having their friends got family get it (I've had four relatives get it, and one died) . Some do not adjust to learning online.  Some do not have good computer connection to learn on line. So what is a good policy? Here are some things I have either seen schools do or heard that they might do.

1) Be more generous with Tuition-Refunds if a student has to withdraw. 

2) Be more generous with Housing-Refunds if a students comes to campus thinking it will be courses on campus and there are no courses on campus. Or if a student has to withdraw. 

3) Make the deadlines for dropping-without-a-W, or taking-it-pass-fail, later in the semester. 

4) Tell the teachers to `just teach them the bare min they need for the next course.'

5) Allow students to take courses P/F in their major and still allow them to count, so a student might get a D in Discrete Math and be able to go on in the major. 

6) How far to extend deadlines? How is this: extend deadline to make it P/F until the last day of classes (but before the final) and then after the final is given, the school changes its mind and says - OH, you can change to P/F now if you want to.

7) Allow either an absolute number (say 7) or a fraction (say 1/3) of the courses to be changed to P/F by the last day of class.

8) Combine 6 or 7 with saying NO- a D is an F for a P/F course. Perhaps only if its in the major, but that maybe hard to work out. since majors can change. Some schools do A-B-C-NO CREDIT, where the NC grade does not go into the GPA.

9) Give standard letter grades and tell the students to tough it out. Recall the following inspirational quotes

When the going gets tough, the tough go shopping

When the going gets tough, the tough take a nap

If at first you don't succeed, quit. Why make a damn fool of yourself. 

If at first you don't succeed, then skydiving is not for you. 

10) Decide later in the term what to do depending on who yells the loudest. 

11) Any combination of the above that makes sense, and even some that don't. 

On the one hand, there are students who are going through very hard times because of COVID and should be given a break. On the other hand, we want to give people a good education and give grades that are meaningful (the logic of how to give grades in normal times is another issue for another blog post). 

What is your school doing? Is it working? What does it mean to be working?

The problems I am talking about are first-world problems or even champagne-problems. I know there are people who have far worse problems then getting a bad grade or dropping courses.

PhDs and Green Cards

Joe Biden's immigration policy has some interesting policies for PhDs. 

Biden will exempt from any cap recent graduates of PhD programs in STEM fields in the U.S. who are poised to make some of the most important contributions to the world economy. Biden believes that foreign graduates of a U.S. doctoral program should be given a green card with their degree and that losing these highly trained workers to foreign economies is a disservice to our own economic competitiveness. 

Biden will submit an immigration plan to congress soon but it is not clear if the above will be in the bill, whether it will survive negotiations or even whether an immigration bill will be passed at all. 

Nevertheless I worry about the unintended consequences of this policy. It will encourage students to apply and come for a PhD who have no interest in research, but want the green card. It gives too much power to professors who may abuse their students who need the PhD. Conversely it will pressure professors and thesis committees to grant PhDs because there would be a big difference between graduating with a PhD and leaving early with a Masters. By making the PhD so valuable, we may devalue it.

The solution is to give green cards to Masters students as well. We shouldn't limit talented researchers and developers who can help the United States keep its technology edge. They don't take jobs away from Americans, but instead help create new ones.

Answers to the Prez Quiz, and some interesting history

I posted a presidential quiz (that is, a quiz about presidents, not a quiz that is so majestic it can be called presidential) on Thursday Jan 21.

Here is the link to the post: The Post

Here is the link to the quiz:The Prez Quiz

 I now post the answers here: The Prez Quiz Answers

Looking back at the quiz and some prior ones I realize that some questions are TRIVIA while others are not- that is- they tell us something interesting beyond the answer. Some thoughts on that: 

a) Asking who the oldest prez ever has to be better defined. But however you define it, the answer is Joltin Joe. I will ask it as Age-when-sworn-in, which for Joltin Joe is 78. Second is Trump who was 70 when sworn in.  INFO: Presidents seem to be getting older. Is this a trend or will 2024 and 2028 feature younger folks? I note that both VP's are in their 50's. 

b) Presidential first via xkcd; here

c) Between Nixon-Kennedy 1960 and Bush-Kerry 2004 EVERY election had one of he candidates being either a former or current Prez or Vice Prez.  (This was a question on the 2008 quiz, but since the streak is broken it is no longer as interesting.) INFO: It was hard to break into the prez business unless you were already somewhat known. Having said that, note that some of the non-VPs and non-Prezs DID win. 

d) Between 1948 and 2008 at least one of the candidates for Prez had served in the Military. How did they do? In many cases both served so I am not going to to a chart of how well the Military ones did. (This was a question on the 2012 quiz, but since the streak is broken it is no longer interesting.) INFO: At one time a lot of people were in the military. At one time a lot of people knew someone in the military. Now it seems like the military is rather far removed from civilians. 

e) Kennedy was our first Catholic Prez in 1960. Biden is our second one in 2020. My impression is that it was an issue for the Kennedy Campaign but not for the Biden campaign. Romney being a Mormon seems to have not been an issue in the General election, but some of the other candidates brought it up in the primaries. INFO: Religious bigotry within different denominations of Christianity is declining. 

f) Reagan was our first divorced Prez, in 1980. Trump was our second in 2016. It don't think it was  an issue for either one. INFO: We need to couple this with Rockefellers divorce being a problem for him getting the nomination in 1964. Peoples attitude on divorce has changed A LOT. 

Contrast: Knowing which president had a fictional street gang named after him  does not tell us anything about History .I will remove that question when I do the quiz in 2025. 

Alan Selman (1941-2021)

From a 1994 Dagstuhl Workshop. Selman is the one wearing a cap.

Alan Selman, one of the early leaders in structural complexity and the co-founder and first chair of what is now the Computational Complexity Conference, passed away this morning. He was one of the true greats in our field both for his research and his service.

Selman was a good colleague and a friend. I hosted his sabbatical at the University of Chicago in 1998 which produced a paper with Selman's student and my later postdoc Pavan Aduri. In Spring of 1999 Selman ran a NSF workshop in Chicago on Challenges for Theory of Computing.

In a desire to create a community of complexity theorists and foster publications of their results, Selman led the efforts to establish the Structure in Complexity Theory conference, first held in 1986 in Berkeley, California. As a student in Berkeley I attended that meeting and the next twenty-five. The conference would join the IEEE in 1987, in 1996 renamed the IEEE Conference and Computational Complexity and in 2015 drop from the IEEE to become the Computational Complexity Conference, still going strong. Alan served as the first conference chair and as PC co-chair of the first meeting. 

For all his service for the field, Selman received the ACM SIGACT Distinguished Service Award in 2002.

Selman joined University at Buffalo in 1990 to serve six years as department chair and then as a professor until he retired in 2014. Lane Hemaspaandra wrote a highly recommended appreciation for Selman and his research in honor of his retirement where he mentions some of Selman's research programs, including his seminal work on P-Selective sets, non-deterministic functions and his work with Joachim Grollman on one-way functions. Selman came down hard on anyone who got the wrong number of l's in Grollman, so we would often joking cite it as Grollman and Sellman or just Grollman et Al.

In my favorite Selman paper, in work with Hemaspaandra, Ashish Naik and Mitsu Ogihara, looks at witness reduction. If there is a set A in P such that for all satisfiable formula φ there is a unique satisfying assignment a of φ such that (φ,a) is in A then the polynomial-time hierarchy collapses. The more general question of whether NP=UP implies the polynomial-time hierarchy collapses remains open.

Also Selman's paper with Steve Homer giving an oracle where Σ₂^P-complete sets were isomorphic was instrumental to my own work on the isomorphism conjecture.

Bill wanted me to mention the Selman paper that most influenced him. Selman and Theodore Baker and gave the first oracle separating the second level of the polynomial-time hierarchy in 1979. It would take Yao and Håstad over five more years to get the whole hierarchy infinite relative to an oracle. The Baker-Selman proof could easily extend to show that AM games did not sit in Σ₂^P, a bit surprising as MA is in Σ₂^P.

In addition to his research, Selman wrote an introductory theory textbook with Homer as well as editor and co-editor of two editions of Complexity Theory Retrospective.

It would be no exaggeration to say the field of computational complexity and my own research within it would have been much different without Alan.

Alan drove a car with a "P NE NP" license plate. One day someone came up to him in a parking lot and said "Yes, but can you prove it?" Possibly the one thing in complexity he couldn't do.

Presidential Quiz: Three ways to take it

 Every four years around the time of the inaugural I post a presidential Quiz! I do that today, and I will post the answers on Monday.  I am tempted to joke that I am posting it AFTER Joltin Joe gets sworn in just in case something happens; however, I always posted it after the prez is sworn in. 

The quiz is here: quiz

 GRADING CRITERIA

33 questions, 3 points each, and 1 free point.

If the answer is a list of L elts and you get x correct, you get x/L points. If any are wrong then 0 points.

You can take the quiz one of three ways.

1) Take it WITHOUT using the web and see how many you can get right. Take 3 hours.

WARNING- its a hard quiz so this may not be fun. 

2) Take it and use the web and try to do it fast. Stop when you want. Your Score is as follows:

If  R is the number if points and  T be how many minutes you took,  your score is 

 (180R/T)+ 1.

If  you get all 33 right in 60 minutes then you get a 100. You could get more than 100 if you do it faster.

3) The answer key has all kinds of other information in it and is fun to read. So do not take the quiz and enjoy  reading the answers on Monday. That's what I did. 

I was curious if Joltin Joe would be sworn in by Joseph or Joe. I was routing for Joe so he would be the second president sworn in by his nickname (Jimmy Carter was the first--his real name is James). When I am sworn in as president I will use Bill and hence join the nickname-club. Lance does not have a nickname, so my veep will be sworn in by his actual name. Would we win? Is  America ready for a 2-Phd ticket? 

The Ethics Board

Back in 2014 I recommended a TCS ethics board mainly to deal with plagiarism and who should get credit for a result. It went the way of most suggestions I make in my blog, a quick road to nowhere. Bill asked "Does any other branch of CS have an ethics board? How have they worked?"

The NeurIPS conference created a such a board for the reviewing process, though with a broader mission.

We appointed an ethics advisor and invited a pool of 22 ethics reviewers (listed here) with expertise in fields such as AI policy, fairness and transparency, and ethics and machine learning. Reviewers could flag papers for ethical concerns, such as submissions with undue risk of harm or methods that might increase unfair bias through improper use of data, etc. Papers that received strong technical reviews yet were flagged for ethical reasons were assessed by the pool of ethics reviewers.

Thirteen papers met these criteria and received ethics reviews. Only four papers were rejected because of ethical considerations, after a thorough assessment that included the original technical reviewers, the area chair, the senior area chair and also the program chairs. Seven papers flagged for ethical concerns were conditionally accepted, meaning that the final decision is pending the assessment of the area chair once the camera ready version is submitted. Some of these papers require a thorough revision of the broader impact section to include a clearer discussion of potential risks and mitigations, and others require changes to the submission such as the removal of problematic datasets. Overall, we believe that the ethics review was a successful and important addition to the review process. Though only a small fraction of papers received detailed ethical assessments, the issues they presented were important and complex and deserved the extended consideration. In addition, we were very happy with the high quality of the assessments offered by the ethics reviewers, and the area chairs and senior area chairs also appreciated the additional feedback.

Without seeing the process I cannot say what criteria were used to reject the four papers. There could be legitimate reasons if the authors did violate professional ethics or even inadvertently based their results on biased data sets. But there's a fine line to rejecting papers because you don't like the outcome or the questions they ask. No easy answers here.

The ACM History Committee call for proposals looks wrong

 In November I (and prob everyone in the ACM) got an email that was a call for proposal from the ACM History committee. Here is a pointer to it. 

One of the passages in it just seems wrong to me:

This fellowship program is designed to support research and/or curatorial projects related to ACM's professional and educational activities and/or to ACM's rich institutional history including its organization, SIG activities, and conferences. 

I will give  examples of history projects that are interesting but do not fit the description. I challenge you to give me a history project that is interesting that does fit the bill. 

1) Compare and contrast how NP-completeness developed in America and in the USSR. For the USSR side it would be an expansion of  Trakhtenbrot's article here. OH, no good, Leonid Levin was not a member of the ACM.

2) Measure how various CS fields have changed by looking at the topics on the CALL FOR PAPERS for a conference. This would be a MASSIVE expansion of my blog post on how CCC call for papers, changed , see here. OH, I could only do ACM conferences. STOC but not FOCS. Okay, it makes perfect sense to study only ACM conference. Oh wait. IT MAKES NO SENSE WHATSOEVER.

3) When did mathematicians begin looking at P vs NP seriously? (e.g.,  In  The Handbook of Mathematical Logic from 1977 only one article mentions P vs NP: Michael Rabin's article on Decidable theories mentioned that even decidable theories may take a long time to decide and noted the importance of the P vs NP problem). How did MATH and CS  interact early on and now? This would need  one to look at math journals. How many of those are ACM? I would guess very few. 

4) When did engineers begin looking at P vs NP seriously, or even notice it? Same issue as the last item. 

5) Looking at how Programming lang design has changed one would have to only look at conference and journals that were ACM. And for those that were ACM but then dropped ACM you might only be able to use them up to the cutoff year. 

6) Which academic studies eventually lead to practical products people can use? This is a really broad topic so one could narrow it down to just academic studies that appeared in ACM journals or conferences. Is that a good way to narrow the focus? Spoiler alert: NO.

7) I recently filled out a survey about the future of theory and if it is insular (the topic of another post surely). Most of the questions were about ``STOC-FOCS'' The people who did the survey clearly do not care that one is ACM and one is IEEE. Does anyone? I ask non-rhetorically. 

Despite the capitol letters, I am not so much mad as puzzled. So I ask non-rhetorically: 

Q1) Did the ACM really mean for people to do history in a way where you can only use ACM materials? 

Q2) If no then please rewrite the Call for Proposals. 

Q3) If yes then give examples of studies that would be interesting. 

I am not asking to be snarky, I really want to know. And I note that interesting is subjective. 

(ADDED LATER- two historians who have worked with this kind of history left comments that indicate the grant is FINE with non-ACM stuff, so Q1 above seems to have the answer NO. Given that they should rewrite the call for proposal next time they do this. ALSO- the historians left pointers to VERY INTERESTING papers they wrote.) 

My Survey on Hilbert's 10th for particular (d,n) is ready for you to help me on!

 Hilbert's 10th problem is  (in modern terminology) to find an algorithm that will, given a poly 

p(x1,...,xn) in Z[x1,...,xn], determine if it has a solution in the integers. 

This was shown undecidable by the combined efforts of Davis-Putnam-Robinson and Matiyasevich. 

I posted here about the question of: For which (d,n) is H10 undecidable when restricted to degree d and number of vars n?

I was  invited to make an article out of the blog post and what I learned from the comments  for  the Algorithms column of BEATCS. That first draft is

here

Did I miss an important reference? (Note- I had meant to get out Matiyasevich's book on H10 but have been unable to because of the Pandemic, so if you have it my have stuff I need to know.) 

Did I misspell Matiyasevich? 

Did I say something stupid?

Please leave comments or email me directly if you have suggestions. 

Incidentally I am usually the one who is asking someone else to do an open problems column, a book review, a guest blog. Nice to be the askee instead of the asker for a change of pace. 

Complexity Year in Review 2020

For the result of the year we go to all the way back to the "before times".

MIP*=RE by Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright and Henry Yuen

A follow up to last year's runner up NEEXP in MIP*, the new paper shows how to prove everything computable and beyond to a polytime verifier with two provers who cannot communicate but share entangled quantum bits. The result had a bit of a scare but fixed up this fall.

Honorary mention goes to beating the Christofides-Serdyukov approximation for traveling salesman by Anna R. Karlin, Nathan Klein and Shayan Oveis Gharan. Nathan got his start in Bill's REU program.

Of course when we think back at 2020 it won't be the theorems but the pandemic, a divisive election and racial reckoning. The word of the year for academia is "virtual", virtual teaching, virtual research collaborations, virtual talks, virtual conferences, virtual panels and virtual job visits. For the most part the the pandemic hasn't really created new trends but accelerated trends already in place. Given the far lower costs in terms of time, money and the environment, how much of "virtual" will remain post-corona?

Bill's published a new book on muffin problems, his second book in two years. I started a new College of Computing. 

We remember Lars Arge, Ed Clarke, John Conway, Richard Dudley, Stanley Eisenstat, Ron Graham, Richard Guy, Katherine Johnson, Vaughn Jones, James Randi, Arianna Rosenbluth, Joy Thomas, Robin Thomas, Alex Trebek, Boris Tsirelson and Paul Young. 

Thanks to our guest posters Gorjan Alagic, Andrew Childs, Tom Goldstein, Daniel Gottsman, Clyde Kruskal and Jon Katz, Antoine Amarilli, Thomas Colcombet, Hugo Férée and Thomas Schwentick, Steve Butler, Varsha Dani, Evangelos Georgiadis and Emily Kaplitz.

In 2021 we will likely see the end of the pandemic and most definitely see the 50th anniversary of P versus NP. Hope for a successful and healthy year for all. 

Slowest Sorting Algorithms

Radu Gigore tweeted "People are obsessed with finding the best algorithms. What about the worst?" So here's a Christmas gift that keeps giving, the slowest of sorting algorithms. 

Before you read on, try to think of the slowest sorting algorithm. No fair spinning its wheels, sleeping or doing unrelated tasks. It should always make progress towards sorting. 

Here are some examples, in particular bogosort that generates all permutations until it finds a sorted one. Takes n! time on average.

But we can do much worse. The following redicusort algorithm I got from Stuart Kurtz back in the 90's.

Generate all permutations and then sort those permutations. The sort of the original permutation will be first on the list.

The running time depends on the sorting algorithm you use after generating the permutations.

If you use bubblesort you get a (n!)² time algorithm.

If you use bogosort you get a (n!)! bound.

What if you just call redicusort recursively? The algorithm will never end. If you want to guarantee termination you need to bound the depth of the recursion. Choose something like an Ackermann function to get a sorting algorithm that always makes progress but not primitively recursive. In general you can get a sorting algorithm that takes longer than any fixed computable function.

Dr Jill Biden

(I was helped on this post by Gorjan Alagic, Andrew Childs,  Tom Goldstein, Daniel Gottsman, Clyde Kruskal,  Jon Katz. I emailed them for their thoughts on the issue and some of those thoughts are embedded in the post. ) 

The First First Lady to have a college degree was Lucy Hayes (Rutherford B Hayes was Prez 1876-1880). Nickname: Lemonade Lucy since she did not serve alcohol. 

Trivia: who was the last first lady to not have a college degree? I'll answer that one at the end of this post. 

The First First Lady to keep her day job was Abigail Fillmore who was a teacher. (Millard Fillmore was Prez in 1850-1852. He became prez  after Zachary Taylor died in office) . 

In recent times it is  uncommon for a first lady to have a day job. So much so that it was notable when Elizabeth Dole said that if her husband (Bob Dole) won in 1996 she would keep her job at the Red Cross. 

For other first lady firsts  see here.

Jill Biden is the First First Lady to have a PhD. (ADDED LATER- one of the comments pointed out that she has an Ed. D, Doctor of Education.)   She says she will keep her day job as a professor.  Four other First ladies had advanced degrees: Pat Nixon (MS in Education), Laura Bush (MS in Library Science), Hillary Clinton (Law degree), Michelle Obama (Law degree). If I missed any, let me know. 

The WSJ had an op-ed  that said Jill Biden should not call herself `Dr'.  Inspired by that, here are thoughts on the use of the word `Dr'

1) At the 1986 Structures Conference (now Complexity Conference ) Lane Hemachandra (now Lane Hemaspaandra) gave a talk. He had just gotten his PhD a few weeks ago, so he was introduced as `DOCTOR Lane Hemchandra' Gee, most of the talks were by people with PhD's but were not introduced as such.

2) Most people I know within academia do not call themselves Dr since it sounds pretentious. However, speaking in public about an issue one might want to use `Dr' to signal that you...know stuff. However, it would be odd for a PhD in (say) linguistics to claim he knows a lot about (say) politics. It has been said: an Intellectual is someone who is an expert in one field and pontificates in another field. 

3) Does the general public think of DOCTOR as Medical Doctor? Probably yes. There are some exceptions: Dr. Martin Luther King and Dr. Henry Kissinger. Also, I think it is  more common in Psychology, pharmacy, education, and counseling to call yourself `Doctor'  

4) The article also criticized her PhD (in education, about community colleges) as `useless' . If that's the reason to not call her doctor than I shudder to think what the WSJ would think of degrees in, say, set theory with an emphasis on Large Cardinals. GEE, you can't call yourself  DOCTOR since your degree is on Ramsey Cardinals. OH, now they found an application, so now you CAN call yourself DOCTOR. OH, the application is to extending the Canonical Ramsey Theory from Polish spaces  to meta- compact  cardinals, so we can't call you DOCTOR after all. Do we really want to go down this path? 

5) I ask all of the following non-rhetorically:  Did the author read her PhD thesis? Is he qualified to judge it? Did he (as one should do) look at her entire body of work to judge her? What point is he trying to make anyway? 

6) Should Dr. Who call themselves a doctor? Are they  a medical doctor? PhD? If so, in what? Is `Who' part of their name? For other TV and movie tropes about the use of the word doctor, see here.

7) I avoid saying I am a doctor since people will then ask me about the medical condition.

I avoid saying I am a computer scientist since people will then ask me how to help them with their Facebook privacy settings. 

I avoid saying I am a mathematician since people will ask me to help their daughter with her trigonometry. 

8) The answer to my trivia question: The last first lady to not have a college degree: Melania Trump. She went to college for a year and then left. The one before her was Barbara Bush who also went for a year and then left. 

ADDED LATER: Many supervillians who don't have a PhD or an MD call themselves `Doctor', see here. Why no outrage about this? Because (1) they are fictional, and (2) imagine the scenario: Not only does Dr. Doom want to take over the world, he also doesn't even have a PhD or an MD!

ADDED LATER: Where does Dr. Pepper fit into this?  

ADDED LATER: The following fictional people have many PhD's:

Reed Richards of the Fantastic Four has 18 PhD's. All are in science. He got them at 5 (maybe more)

schools: Harvard, Cal Tech, Columbia, Empire State (fictional but prob NYU), and State Univ (fictional) 

Mr. Terrific (DC Universe) has 14 PhD's. There is more detail on his than others, possibly because his genius IS his superpower (unlike Reed Richards and Tony Stark).  They are in Engineering, Physics, , Math, CS, Chemistry, Medical Science, Psychology, Law, Poly-sci, Bio, Genetics, History, Philosophy. I looked up if he is the smartest person in the DC universe. He is third to Lex Luthor and Bruce Wayne among humas, and any member of the Brainaic Species.

Ford Pines on the TV show Gravity Falls has 12 PhD's. All theoretical, he has a hard time doing practical things. All Science. He may be broader than Reed Richards since some are in biology. 

Tony Stark has 7 PhDs. I assume all in science, tending towards engineering. 

Optiland

Many of you have heard of Russell Impagliazzo's five worlds from his 1995 classic A personal view of average-case complexity In short 

• Algorithmica: P = NP or something "morally equivalent" like fast probabilistic algorithms for NP. 
• Heuristica: NP problems are hard in the worst case but easy on average.
• Pessiland: NP problems hard on average but no one-way functions exist. We can easily create hard NP problems, but not hard NP problems where we know the solution. 
• Minicrypt: One-way functions exist but we do not have public-key cryptography.
• Cryptomania: Public-key cryptography is possible, i.e. two parties can exchange secret messages over open channels.

Impagliazzo's world has an explicit "you can't have your cake and eat it too", either you can solve NP-hard problems on average, or have cryptography but not both (neither is possible). That's the mathematical world of P v NP. 

The reality is looking more and more like Optiland, where we can solve difficult NP problems and still have cryptography thanks to vast improvements in machine learning and optimization on faster computers with specialized hardware.

Back in 2004 I gave my guess of the world of P = NP

Learning becomes easy by using the principle of Occam's razor--we simply find the smallest program consistent with the data. Near perfect vision recognition, language comprehension and translation and all other learning tasks become trivial. We will also have much better predictions of weather and earthquakes and other natural phenomenon.

Today you can take your smartphone, unlock it by having the phone scan your face, and ask it a question by talking and often get a reasonable answer, or have your question translated into a different language. You get alerts on your phone for weather and earthquakes, with far better predictions than we would have though possible a dozen years ago.

We have computed the shortest traveling-salesman tour through nearly 50K UK pubs. AlphaFold can simulate protein folding with an accuracy nearly as good as what we get with real-world experiments. You can view GPT-3 as generating a form of a universal prior. Even beyond P = NP, we have self-trained computers easily besting humans in Chess, Go and Poker.

Meanwhile these techniques have done little to break cryptographic functions. Plenty of cybersecurity attacks but rarely by breaking the cryptography. 

Not all is rosy--there is still much more we could do positively if P = NP  and we are already seeing some of the negative effects of learning such as loss of privacy. Nevertheless we are heading to a de facto best of both worlds when complexity theory tells us those worlds are incompatible. 

Quarterly Th. Wksp `at' Northwestern, and thoughts inspired by it

 On the Northwestern CS Theory Group there is a set of Quarterly Theory Workshops. There is one coming up on Dec 17-18, 2020, called the Junior Theorists Workshop. Take a look and possibly go to it! Because it is virtual you do not need to plan that much ahead- though they do want you to register. 

1) I notice broadly two kinds of meetings:

Based on WHO will be there, e.g., JUNIOR theorists

Based on TOPIC: e.g., there was a meeting on ALGORITHMIC FAIRNESS.

2) These types of meetings (NY Theory day is another) are, I believe, intended to be for people that are local (more on that later). But because the meeting will be on zoom, geography is no longer an impediment for either the attendees or the speakers. 

3) Before covid there was some talk of  `Gee, flying off to STOC, FOCS, other conferences is bad for the environment, what to do about that?'. With that in mind, here is a history which might not be true but makes a point:

In an earlier era FOCS/STOC were attended by mostly Americans and ICALP was attended by mostly Europeans.  I do not think there was any policy of discrimination on admissions, but it was more like Americans just did not submit to ICALP as much, nor Europeans  to FOCS/STOC. But over time when these conferences got to be considered prestigious people would routinely submit to either one depending on timing. If your paper was done in time for Conf X deadline, that's where you submit. If it does not get in then  you edit it some, perhaps add some new results, and submit to Conf Y. 

So one solution to the air-travel-global-Warming   problem of conferences is go back to a time (which may not have ever existed) where it was just understood that you go to LOCAL conferences. Math does this, but it helps that their regional conferences are not  prestigious. But even they don't quite get it right: the joint AMS-MAA meeting alternates coasts. One year when it was in California they invited me to be a guest speaker (on the Muffin Problem). The following year it was in Baltimore. Note that I live in Maryland, so perhaps they should have waited a year. 

How to encourage people to submit locally. I DO NOT want to have a rule or a diff standard for those who don't. As such... I have no idea. 

4)  Are virtual conferences a good idea? This is a hot topic now so I won't dwell on it, just to say that there is still something about being there IN PERSON, meeting people, serendipity that makes live confs better.

However, to have it at the same time be virtual and recorded will be VERY HELPFL to those who can't afford to go for whatever reason. 

And of course there is the whole issue of if we should have prestigious conferences, which I won't get into now. Or later. That's more Lance's issue (he thinks no). 

Shuffling Around

In the fall of 1983 as a junior at Cornell I took CS 481, Introduction to the Theory of Computing, from Juris Hartmanis. Needless to say this was the course that changed my life and led me down a path that would have me teach a variation of this course myself more than twenty times.

For some reason one of the final exam questions is still stuck in my head.

Let the permutation of a language L be the set of strings x such that there is a string y in L which is a permutation of the letters in x. For example perm({01},{001})={01,10,001,010,100}. 

Are regular languages closed under permutations?

There's a short and easy answer that's not so easy to find. Just in that P v NP spirit a Hartmanis test question should have. Give it a try before you read more.

If you ask Chegg you end up with 

And for $15 you can get the answer. Back in 1983 we called that cheating.

The hint only works if there are regular languages whose permutations are not context free. Which is true in general but oddly enough the permutation of every context-free language over a two-letter alphabet is context free. The proof uses Parikh's theorem which implies that the permutation of any context-free language is the permutation of a regular language (over any alphabet size).

Some other fun permutation questions:

• Show that NP is closed under permutations.
• Show that P is closed under permutations if and only if E = NE.
• What about NL and L?

Go ahead and chegg on those. 

In 1974 Planarity was O(V) time and could do 900 node graphs in 12 seconds! Fast then...

In 1974  Hopcroft and Tarjan showed that Planarity is in polynomial time. That is an understatement- they actually have an O(V) algorithm which one can actually code up. See their paper here.

It has the curious line:

An Algol implementation of the algorithm successfully tested graphs with as many as 900 vertices in less than 12 seconds.

900 nodes in 12 second was fast then but it slow now. 

1) How would their algorithm do on todays machines? How does that compare to what Moore's law (for time) would have predicted? Can this help us determine an x such that Moore's law stopped working at year x. I've  heard various candidates for x including the notion that the death of Moore's law has been greatly exaggerated. Moore himself is still alive, at the age of 91. 

2) Are there better algorithms now? Nothing can beat O(V); however, is there an algorithm  with a better constant? 

3) Is there a modern implementation of it (or perhaps even an old implementation run on a modern machine)? If so, how fast does it run on 900 nodes? 9000 nodes? 90,000 nodes? 900,000 nodes? Not sure where to stop.

4) The people in the real world who really need to solve this problem fast: (a) do they exist, and (b) if they do exist then what do they use? 

Chess is Back

Back in 2005, I wrote a post titled Chess and Poker. Not really comparing the two but noting that Chess had lost its mojo while poker had high-stakes prime time tournaments. The inspiration was an NYT Op-Ed that started "CHESS in America is having a crisis". I suggested that computers getting better than humans may have reduced interest in the game. 

Now chess is booming again, due to all of us being stuck at home and the Netflix limited series The Queen's Gambit (highly recommended). 

The fictional show takes place in the 1960's when interest in chess in the US started to pick up due to Bobby Fischer's exploits and well before computers played a decent game. Fischer isn't mentioned in the Netflix series, the main character Beth Harmon sort of plays his role. The games themselves, created by Gary Kasparov and others, are even a joy to watch. Check out this analysis of the final game (spoiler warning). 

The New York Times started a chess column in 1962 and ran its last column in 2014, though that might be saying more about the state of newspapers than the state of chess.

What about the computers? They have just gotten so good and with AlphaZero mastering the game with just machine learning on top of the rules of chess, it's not even fun to watch computer versus computer anymore. Now we're back to watching humans and getting back into the games ourselves.

Computers have opened the door to cheating. Complexity theorist Ken Regan has a side gig reviewing games to determine if a player punching above their weight secretly used a computer algorithm. 

Microsoft just announced chess programs that play as a human at various levels of strength. I suppose someone could use a program like this to cheat in a way that even Ken couldn't detect. But mostly it would be like Googling in pub trivia--just takes the fun out of the game.