Alternate Histories

Imagine if we had a machine that let us change some earlier moment in history and see what developed. We wouldn't actually change history--that would lead to paradoxes and other disasters, but we could see what would develop. Suppose Archduke Ferdinand was never assassinated. How would that have changed 20th century history and beyond?

The same could be said for an academic field. Science flows like a story, with building results from other results, "standing on the shoulders of giants" so to say. But not all theorems are dependent on earlier ones and suppose that things happened in a different order. How would that have changed our field?

Points to ponder:

Suppose Gauss was alive today instead of two centuries ago. Would he still be as famous? Would there be a big hole in mathematics that would have been left for the current Gauss to solve?

Suppose Fermat's last theorem was still a conjecture. Would there be more budding mathematicians inspired by the wonders of  that  famous open problem like my generation was? Would the Clay Mathematics Institute still have produced a list of those millennial problems, including P v NP?

Suppose we knew Primes in P back in 1975. Would randomized algorithms and subsequent derandomization techniques have happened without its prime example? Same for Undirected connectivity in log space. These both are small cheats as the AKS and Reingold proofs are at their core derandomization arguments and may not have happened if we didn't think about randomized algorithms.

What if someone settled P vs NP right after Cook? Would it have stopped most of the research in computational complexity theory? Would it depend on whether P = NP was answered positively or negatively?

What if you were never born? Ultimately that would be the only true measure of your influence in the world. What if your research somehow prevented other great theorems from happening? Would you even want to see the results of that experiment?

This university does not discriminate based on....

I recently came across the following (I delete the name of the school)
and also add my own comments in caps as they relate to UMCP hiring
of professors.


X-University, located in YZ, in hiring professors
does not discriminate on the basis of

race

color . COULD AN HBCU DISCRIMINATE AGAINST WHITE PROFESSORS?

religious HAS NEVER COME UP. COULD A RELIGIOUS SCHOOL DISCRIMINATE ON THIS BASIS?

creed  I LOOKED UP HOW IT DIFFERS FROM RELIGIONS- CREED COULD BE ANY SET OF BELIEFS. WHAT IF ONE OF CANDIDATES BELIEVES ARE REPREHENSIBLE BUT WOULD NOT INTERFERE WITH THEIR JOB? JUST ASKING.

age  WHAT IF YOU WANT SOMEONE THERE LONG-TERM?

gender. COULD A WOMEN'S (MEN'S) SCHOOL DISCRIMINATE AGAINST MEN (WOMEN)?

gender identity or expression  I THINK THIS IS A NEW ONE TO

COVER TRANS AND SIMILAR THINGS. ITS NOT SEXUAL ORIENTATION WHICH IS BELOW.

national origin  HAS NEVER COME UP. EVEN SO, I WOULD NEVER HIRE A WISIAN. See here for why.

marital status WE CAN"T EVEN ASK IF THEY HAVE A SPOUSE WHICH MAY BE HARD FOR TELLING THEM SOME SELLING POINTS OF THE AREA- GOOD FOR MARRIED PEOPLE? GOOD FOR SINGLE PEOPLE? COULD A RELIGIOUS SCHOOL HOLD AGAINST THEM THAT YOU WERE LIVING WITH SOMEONE BUT NOT MARRIED TO THEM?

ancestry I CAN"T IMAGINE THIS COMING UP.

present or past history of mental disorder I"M SURPRISED ABOUT THIS ONE SINCE I WOULD THINK HAVING A PRESENT MENTAL DISORDER IS RELEVANT TO THE JOB. THEN AGAIN, GODEL AND CANTOR PROB HAD SOME MENTAL DISORDERS.

learning disability  HAS NEVER COME UP.  I HAVE NEVER EVEN SEEN A GRAD STUDENT WITH A LEARNING DISABILITY SO IT MIGHT NOT COME UP FOR A WHILE. 

physical disability HAS NEVER COME UP. I CAN SEE IT COMING UP.

political belief  HAS NEVER COME UP. I WONDER IF IT COMES UP MORE IN A POLYSCI OR HISTORY DEPT.

veteran status HAS NEVER COME UP. COULD A MILITARY SCHOOL PREFER VETERANS?

sexual orientation  HAS NEVER COME UP. BUT COULD.

genetic information THIS MAY BE RELEVANT  IN THE FUTURE. PERHAPS THE NEAR FUTURE. SCARY?

non-position-related criminal record. WHICH CRIMES ARE POSITION RELATED? THIS COULD BE A LEGAL QUAGMIRE . MURDERING SOMEONE IN A BAR FIGHT IS NOT POSITION RELATED BUT MURDERING A STUDENT WHO COMPLAINED ABOUT A GRADE IS. OTHER CASES MAY NOT BE AS CLEAR CUT.

2016 Fall Jobs Post

The weather cools down, the leaves change color and you start thinking about what you plan to do after you graduate. As a public service every year about this time, we offer up links and advice for the job search for PhDs in theoretical computer science.

For computer science faculty positions best to look at the ads from the CRA and the ACM. For theoretical computer science specific postdoc and faculty positions check out TCS Jobs and Theory Announcements. If you have jobs to announce, please post to the above and/or feel free to leave a comment on this post.

It never hurts to check out the webpages of departments or to contact people to see if positions are available. Even if theory is not listed as a specific hiring priority you may want to apply anyway since some departments may hire theorists when other opportunities to hire dry up. Think global--there are growing theory groups around the world.

I expect hiring this year to be similar to recent years. Most departments looking to dramatically expand in computer science but with a preference for the more applied areas that the students and the companies that will hire them desire. You can make yourself more valuable by showing a willingness to participate and teach beyond core theory. Machine learning, data science and information security are areas of great need where theorists can play a large role.

A bad job talk can sink your job prospects. Know your audience and particularly for faculty positions create a talk that can express your results and importance to those outside of theory. Pictures help, complex formulas and heavy text work against you. Practice your talk in front of non-theory students and faculty. You will greatly increase your chances if you can sell yourself as a valuable colleague, not just a smart theorist who will prove great things in isolation.

Good luck out there and I look forward to seeing your names on the Spring 2017 jobs post.

Ideal courses for a comp sci dept/I'm glad we don't do it

I once heard it said:

In our data structures course we read Knuth and ignore the proofs

In our algorithms course we read Knuth and ignore the code.

And indeed, there are many topics where the theory-part is in one course and the programming is in another.

With that in mind, here is a different way to offer courses (some depts already do some of this).

1) A course in Algorithms AND Data Structures. Actually I have seen this title on courses but its usually just a theory course. I mean you REALLY PROOF and CODE. Might need to be a year long.

2) Crypto AND Security. You learn crypto and security together and do both. If you only did the crypto relevant to security it might be a semester long and in fact it might already be being done. But I am thinking of really combining these courses--- code and prove. Might be a year long.

3) Formal lang Theory and Practice of Compilers. You do DFA, NDFA, CFG, but also do compiler design. If you also want to do P, NP, and decidability (spell check thinks that decidability is not a word!) then might not quite connect up with compilers, then again in might with theorems like: CFG equivalence is undecidable. Might be a year long.

4) Machine learning AND Prob/stat.

PROS: Theory and Practice would be more united.

CONS: Having year-long courses is hard for the students scheduling their courses. Would having just one semester of any of the above courses make sense?

CONS: Harder to find someone to teach these courses. I'll confess that I prefer to teach formal lang theory without any programming in it.

CAVEAT: I think theorists coming out now know more of the practical counterpart of their area than I did and perhaps than people of my generation did.

CAVEAT: A much less radical thing to do is to put more security in to crypto, more about compilers into Formal Lang Theory, etc. But thats a bit odd now since we DO have a course in security, and a course in compilers. Even so, seems to be a good idea and this I know many schools are doing.

Typecasting from Avi's 60th Celebration

Lance: Live from the Institute for Advanced Study in Princeton, New Jersey, Bill and I are doing a typecast from the Avi Wigderson 60th Birthday Celebration. Hi Bill.

Bill: Hi Lance. Last time I saw you was for the Mike Sipser 60th birthday and next for Eric Allender and Michael Saks.

L: New Jersey again. The Garden State.

B: New Jersey rocks! After all Bruce Springsteen is from here.

L: So was I.

B: You both rock! You are better at math. But I don’t want to hear you sing. I got in last night. How was the first day of the conference.

L: There’s two kinds of talks at a celebration like this. Some people who survey how Avi has shaped the field and their research in particular. Scott Aaronson gave a great talk titled “Avi’s Permanent Impact on me” on how a survey talk on the permanent problem eventually led to Scott’s work on boson sampling.

Most people though are giving talks on their latest and greatest research. At least they have some good Avi jokes to start their talks.

B: So what category did Oded Goldreich’s “Canonical depth-three Boolean circuits for multi-linear functions, Multi-linear circuits with general gates, and matrix rigidity” talk fit in.

L: Oded did say the title was a joke and he did start with an Avi story, Actually it was a Silvio and Shafi story about when to leave for the airport.

B: I liked Alex Lubotzky’s ruminations on pure math versus computer science. He went into pure math thinking computer scientists had to know how to use computers. Had he known that they didn't he may have gone into computer science. He likes that his work on expanders has “applications” to computer science.

L: How did you like the rest of his talk.

B: I don’t know--it’s still going on.

L: Once I gave a talk at a logic meeting about generic oracles. People came up to me afterwards so excited that logic had such applications in computer science.

B: I liked Noga Alon’s talk “Avi, Graphs and Communication”. There is a fixed graph G on k nodes with k players each with an n bit string. How many bits do they to communicate over the edges to determine if all the strings are identical? Basically if the graph is 2-connected you need half of the trivial kn bits.

[Intermission]

L: A day has passed and we find ourselves talking again on Friday at IAS. It’s a beautiful day and we are sitting on lawn chairs on the IAS grounds. This is how Einstein must have felt.

B: Although he wouldn’t be skipping a talk on quantum physics.We are listening in through the magic of the Internet.

I liked Noam Nisan’s talk on the complexity of pricing. Perhaps surprisingly a seller can do better bundling two independent items than using individual prices. I plan to use that example in my fair division class. I may even do a short blog post.

L: I can’t wait. Madhu gave a great talk on low-degree polynomials and how error-correcting have gone from obscurity in theoretical computer science in the early 90’s to a standard tool just a few years later. And we have Madhu (inspired by Avi) to thank for that.

B: Eyal Wigderson, son of you know who, is a grad student studying neuroscience. Talked on “Brains are Better Computers than Computers” which I found flattering.

L: He was talking about rats, not your brain Bill.

B: Should I feel complemented or insulted?

L: Yes.

B: It’s kind of nice to have a birthday person’s child talk a conference like this. In TCS there are several including Paul Valiant who talked at Les Valiant’s 60th, Tal Rabin talked at Michael Rabin’s 80th, father and son Edmonds are here, and Molly Fortnow will surely talk at Lance Fortnow’s 60th. I remember when she interviewed us.

L: Let’s leave it at that.

B: Silvio Micali “knighted” Avi and used Byzantine agreement to improve bitcoin-like chains. I was  enlightened to learn bitcoin is so expensive only five companies do it regularly.

L: When people claim to solve P = NP I asked them to mine a few bitcoins and get back me. I have yet to see any bitcoins.

B: I asked them to find Ramsey of 5. I’m still waiting.

L: On that note time to say goodbye until we meet again.

B: Allender/Saks New Jersey again! Take us out Lance.

L: In a complex world, best to keep it simple, and

B/L: HAPPY BIRTHDAY AVI!

A Second Order Statement true in (R,+) but not (Q,+)

In my last post I asked

Is there a first order statement true in (R,+) but false in (Q,+)

Is there a second order statement true in (R,+) but false in (Q,+)


First order: No.

One reader said it followed from the Lowenheim-Skolem Theorem. I can believe this but don't see how.

One reader said the theory of (Q,+) is complete. True, but would like a reference.

I would prove it by a Duplicator-Spoiler argument.

Second order: Yes

Some readers thought I had < in my language. I do not so I think that examples using < do not work- unless there is someway to express < in my language.

Some readers thought that second order meant you could quantify over functions. My intent was just to be able to quantify over sets.

Some had the right answers. The one I had in mind was

There exists A,B such that A,B are subgroups with at least two elements that only intersect at 0.

Here is a writeup.

Give a second order statement true in (R,+) but false in (Q,+) or show there isn't one

Here is a logic question I will ask today and answer next week. Feel free to leave comments with
the answer- you may come up with a different proof than me and that would be great!

Our lang will have the usual logic symbols, quantification over the domain, quantification over subsets of the domain (so second order) the = sign, and the symbol +

Examples of sentences:

(∀ x)(∀ y)[ x+y=y+x]

true over Q,R,N.  False in S_n for n\ge 4 (group of perms of n elements)

(∃ x)(∀ y)[ x+y=y]

true in Q, R by taking 0. not true in {1,2,3,...}

Lets assume it is true and call the x 0

(∀ x)(∃ y)[x+y=0]

True in Q, R, Z, not true in N.

QUESTION ONE: Is there any sentence in the first order theory that is TRUE over (Q,+) but
FALSE over (R,+)?

QUESTION TWO: Is there any sentence in the second order theory that is TRUE over (Q,+)
but false over (R,+)?

Who's Afraid

The playwright  Edward Albee passed away earlier this month at the age of 88. I had never actually seen any of his plays so I took the opportunity to watch the 1966 movie Who's Afraid of Virginia Woolf? based on the play.

The play has four characters, George, a history professor in his forties and his wife Martha, the daughter of the University's president. Joining them for a late get together is a young biology professor and his wife. The movie had an incredible cast: Richard Burton, Elizabeth Taylor, George Segal and Sandy Dennis. All nominated for academy awards. The women won.

The movie covers many themes, mostly revolving around the disintegrating relationship between George and Martha. George did not live up to his potential as a drunk Martha did not hesitate to point out in front of all of them.

I actually fell for him. And the match seemed practical too. For a while Daddy thought George had the stuff to take over when he was ready to retire. Anyway, I married the S.O.B. I had it all planned out. First he'd take over the History Department, then the whole college. That's how it was supposed to be! That was how it was supposed to be. All very simple. Daddy thought it was a good idea too. For a while!

Until he started watching for a couple of years and and started thinking it wasn't such a good idea. That maybe Georgie-boy didn't have the stuff! That he didn't have it in him! George didn't have much push. He wasn't aggressive. In fact, he was sort of a flop! A great big, fat flop! 

So here I am, stuck with this flop, this bog in the History Department. Who's married to the president's daughter who's expected to be somebody. Not just a nobody! A bookworm who's so goddamn complacent he can't make anything out of himself.

The movie reflects an earlier time in academics, when all the professors were white males and success was measured by taking charge of a department.

Nevertheless Albee captures a fear many academics have. That one day we may wake up and realize we've become that bog. Stuck in our job because we can't afford to give up tenure, but just going through the motions. It's a fear that motivates us, to make us continually try to do new things and achieve new heights. But it's also a reminder that academics is a long career and one that could bog down before we even notice.

Edward Albee writes a play incredibly uncomfortable to watch yet impossible not to. So rare to see that in mainstream plays or movies today.

Boris Trakhtenbrot (1921-2016)

I woke up this morning to two pieces of news. Subhash Khot has just been named a MacArthur Fellow, the "genius" award, for his work on unique games. But I also just learned about Monday's passing of the great Russian theorist Boris Trakhtenbrot at the age of 95.

Trakhtenbrot has a number of important results in automata theory, model theory and logic to name just a few areas. In computational complexity we know him best for the Gap Theorem which he proved independently with Allan Borodin. Roughly the gap theorem states that for any computable f there is a computable time-bound t such that DTIME(t) = DTIME(f(t)), every problem solvable in time f(t) can also be solved in time t. For example there is a time bound t such that DTIME(t) = DTIME(2^t). This doesn't violate the time hierarchy since t may not be time-constructible. There is nothing special about time here, it works for space or any abstract complexity measure.

Borodin and Trakhtenbrot worked independently because they sat on different sides of the iron curtain during the cold war which very little communication in between. Boris Trakhtenbrot wrote A Survey of Russian Approaches to Perebor (Brute-Force Searches) Algorithms (PDF) that traces this early history of Russian theory. He didn't hold back discussing some of the academic politics in Russia that stifled research into computational complexity and gave us a window into the Russian scientific community in the mid-20th century.

Thankfully the iron curtain has been replaced by a global internet and we can do science together. Let's hope it stays that way.

Academic Rankings Foster Competition

This week US News and World Report released their undergraduate rankings earlier this week. A time for schools to brag. US News and World Report used to publish an actual weekly news magazine, now they mostly just focuses on rankings. Besides various categories of undergrad institutions USN&WR ranks engineering and business programs. There are many other ranking systems of varying quality but in the US we take the USN&WR rankings the most seriously.

Computer Science does not get an undergraduate ranking. Computer engineering does--not the same. CS does get rankings as a PhD program, last time in 2014. I've posted on rankings in 2005, on the failed NRC rankings of 2010, on using metrics for rankings, and Bill had his own so-called non-controversial thoughts on rankings.

In the September CACM, Moshe Vardi wrote his editor's column entitled Academic Rankings Considered Harmful

Academic rankings, in general, provide highly misleading ways to inform academic decision making by individuals. An academic program or unit is a highly complex entity with numerous attributes. An academic decision is typically a multi-objective optimization problem, in which the objective function is highly personal. A unidimensional ranking provides a seductively easy objective function to optimize. Yet such decision making ignores the complex interplay between individual preferences and programs' unique patterns of strengths and weaknesses. Decision making by ranking is decision making by lazy minds, I believe.

No potential grad student should decide based solely on rankings but neither can we expect them to solve a highly-complex multi-objective highly-personal optimization problem over all 266 PhD-granting CS departments. They will find ways to narrow down their list of schools somehow and a reasonable independent ranking of CS departments can certainly help.

More importantly rankings cause us to compete against each other. Every CS department wants to raise their rankings (or stay on top) and use that goal to work on strengthening their departments and use rankings to make the case to upper administration and alumni to get the resources needed to continue to grow. By the nature of rankings, not everyone can rise up but we all get better in the process.

Noam Nisan wins the Knuth Prize

Noam Nisan will receive the 2016 Donald E. Knuth Prize, the highest honor in theoretical computer science, for his work in computational complexity and algorithmic game theory. He'll receive the award at the upcoming FOCS conference.

I've known Noam since we were fellow grad students in Berkeley in 1985 and we have become good friends and colleagues. Noam Nisan started his career in computational complexity and Luca posts about several of his seminal works in derandomization, interactive proofs, communication and circuit complexity. I'll focus this post on Nisan's 1992 paper with Mario Szegedy, On the degree of boolean functions as real polynomials.

Let f be a Boolean function on n variables and p a n-variate polynomial over the reals and suppose for every x in {0,1}ⁿ, |f(x)-p(x)| ≤ 1/3. Nisan and Szegedy show that the decision tree complexity of f is bounded by a polynomial in the degree of f. The decision tree complexity of a function is the number of bits one has to query to determine whether f is 0 or 1.

The theorem didn't have an immediate application but soon afterwards I told Noam we found a result that followed from his paper. His ears perked up until I told him the actual result (if P = PSPACE then P = AWPP relative to a Cohen generic oracle).

Later on Nisan-Szegedy would have direct applications to quantum computing, showing that quantum, random and deterministic decision tree complexity for total functions are polynomially related. Just last STOC we saw two papers giving tighter bounds on these relationships.

In 1997 Noam Nisan walked away from complexity and soon thereafter became one of the founding players in algorithmic game theory, co-organizing the 2001 DIMACS workshop that would kickstart this field. He won the 2012 Gödel Prize for his early work on algorithmic mechanism design with Amir Ronen.

Nisan and Szegedy begat one of my most frustrating open questions on the relationship between rational functions and decision tree complexity. I have an application for it but I don't think you really want to know.

Why I Don't Believe in ET

Is there life on other planets? The naive answer is "of course, why should Earth be special." What makes a lottery winner special? We could have just won the life lottery and the losers aren't around to complain.

The more scientific approach uses variants of the Drake Equation. I'll use the variant from the recent paper A New Empirical Constraint on the Prevalence of Technological Species in the Universe by Frank and Sullivan.

We define the ‘‘A-form’’ of the Drake equation, which describes the total number of technological species that have ever evolved anywhere in the currently observable Universe:

A  = Nf_pn_pf_lf_if_t

where N is the total number of stars, f_p is the fraction of those stars that form planets, n_p is the average number of planets in the habitable zone of a star with planets, f_l is the
probability that a habitable zone planet develops life, f_i is the probability that a planet with life develops intelligence, and f_t is the probability that a planet with intelligent life develops technology (of the ‘‘energy intensive’’ kind such as that of our own civilization).

I first encountered the Drake equation in high school from Carl Sagan's book Cosmos. The conclusion that there must be life out there from the equation never satisfied me because of our inability to measure the f values.

Frank and Sullivan's computations show that we expect there to only be life on Earth, say A=0.01 then f = f_lf_if_t must be at most 2.5 x 10^-24.

Seems small but is it really? 2.5 x 10^-24 is roughly the probability of flipping 78 coin tosses and having them all come up heads. Maybe life requires 78 50/50 independent events to occur. Or 1060 independent events each with 95% probability. 1060 doesn't seem that big.

Our attempts at finding life, whether by SETI or Mars soil or UFOs have so far turned up nothing substantial. We just might be the only ones out there.

By no means am I arguing that we give up the search. I could be wrong, f could be much larger than 2.5 x 10^-24. Let's keep looking, perhaps exploring the ice caps of Mars or the moons of Jupiter, keep listening the the stars, explore new ways to probe the galaxy. It would be incredible to find extraterrestrial life, but just don't be surprised if we don't.

I have consulted four times. Really!

Those who know me know that I work on stuff that is not readily applied. Or perhaps not applied at all. Certainly my current state of knowledge does seem like it would be useful to  a company. Many theorists, at one time in their lives, were excellent programmers. (For example, Lance helped write a program that played Othello see here and an email system see here.) I have no such stories. I took ugrad compiler design and ugrad Operating Systems as a grad student (not at the same time!).
I was taking Operating systems and TAing Aut theory. Dave (I forget his last name) was taking Aut theory and TAing Operating systems. We both got B's which you can regard as either very good or very bad planning.

Anyway, I never was a programmer. Could I have been a good programmer? Irrelevant! Would real world experience have helped my research? Very hard to know, but prob yes.

Four times I have worked for a real world company of some sort (never for more than a two months) and I always wondered Gee, I don't know anything they would care about. But in all four cases they seem to like what I did for them.  Why?

For two of the four I signed an NDA (Non Disclosure Agreement) so I will need to talk in general terms.

1) I was hired to find out which of two ways to schedule jobs was better. I did some easy math, did some easy simulations, found out that it didn't matter much. I think they knew that, but having someone with a PhD tell them comforted them.

2) I was hired to find out why the Operating System was so slow. I had already had the course in OS but it didn't help at all. I did some easy math that identified some of the problems, but told them that they had a more overwhelming problem and what it was. I think they sort-of knew this, but I clarified it for them. AFTER the course I took a course in queuing theory since the job peaked my interest. If I had the course before the job it would  not have helped.

3) I was hired to do some statistical work. I wrote a report detailing the methods that I used, and then I used them. Everything I did was elementary statistics. The techniques were standard. But they really appreciated having it all laid out for them. Originality was not needed, just using known stuff.

4) A company working on SAT Solvers- I helped clear up some misconceptions they had.

I suspect that 3/4 of the people reading this blog could do 3/4 of the consulting I've done. What I learned from these experiences is

(1) just knowing math in a general sense may be all they need,

(2) you can pick up what you need,

(3) sometimes they just need someone with a degree to tell them what they already know.

In all four cases I was intrigued by having to solve a REAL problem as opposed to a CLEAN math problem.

1956 Was a Fine Vintage

Bill sends me an email last week with the subject "Our field is getting old!" and talks about upcoming 60th birthday/retirement conferences he's invited to, all of which I've been invited to as well. Bill's subject line should have read "We're getting old."

Here's what's upcoming:

• Avi Wigderson's 60th celebration before FOCS in New Jersey, October 5-8. Avi is a giant in computational complexity and the event has an amazing line-up of speakers.
• Albert Meyer's retirement celebration at MIT, November 11. 
• Rod Downey's 60th symposium in New Zealand, January 5-8.
• Eric Allender and Michael Saks will have a joint 60th celebration at DIMACS in New Jersey, January 26-27.

Surely I've missed some. Feel free to add in the comments.

Theoretical computer science has a tradition of holding a symposium in honor of a 60th birthday, typically organized by the PhD students. I co-organized such a celebration for my advisor Michael Sipser two years ago. 60 is a good age, a time to look back but not quite the end of a career. 

Janos Simon giving toast to Juris Hartmanis (center).

The first 60th I attended was for Juris Hartmanis, one of the founders of computational complexity, back in 1988 at the 3rd Structure in Complexity Theory (now Computational Complexity Conference) meeting in DC. The conference announcement required everyone to wear a jacket and tie, the first and probably last time I will see a bunch of complexity theorists all dressed up.

Juris was also the first faculty retirement I attended in 2001 at Cornell. Retirement celebrations are generally organized by the department.

Many of the first generation of CS theorists from the 60's and 70's are hitting retirement age. The 1980s saw an explosion in CS theory PhDs and many of them turn 60 in the near future. Expects lots of celebrations, great speakers and remembering many great careers. 

Chrisitan Comment on the Jesus Wife Thing misses the important point

In 2012 a Professor of Divisinity at Harvard, Karen King, announced that she had a fragment that seemed to indicate that Jesus had a wife. It was later found to be fake.  The article that really showed it was a fake was in the Atlantic monthly here.  A Christian Publication called Breakpoint  told the story: here.

When I read a story about person X being proven wrong the question upper most in my mind is: how did X react?  If they retract then they still have my respect and can keep on doing whatever work they were doing. If they dig in their heels and insist they are still right, or that a minor fix will make the proof correct (more common in our area than in history) then they lose all my respect.

The tenth paragraph has the following:

Within days of the article’s publication, King admitted that the fragment is probably a forgery. Even more damaging, she told Sabar that “I haven’t engaged the provenance questions at all” and that she was “not particularly” interested in what he had discovered.

Dr. King should have been more careful and more curious (though hindsight is wonderful)  initially. However, her admitting it was probably a forgery (probably?) is ... okay. I wish she was more definite in her admission but... I've seen far worse.

A good scholar will admit when they are wrong. A good scholar will look at the evidence and be prepared to change their minds.

Does Breakpoint itself do this when discussing homosexuality or evolution or global warming. I leave that to the reader.

However, my major point is that the difference between a serious scientist and a crank is what one does when confronted with evidence that you are wrong.

Predicting in Changing Environments

The New York Times yesterday ran a story connecting climate change to the Louisiana flooding.

The National Weather Service reports that parts of Louisiana have received as much as 31 inches of rain in the last week, a number Dr. Easterling called “pretty staggering,” and one that exceeds an amount of precipitation that his center predicts will occur once every thousand years in the area.
Dr. Easterling said that those sorts of estimates were predicated on the idea that the climate was stable, a principle that has become outdated.
The third National Climate Assessment, released in 2014 by the United States Global Change Research Program, showed that “the amount of rain falling in very heavy precipitation events” had been significantly above average since 1991.
However, the research did not identify the South as one of the areas of greatest concern; the increase was found to be greatest in the Northeast, Midwest and Upper Great Plains regions of the United States.

In short climate change means our old prediction models of the weather no longer apply. On top of that, new models that tried to take into account climate change predicted heavier rains but not in that area of the country.

The weather is hardly the only predictions gone bad this year. From Nate Cohn's What I Got Wrong About Donald Trump

Did he have a 1 percent chance to win when he descended the escalator of Trump Tower last June? Twenty percent? Or should we have known all along?
Was Mr. Trump’s [republican nomination] victory a black swan, the electoral equivalent of World War I or the Depression: an unlikely event with complex causes, some understood at the time but others overlooked, that came together in unexpected ways to produce a result that no one could have reasonably anticipated?
Or did we simply underestimate Mr. Trump from the start? Did we discount him because we assumed that voters would never nominate a reality-TV star for president, let alone a provocateur with iconoclastic policy views like his? Did we put too much stock in “the party decides,” a theory about the role of party elites in influencing the outcome of the primary process?
The answer, as best I can tell, is all of the above.
I do think we — and specifically, I — underestimated Mr. Trump. There were bad assumptions, misinterpretations of the data, and missed connections all along the way.

We also had bad predictions on Brexit and one factor in the 2008 financial crisis was a heavy reliance on historical patterns of housing prices.

We have at our fingertips incredible prediction tools from machine learning models to prediction markets. Not all things change, our models trained to recognize cat pictures will continue to recognize cat pictures for a long time running. But as we continue to rely more and more on data driven predictions and decisions, be prepared for more and more surprises as underlying changes in the environmental, political and financial climates can pull the rug out from under us.

Is the examiner being pedantic? Whats really going on here?

The following are two real conversations.  For each one: (1) Is the examiner correct?, and
(2) Where and when do you think this conversation took place?

I give the answers below so you may want to read, stop and think, and then read on.

CONVERSATION ONE:

EXAMINER: What is the definition of a circle?

STUDENT: The set of points equidistant from a given point.

EXAMINER:  Wrong! It is the set of ALL points equidistant from to a given point.

CONVERSATION TWO:

EXAMINER: What is the definition of a circle?

STUDENT:  It is the set of all points equidistant from a given point.

EXAMINER: WRONG! You did not specify that the distance is nonzero.









AN ANSWER I HAVE HEARD FROM SOME NON-MATHEMATICIANS: Since math people are  pedantic and formal to an absurd level the examiner is correct.

REAL ANSWER: Nobody in math would be that pedantic.  In the old USSR, entrance exams for
Moscow State University were rigged so that Jewish students could not pass.  The following is a quote from


Bella Abramovna Subbbotovskaya and the Jewish People's university, By G. Szpiro. Notices of the AMS Vol 54,  . 1326--1330. Article is here


The first story in it happened to Edward Frenkel when he was a 16-year-old taking the oral entrance exam to Moscow State University in 1984, recounted in his book Love and Mathematics, which I reviewed here.


BEGIN QUOTE
Jews -- or applicants with Jewish-sounding names -- were singled out for special treatment.  On one occasion a candidate was failed for answering the question what is the definition of a circle with
 the set of points equidistant to a give point. The correct answer, the examiner said, was the set of all points equidistant to a given point.  On another occasion an answer to the same question was deemed incorrect because the candidate had failed to stipulate that the distance had to be nonzero.
END QUOTE

A different technique used on the entrance exams was to give Jewish students problems that had simple solutions which were extremely difficult to find.  The simplicity of the solution made appeals and complaints difficult.  Some of these problems and their history is in this article:

Killer Problems by Tanya Khovanova and Alexey Radul, The American Mathematical Monthly ,Vol 119, pp. 815--829.Article is here (link is to arxiv version where title is Jewish Problems.)

This is of course apalling; however, I have another issue to raise. Not allowing some part of your population to contribute  is just... idiotic. What I really want to know is why did they do this when it so clearly worked against their interests. How would an intelligent defender of this system defend it? By intelligent I mean someone who knows that Jews are not FILL IN ANY FALSE NEGATIVE BELIEF ABOUT JEWS.  By intelligent I also  mean someone who actually sees the downside. In other words, how would they fill in the following sentence:


The downside is that people who are talented in math and other fields do not get to contribute to our society, but the upside is FILL IN THE BLANK.

For more information on this, but not really an answer to my question, see the Wikipedia entry on anti-semitism in Russia here.

Robin Hanson's Ems

Robin Hanson an economist at George Mason and author of the Overcoming Bias blog, has a new book The Age of Em: Work, Love and Life when Robots Rule the Earth. Em stands for brain emulation, a computerized version of a human brain processes including consciousness, all the wants, desires and faults of a human brain. An em can be created by copying from a human brain or another em, it can be stored and restored, slightly tweaked and can run faster or slower depending on the power consumption. Reminds me a bit of the cookies in the White Christmas episode of Black Mirror. Hanson also talks about clans, the collection of all the ems that descend from a particular human.

The book has two distinct parts. The first gives a plausible physical and social explanation as to how and why the em world will come to be. Hanson gives the odds of such a world developing at one in a thousand. I put the odds much lower, especially since we have no true understanding of consciousness. For example if consciousness requires quantum entanglement, we would have no hope of copying into an em without destroying the old em (or human) it came from. More likely I expect we would have em-like machines that can perform a number of human-like tasks but won't have any true consciousness. Nevertheless I'd acknowledge that Hanson's world is at least possible given our current knowledge of the brain.

I enjoyed more Hanson's discussion about the world of the ems given that they exist. Hanson takes a science, as opposed to a science fiction, approach to the topic, carefully thinking about how ems would work as a clan, how they interact politically, socially and economically. You can see the variety of topics in the table of contents on the book's website. For example, Hanson argues that it makes economic sense to have ems split off as "spurs" to do more menial tasks in parallel in exchange for a short working life, as short as a few minutes, and a long retirement, with the retirement in a slower low-powered mode. Hanson has done some strong research and advocating for prediction markets (we even have a joint paper on the topic) that there is no surprise that markets show up as a decision making systems for ems.

I don't agree with all of Hanson's conclusions, in particular he expects a certain rationality from ems that we don't often see in humans, and if ems are just human emulations, they may not want a short life and long retirement. Perhaps this book isn't about ems and robots at all, but about Hanson's vision of human-like creatures as true economic beings as he espouses in his blog. Not sure it is a world I'd like to be a part of, but it's a fascinating world nevertheless.

A Game Theory Conference! That sounds like fun!

Bill: Lance just came back from Games, a conference on Game Theory.

Darling: That sound like fun! From what you tell me there is some nice math behind
Monopoly(see here for a paper on Monopoly as a Markov Process), Risk (see here for a paper on using Markov chains in the game Risk. Was Markov a game player?) and other FUN games.

Bill: Uh, I don't think they talked much about those kinds of games.

Darling: Darn. Did they talk about those really boring math games like Dup-Spoiler games, those games that AD is about, Gale-Stewart Games, Banach-Mazur games, Martingales, Pebble games, Communication complexity games. Oh, and Combinatorial games like NIM which can be sort of fun.Or did they talk about computers that play Chess and similar games? Or did they have talks on things like Chess being EXPTIME complete. Or did they talk about the Unique Game Conjecture.

Bill: Most of the talks were about setting up a system so that all players acting in their own best interest is also good for the system. Like an auction system where it is a players best interest to bid what they actually think the item is worth.

Darling: So there are no Games at a Game Theory conference?

Bill: There were a few papers on Poker, but that's it.

Darling: They should change the name of the field.

Bill:  Lance tells me that Ehud Kalai has suggested  Game Science. 

Darling: So long as the word Game is in the title they should have fun games there. Oh well.

Seymour Papert (1928-2016)

Seymour Papert, the great AI pioneer, passed away Sunday at the age of 88. In the theory community we best know him for his 1969 book Perceptrons with Marvin Minsky, who died earlier this year. In that book they show that a perceptron (what we now call a weighted threshold function) cannot compute parity, one of the first examples of a circuit lower bound.

In the summer of 1982 I worked a a computer camp in Los Olivos, California and we taught the kids programming with the Logo programming language, a simple functional language co-created by Papert and Wally Feurzeig. In Logo you controlled a virtual turtle that carried a pen and you could give simple instructions like raising and lowering the pen, moving forward and backward and turning. With simple functions one could create complex diagrams, like the one above. Normally you would see the diagrams on a screen but we also had a physical electronic turtle that would move and draw on a sheet of paper. The kids loved it since they could see the results of their programs as a picture while they learn programming functions and recursion without realizing it.

You can play with Logo at Turtle Academy  Logo set the stage for control of actors in many other computer languages designed for children including the tasks for the popular Hour of Code.

Let's raise our turtle pens to honor Papert and the many that he brought into the world of computing.

GAMES/EC 2016

This week I report from Maastricht in the Netherlands from the GAMES 2016, the 5th World Congress of the Game Theory Society. By having their congress every four years, everyone who is anyone in the game theory community makes a strong effort to be here, including three Nobel laureates, Robert Aumann, Roger Myerson and Eric Maskin. The conference has about 750 participants and up to 19 parallel sessions.

This year the conference is co-located with the Economics and Computation conference that comes more from the CS community. By co-located we are sharing the same buildings and many of the events, effectively one larger conference (which means in reality 21 parallel sessions).

EC keeps growing, accepting 80 papers out of 242 submissions, all of which are freely downloadable.

My favorite EC talk was the best student paper, Deferred Acceptance with Compensation Chains by
Piotr Dworczak, a graduate student in the Stanford Business School. He gives an algorithm for finding stable matchings with the property that every stable matching can be found by changing the order that the agents get to choose. The paper Which Is the Fairest (Rent Division) of Them All? by
Kobi Gal, Moshe Mash, Ariel Procaccia and Yair Zick won best paper.

Also a shout out to the talk Cadet-Branch Matching in a Quasi-Linear Labor Market solely authored by Ravi Jagadeesan, a rising junior undergraduate at Harvard. I went to grad school with Ravi's mother Lalita, and yes that makes me feel old.

Tim Roughgarden gave the Kalai prize talk for his work on Intrinsic Robustness of the Price of Anarchy. The talk, attended by a good number of the game theorists, gave a general approach to generalizing bounds price of anarchy results to broader classes of equilibria. Tim followed Keith Chen who heads the analytic team for Uber and discussed how game theory and optimization ideas are driving a major e-commerce company. No major surprises but here's one trade secret: Uber covers its maps with hexagons while Lyft uses squares.

All is all a great week, with packed schedules and crowded activities, but great to see all these game theorists and computer scientists talking with each other.

The College Issues that are talked about/College issues that are important

The following college issues get lots of attention:

Admissions-  high school students PLAN to do things JUST to get them into an elite college. For example nobody takes the SATs just for the fun of it anymore.

Admissions- Some High School Students are stressed out about college admissions, see here

Admissions- Affirmative action.

Admission- Lower standards for athletes?

Sports- too much money spend on it?

Sports- the players treated unfairly?

Are other out-of-class activites also an issue? See here.

Jock Culture.

Free speech- Speech codes, triggers. (I've heard this talked about for about 30 years now.)

A  common core. How to make it not just dead white males.

A common core. How to get this to work when profs are overly specialized.

Professors are rewarded for research more than teaching- how to induce them to be better teachers.(I've heard about this one for about 40 years.)

Renaming buildings that are named after racists. ( Byrd Stadium at UMCP is now Maryland Stadium  see here) (If we find out that Fields or Abel was a racist will we rename the awards? Why bother naming things after Justice Scalia or Bobby Kennedy when they will be renamed at some point because of their views on gay people?)

Renaming buildings that are named after the things racists do (see here)

White privilege  (If I was black then whenever my blog had   bad spelling or grammar it would be connected to my race and assumed upbringing.)

The crushing debts of $100,000 or so that some students face after college. (Which is why some students Feel the Bern!, though others Feel the Bern! for different reasons.)

Hookup culture on campus

Lack of diversity in some majors (I've heard this talked about for about 30 years now.)
(Examples: Across the country CS is at around 15% female.  Art History is around 80% female. Why does the CS one generate much discussion and some outrage but the art history one... not so much? I would guess jobs. But the point of this list is just that these are the issues people ARE talking about.)

Should college be vocational vs intellectual? Are these two disjoint?

MOOCS: How will they affect education?

These are important issues. But they affect  few people. 65% (and dropping) high school seniors goto college. Over 1/2 of all college students go to community college. Many of those students are part timers who also work. Speech codes are not at the top of the things they have to worry about. These people face other problems that do not get attention. See the following excellent articles

Shut up abour Harvard by Ben Cassleman

and

The other 75% by Paul Attewell and David Lavin

To give one example: The number of students going to community college who need to take part time jobs to finish and end up with a crushing (to them) debt of $10,000 is a far more common problem then any of the ones above. Why so little coverage?

The articles give other examples of problems that are NOT being talked about.

The other 75% is from the excellent book What is college for edited by Lagemann and Lewis, and reviewed by me, for SIGACT News, here.   Most of the other chapters are about the issues above. We need a book, or at least  a conversation, about issues of education affecting many more people.

The Morrill Land-Grant Acts established many colleges. It was passed in a time when it was realized that its important to have an educated public. It must have been passed in a time where there were not the pressing issues we have today (like bathrooms for transgender people) so they could think about these issues clearly. It was passed in 1862 in the middle of the Civil War.

A meta-thought--- Every comment on this blog about the issues I list above as NOT affecting that many people will prove my point that those issues are discussed far more than issues that affect far more people. Even so, feel free to comment on whatever issues you want.

Snowbird 2016

Earlier this week I attended the 2016 CRA Snowbird Conference, a biennial meeting of CS chairs and other leaders in the the computing community. I’ve attended every meeting since 2010, the first as a panelists on journals in CS and the last three as department chair. I enjoy this meeting mostly for the networking with other chairs across the whole CS discipline.

Computer science sits in an enviable position with booming enrollments, our graduates easily finding quality jobs in the field and computing literally changing society. Success breeds challenges, top of this list is how do we cover the dramatically increasing course load. Right now most schools are applying a variety of short-term solutions from increased use of larger classrooms, sometimes with recorded video, instructors and teaching faculty, PhD, MS and undergrad TAs and less small specialized graduate classes to allow for more core courses. What we haven’t seen is increased teaching loads. We need to be careful not to drive faculty into industry’s waiting arms.

The confrence had some interesting speakers such as John Markoff from the New York Times on the excitement and concerns over machine learning and automation, Cynthia Dwork on the theory approach to privacy and fairness in the big data era, and Robert Morse from US News on how they rank CS departments. I'm generally fine with the US News Ranking, they measure reputation and reputation is in the end what matters in recruiting students and faculty. Many at the meeting had other, less friendly, opinions towards US News and rankings in general.

One popular session discussed schools and colleges of computing beyond the department. Most of the successful transitions came out of CS programs in colleges of science, such as at CMU and Georgia Tech. Rarely do we see the transition out of engineering. With both the growth of computer science and its increasingly central role in many academic disciplines, now is a good time to make the argument for more colleges of computing. Perversely the growth makes such changes more challenging as an engineering dean would not want to give up such a large and growing part of their college.

The Snowbird conference moves the conversation away from the weeds of current results to look back at what our field has achieved and where it is going. It's never been a more exciting time to be a computer scientist and while chairs always love to grumble when we get together, we all know how lucky we are to be leaders in this era.

Solution to the Alice-Bob-Box problem.

In my last blog I solved one problem and asked another (when will it end!).  Damien Roberts provided an answer in the comments to the last blog, so kudos to Damien! I summarize the question asked and answer it (same answer as Damien, though more long winded)  and then some comments and further questions.

Peter Winkler told me this problem at the Joel Spencer 70th Bday conference.  He got it from Sergui Hart who does not claim to be the inventor of it.

PROBLEM:
Alice and Bob play the following (non-fun) game:

Alice puts natural numbers in the boxes 1,2,3,4,... She can put any number in any box. She can put the number 12 in two boxes. She can refuse to put primes in any box. The world is her burito!

Bob opens all but a finite number of  boxes. He chooses one of the boxes that is NOT opened and guesses what is in it and then opens it.  If his guess is correct he wins! (not clear what he wins, but he wins).If not he is, as one of our candidates for prez would say, a LOOOOOOOSER.

Would you rather be Alice or Bob?
END OF PROBLEM

(Added Later- below is an answer that I believe to be correct. Some people disagree. See some of the comments below and also the comments on this)

ANSWER:
I would rather be Bob:

As you might have guessed, Bob takes the set of all infinite sequences of natural numbers and looks at the equivalence x==y iff x and y differ on only a finite number of positions. He then picks a representative from each part of the induced partition.

(When I tried to solve the problem thats as far as I got. Some commenters thought that was the solution, or the solution was obvious from that point. I don't see how.)

Here is what Bob does;

STEP 1: (I used to have Bob takes a random perm of the naturals and relabels the boxes but commenters pointed out that this was not needed and might not even make sense. So there is no Step 1, but I keep
it in case someone sees this post and wonders what happened to step 1.)

STEP2: Bob rearranges the boxes into (say) 100 rows.  We'll say

ROW1: BOXES: 1,101,201,301,...

ROW2: BOXES: 2,102,202,302,...
...
ROW100: BOXES: 100,200,300,...

STEP3: Bob picks a Row AT RANDOM (second use of probability). We will say ROW8:

STEP4: Bob opens the boxes in all of the rows EXCEPT ROW8 (he will open some in ROW8 later). For each Row i NE 8 he does the following:

ROWi is in one of the parts of the partition. Bob had ahead of time chosen a representative in that part. Let x(i) be such that from the x(i)th element on ROWi and the Representative AGREE.

Let x  = max of the x(i).

So, for all rows i NE 8, if you look past the xth position, ROWi and the represenative for that equivalance class are the same.

STEP5: Bob opens up, in ROW8, the boxes x+2, x+3,...

STEP6: Bob knows the part that ROW8 is in the partition. Bob looks at the representative. Bob guesses that the number in box x+1 of ROW8 is the same as the number in the (x+1)th position of the representative.

WHY is this a good idea?

Consider ROWi. Let x(i) (as above) be such that past x(i) ROWi agrees with its rep. In order for the guess to be wrong you would need x(8) > all other x(i). How likely is that? Since we began with a random perm, the prob that x(8) is the largest (for that matter, the prob that any particular i_o has max x(i_0) ) is 1/100. So Bob wins with prob 1- 1/100

Bob can do even better with 1000 rows or 10,000 rows, etc. For any eps>0 he can make the prob that he'll win 1-eps.
END OF ANSWER

One issue I've been trying to get at in this post and the last one was, is this a real solution? I think so, but some students don't like it. Even those that understand it. What do you think?

There is no deterministic solution to the Alice-Bob-Box problem since if the adversary knows what box Bob will guess he can make it so that Bob gets it wrong.

Some asked if there was a computable solution. For both the infinite-hats problem and the box-problem if the players have a strategy depending on only a finite number of inputs they can't win. There are ways to measure the complexity of a strategy and it would be interesting to get upper and lower bounds on the complexity of a strategy. And one can look at the following: If the adversary's strategy is of complexity BLAH then what complexity do the players need to beat it?

Also, in both games AC is used by the players. Eddie Fisher pointed out that if you toss out AC and instead have the AC AM (all sets are measurable) then the Adversary wins the hat game. Not sure about the box game.  One can look at what happens in various axiom systems.

So many open questions, so little time!

Solution to the infinite hat problem/a point/a new problem

In my last post I asked the following question (I've shortened it here but its the same really.)

An infinite number of people, labelled 1,2,3,... have hats on their head, RED or BLUE. They must all shout at the same time a color. They want only a finite number of people to not guess their own hat color correctly. Can they do this?

YES they can manage to get all but a finite number of hats wrong. Here is what they do in their strategy meeting:

1) Define an equivalence class on infinite strings of R's and B's:  x==y iff x and y differ on only a finite number of places. It is easy to show that this is an Equiv relation. We think of the string as telling us the hat colors of all the people.

2) An Equiv Rel induces a partition. Choose, for each part of the partition, a representative. They all know all of the representatives.

NOW, once the hats are put on here is how each person reacts:

Gee, I see all of those hats out there! Since I see all but my hat I KNOW which partition the hat coloring is in. I look at the representative of that partition. I guess the hat color that I have in that representative.

Note that ALL of the people will use the SAME rep, and that rep differs from reality in only a finite number of hats. Therefore the number of incorrect answers is finite.

POINT- when I show this to my class they often don't like it. Some of course do not understand it, but even those who do sometimes say that's not practical  or the people would need infinite brains!  These are both true, but I like it anyway. HOW ABOUT YOU- does the solution satisfy?

NEW PROBLEM (are any problems really new?) I heard this from Peter Winkler who heard it from Sergui Hart who does not claim to have invented it.

Alice and Bob play a game (My darling complains that when a math puzzle uses the word game its usually not a fun game. This problem will not be a counterexample.)

There are an infinite number of boxes labelled 1,2,3,....

Alice puts into each box a natural number.

(CLARIFICATION based on a comment- Alice an put any number she wants in any box.
She wants to put 18 in both box 199 and box 3999 she can do that. NO restriction on what
Alice puts in the boxes except that every box has SOME natural number. ALSO- she need not
use all the naturals- if she wants to put a 1 in every box, she can.)

Bob opens all but a finite number of boxes.

For one of the boxes Bob does NOT open he guesses what the number in it is.

They then open that box. If Bob is right, he wins. If not then Alice wins.

Would you rather be Alice or Bob?

Feel free to post thoughts in the comments, though if you want to solve it without help you may want to avoid the comments. I'll post the answer next week.

An infinite hat problem and later a point

Problem: There are an infinite number of people. They are labelled 1,2,3,... (I am not a number, I am a free man!) There is the Master who I call The Master. The Master will, at the same time, put a hat on each persons head. Some of the hats are RED, some are BLUE. (Clarification  added later- everyone can see all the peoples hat colors except their own.)

The people will then all, at the same time, yell out a hat color. (Clarification added later- NO other form of communicationis allowed.)

If  only a finite number of them get their own hat color wrong they win (not sure what they win, but they win!)

If an infinite number of them get their own had color wrong, then they lose.

They can discuss strategy ahead of time; however, The Master overhears all conversation.

Assume that the people and The Master are experts at this game.

Who would you bet to win? How much and at what odds?

I'll post the answer, a meta question about it, and another math question, on Thursday.

Feel free to post your answer as comments. If you do then please also comment on if you've seen the problem before since I'm curious (1) how well known the problem is, and (2) how hard it is to solve if you haven't seen it.

If you want to try to solve it yourself, don't look at the comments in case the right solution is there.

Is determining if a poly over a finite field is 1-1 hard? Sure seems so.

When I teach cryptography  to High School  students I begin with shift and linear  ciphers which are

x --> x+s mod 26 (s is a shift, x is a letter of the alphabet. Hmm- x really IS a letter of the alphabet!)

x--> ax+b mod 26 (Note that a has to be rel prime to 26.)

I then ask why nobody seems to have ever used

x --> ax² + bx + c mod 26.

I then tell them that this is because there is no quick test that will, given (a,b,c), tell if
f(x) = ax² + bx + c mod 26 is 1-1 (and hence onto).

It recently dawned on me that I don't really know that its hard to test.
(ADDED LATER) In fact its not true. Algebra shows that f(x) is NOT 1-1 iff

a(x+y)+b =0 mod 26

has a solution. If a is rel prime to 26 then clearly there is a solution (many in fact).
If a is not rel prime to 26 then I suspect this is not hard.

How hard is the following problem?

Given a poly f(x) of degree d, and n,  is f(x)  mod n 1-1?

We will assume all coefficients are between 0 and n.. We can also assume that c is 0 since f(x) is 1-1 then f(x)-c is 1-1.
 The coefficients are  given in binary so the length of the input is roughly dlog(n).

One can of course compute f(0), f(1),...,f(n-1) and see if there are any repeats, but this takes O(n) steps which is exp in the input of length log n.

I suspect that this is either a well known solved problem (either in P or NPC) or a well known open problem. Any help or references will be more appreciated than usual-- see next paragraph.

I am the new SIGACT News Open Problems Column editor. In the future I will be soliciting people to write columns for me, but the first one I'll do myself and this might be a good topic- if its open! And if it is open, would be good to know references and what is known.