Thursday, May 28, 2015

Who wins this bet?

Alice and Bob are at an auction and Alice wants to buy an encyclopedia set from 1980. Bob says don't buy that, you'll never use it. In this age of Wikipedia and Google and THE WEB.  Alice says you don't know that.  They agree that Alice will spend no more than $20.00 on it (and she does win it at $20.00) and that:

If Alice does not use the encyclopedia within 5 years then she owes Bob $10.00. If she does use it then Bob owes Alice $10.00.

3 years later its really cold outside. Alice's house is not that well insulated. So she takes carpets against the bottom cracks in the door and weights them down with the volumes of the encyclopedia. This helps her keep warm and cuts down on her heating bill.

Alice says I used the encyclopedia. Pay up! In your face! I used them! You were wrong!

 Bob says You didn't use them to look anything up. So that doesn't count.

Alice and Bob are asking YOU do decide. So leave comments either way and whoever has more votes before my next post wins! Feel free to leave reasons as well to persuade the other readers.

Monday, May 25, 2015

John Nash (1928-2015)

John Nash and his wife Alicia died in a taxi accident, returning from the airport after he received the Abel prize in Norway. The public knew John Nash as the "Beautiful Mind" of book and screen, but we knew him as one of the great geniuses of the 20th century. Rakesh Vohra captures Nash's life and work, including his amazing letters to the NSA.

I briefly met John Nash at some MIT alumni events in New Jersey when I lived there (even though neither of us were MIT undergrads). He would come with his wife and son, the son wearing a winter coat no matter the season. Nash just seemed like any other introverted scientist and was happy to talk though understating his research ("I did some work in game theory") and never revealing the challenging life he led.

Now that John and Alicia have found their final equilibrium, may we remember them and Nash's vision of using mathematics to understand the world we live in.

Thursday, May 21, 2015

An Intentional and an Unintentional teaching experiment regarding proving the number of primes is infinite.

I taught Discrete Math Honors this semester. Two of the days were cancelled entirely because of snow (the entire school was closed) and four more I couldn't make because of health issues (I'm fine now). People DID sub for me those two and DID do what I would have done. I covered some crypto which I had not done in the past.

Because of all of this I ended up not covering the proof that the primes were infinite until the last week.

INTENTIONAL EXPERIMENT: Rather than phrase it as a proof by contradiction I phrased it, as I think Euclid did, as

Given primes p1,p2,...,pn you can find a prime NOT on the list. (From this it easily follows that the primes are infinite.)

Proof: the usual one, look at p1xp2x...xpn + 1 and either its prime or it has a prime factor not on the list.

The nice thing about doing it this way is that there are EASY examples where p1xp2x...xpn+1 is NOT prime

(e.g., the list is {2,5,11} yields 2x5x11 + 1 = 111 = 3 x 37, so 3 and 37 are both not in {2,5,11})

where as if you always use the the product of the first n primes then add 1, you don't get to a non-prime until 2x3x5x7x11x13 + 1 = 30031 = 59x 509.

They understood the proof better than prior classes had, even prior honors classes.

UNINTENTIONAL: Since I did the proof at the end of the semester they ALREADY had some proof maturity, more so than had I done it (as I usually do) about 1/3 of the way through the course.

They understood the proof better than prior classes had, even prior honors classes. Hence I should proof all of the theorems the last week! :-)

But seriously, they did understand it better, but I don't know which of the two factors, or what combination caused it. Oh well.

Monday, May 18, 2015

Theory Jobs 2015

In the fall we list theory jobs, in the spring we see who got them. Like last year, I created a fully editable Google Spreadsheet to crowd source who is going where. Ground rules:
  • I set up separate sheets for faculty, industry and postdoc/visitors.
  • People should be connected to theoretical computer science, broadly defined.
  • Only add jobs that you are absolutely sure have been offered and accepted. This is not the place for speculation and rumors.
  • You are welcome to add yourself, or people your department has hired.
This document will continue to grow as more jobs settle. So check it often.


Thursday, May 14, 2015

Fiftieth Anniversary of the Publication of the seminal paper on Computational Complexity

Juris Hartmanis and Richard Stearns in a photo dated May 1963. The main theorem from their paper is on the board later improved by Hennie and Stearns. Photo courtesy of Richard Stearns.
The seminal paper of Juris Hartmanis and Richard Stearns, On the Computational Complexity of Algorithms, appeared in the Transactions of the American Mathematical Society in the May 1965 issue. This paper gave the name to the field of Computational Complexity which I took for the name of this blog. Hartmanis and Stearns received the Turing Award in 1993 for this work.

I've mentioned this paper several times in the blog before, including as a favorite theorem. Hartmanis and Stearns first formalized and truly popularized the idea of measuring time and other resources as a function the problem size, laying the foundation for virtually every paper in computational complexity and algorithms to follow.

Both Hartmanis and Stearns wrote about those early days. The main breakthroughs for their paper started in November 1962 and on December 31 Hartmanis wrote in his logbook "This was a good year," A good year indeed.

Monday, May 11, 2015

The law of the excluded middle of the road republicans

In the book Hail to the Chiefs about the presidents, when pointing to a race between two people who had no business being president (I think it was Franklin Pierce vs Winfield Scott) wrote something like That's the thing about elections, someone has to win. 

 Looking at the republicans running for the nomination I can (with the help of reading many of Nate Silver's Columns) tell you why, for each one, they can't win the nomination. Note that this is not a partisan thing. But again, someone has to win. Is it possible to have the statements

A1 can't win AND A2 can't win AND .... AND An can't win

and yet someone wins?

Here is a list of the candidates and why they can't win.

  1. Jeb Bush is what passes for a front runner nowadays. Has the money, does not have the party (very few endorsements), and not doing well in polls in Iowa or New Hampshire, the first two states. Possibly because he does not hate Obama enough.  Its an interesting question of whether the party, the people, or the money pick the candidate. In the past its been the party and the money together, but that might be changing. CAN"T WIN: Viewed as too moderate by the people who go to Caucus's. Also some people may be put off by the family name. THOUGHT: If H CLINTON was not the likely Democratic candidate, thus making the family name thing a less of an issue in the general, I don't think he would have run at all. UPDATE- Dropped out after the South Carolina Primary. 
  2. Marco Rubio. Running for president for the first time is hard. The republicans rarely nominate someone who hadn't run before  (W in 2000, Ford in 1976 which is an out lier since he was prez). Perhaps the voters/party/money like someone they are familiar with OR perhaps first timers make mistakes. CAN"T WIN: Will make some mistake and some might think its not his turn since he's so young. Also, Senators have it rough since they sometimes vote for bills in funny ways, TRIVIA: The electoral college reps from state X cannot cast both the Prez and Veep vote for people both from state X. So you won't see a Bush-Rubio or Rubio-Bush ticket  UPDATE- Dropped out in Mid March after losing his home state to Trump. Very odd in that various pundits pointed to him as being the hope to beat Trump even though he only won Minnesota and Puerto Rico. Looks like running for prez IS hard for first times- though Trump seems to be managing..
  3. Rick Perry. He lost in 2012 because either he was too soft on immigrants (he supported some sort of Dream Act) or because of his Whoops moment in a debate. Frankly I have sympathy on that one--- I also sometimes  forget which cabinet positions I want to get rid of. He has tried to cure both of his problems by flip-flopping (or evolving) on immigration and by wearing glasses so at least he looks smart. CAN"T WIN:  His past failure makes him not look that serious this time around, and he will likely have another WHOOPS moment. UPDATE: He dropped out in September. His speech saying he was dropping out was pretty good (I am not being sarcastic).
  4. Scott Walker. Like Rubio but might be in better shape because he's a governor. Still, people in his home state are beginning to turn on him, a bad sign. He may soon have the same problem that Gov Brownback of Kansas has--- you promise to cut taxes, close loopholes, and cut spending, and that might actually be a good idea if done right, but you cut taxes , don't close loopholes, cut spending in stupid places like education, run up a big debt, destroy your states economy, and ... get re-elected. CAN"T WIN: Having not run before Walker will   say something stupid. And as a gov of a moderate state I suspect some moderate things he did may be a problem for hard core republican voters.Plus his states current problems may be an issue. UPDATE: He dropped out in September. See Note on Bobby Jindal.
  5. Ted Cruz. Which of these quotes did Ted Cruz really say and which did I hear in some satirical setting (as a fan of The Daily Show, The Colbert Report, John Oliver's show, The Capitol Steps, others, I lose track of where I heard what) (1) There was no ebola before Obamacare, or (2) Net neutrality is Obamacare for the internet. I'll answer at the end of the post. He is now using Obamacare himself. . CAN"T WIN: A niche candidate with a small but loyal set of supporters. Not enough CAVEAT:  Might be running to get some points of view out there like Adlai Stevenson and Barry Goldwater, though they got all the way to the nomination which I doubt he can.
  6. Rand Paul. Interesting mathematically: an authenticity -  electability trade off.  Some people are attracted to his libertarianism, authenticity and consistency,  but not enough to get him anywhere close to the nomination. So he changes some of his positions to be more mainstream but less libertarian, authentic, and consistent. Alas, those who trade their integrity for electability end up with neither. CAN"T WIN: His evolving views might lose him his base but not gain him any establishment cred. Also he has the chicken-egg problem in that even people who like him don't think he can win the general election. (Ted Cruz and others on this list may also have this problem.) ADDED IN FEB- DROPPED OUT AFTER IOWA CAUCUS.
  7. Mike Huckabee. Won't get much  support beyond his Social Conservative base. His stance on Same Sex Marriage will hurt him outside of the social conservatives, especially in 2016 (as opposed to his last run in 2008).  I'm surprised he's running- he got what he wanted from his last run, a show on FOX News. CAN"T WIN: Was a moderate on some economic and immigration issues (he may also `evolve' which won't work), a conservative on social issues, is just the wrong mix for current  republican primary voters. Note that many candidates are trying to avoid the Same Sex Marriage question as they know that being opposed to it will hurt them  in the general. Plus some don't want to be on the wrong side of history (or as the kids say WSOH) .  Politics- sometimes you're forced to have a public opinion that you disagree with and know will make you be on the WSOH , but you're stuck with it. I think Huckabee is sincere in his opposition to same sex marriage but he must surely know he's on the WSOH. He's hoping for a win in Iowa like he had in 2008. I predict that if he doesn't get it he'll drop out. ADDED IN FEB- DROPPED OUT AFTER IOWA CAUCUS
  8. Ben Carson. I suspect he is actually running to be a FOX News commentator. At that he might succeed. CAN"T WIN:  First timer, never ran for anything, he'll be a curiosity not a candidate. Which of the following did he say:  (1) Obama is a sociopath (2)  Obamacare is like slavery. This may even hurt him with the Republican hard core who want someone who can win. CAVEAT: is being African-American going to help or hurt? I doubt his campaign will get far enough to tell. The other candidates and the debate panelists (in the sanctioned debates) will treat him with kid gloves to avoid being called racist. UPDATE- saying absurd things about Obama seems to not hurt any candidate. ADDED LATER- he dropped out in early March, prob because of his poor showing on Super Tuesday.
  9. Carly Fiorina. She said that our founding fathers did not intend there to be a political class, what we now call politicians. They intended for ordinary people (like the president of HP), for the good of their community, to serve in office. She left out that the founding fathers also did not intend for women to be president. CAN"T WIN: First timer, never ran for anything.  (correction added later: She ran for Senator of California . She got the nomination but lost to Incumbent Barbara Boxer.) CAVEAT: is being female going to help or hurt? I doubt her campaign will get far enough to tell. The other candidates and debate panelists (in the sanctioned debates) will treat her with kid gloves to avoid being called sexist. HER HOOK: She claims that as a women she can neutralize H Clinton' women-advantage in the general. Interesting that she is making an electability argument instead of a policy argument, given that she has no chance of being elected. ADDED LATER- Dropped out after the NH primary.
  10. Chris Christie. CAN"T WIN: Hated inside of New Jersey. Hated outside of New Jersey.ADDED LATER- Dropped out after the NH primary.
  11. Bobby Jindal. Once said the Republicans have to stop being the stupid party. Later said some stupid things about Muslims in America. CAN"T WIN: If he ran as the moderate sane voice who will rescue the party from itself, he might get some traction. If he runs as anything else he has too much competition. Also a first-time-runner which is hard. UPDATE- He dropped out in November. He was in double digits (not his percent- the number of people who wanted him to be Prez- his wife, one of his kids, the other is in the Cruz-Camp.) He tried to be BOTH Tea-party AND establishment which was foreshadowed by saying Reps shouldn't be the stupid party (an establishment thing to say) and being racists towards Muslims (a Tea-Party thing). Was not liked by either. Scott Walker may have had the same problem. Rubio may have the same problem.
  12. Lindsey Graham. CAN"T WIN:  Has  worked with democrats which should be a PRO but it's a CON. He's running to be the voice of more troops-on-the-ground in the mideast, not to win. Since Americans don't really want troops on the ground I doubt this will go anywhere; however, instead of troops the other candidates will show their machismo by deporting Muslims. UPDATE- He Dropped out.
  13. John Kasich. Gov of Ohio. CAN"T WIN:  Not that well known. Democrats have nominated unknowns (B Clinton, B Obama) but republicans almost never do (W might count).
  14. Donald Trump (ADDED LATER). Some people say that corporate America controls this country. If we make Donald Trump Prez we're just cutting out the middle-man. Won't get the nomination--- over half of republicans say they would never vote for him--- but his running is a farewell gift to Jon Stewart.
  15. Rick Santorum (ADDED LATER). Against Birth Control? Really? Google him to find other reasons he can't win. Odd thing- usually the person who came in second the last time around has a pretty good shot at getting the nomination this time around, but the fact that he came in second last time may be a sign of how weak the field was last time. Is hoping for a win in Iowa like he had in 2012. I predict that if he doesn't do well in Iowa he'll drop out. UPDATE- DROPPED OUT AFTER THE IOWA CAUCUS.
  16. George Pataki (ADDED LATER) Would be at home in the democratic party. I want to see him challenge Hillary from the right. Oh, he's a republican? But he's pro-choice, pro-gay, anti-gun, and doesn't seem to hate Obama. UPDATE- DROPPED OUT.
  17. Rob Portman. (ADDED LATER) I saw him on a list of possible nominees. Not under `declared', not under `exploratory committee) (Chris Christe and Scott Walker are still exploring, which surprised me- I thought they already declared), but on a list of `No indication'. Not sure why he's on any list; however, as Rick Perry taught us last time (and this is not a joke) you need to start early.UPDATE- never got into the race.
  18. Jim Gillmore (ADDED LATER).  Never made it into the adult debates nor even the kiddie table. Do not know why he is running since he does not have a chance and is not trying to push some sort of issue (that may be why Rand Paul is running- to promote a viewpoint) UPDATE: In the least important political story of the year, Jim Gillmore dropped out after the NH primary. 

UPDATE IN FEB: After NH primary its  down to 6. Maybe now they CAN all fit on a debate stage!

UPDATE IN FEB: After SC primary Jeb dropped so its down to 5. They can now all fit into a car!

UPDATE ON MAR 2. Ben Carson dropped out so its down to 4. Now they can have an intelligent debate.

UPDATE ON MAR 4. They spend some of the debate talking about the size of Donald Trump's... hands.

UPDATE ON MAR 16. Rubio Dropped out. Now its just Trump, Cruz, Kasich. They can all fit in
a Volkswagon.

There may be more (Donald Trump anyone? ADDED LATER- When I first posted this mentioning Donald Trump was supposed to be a joke. But political satire and reality may have finally merged with a reality-TV star running for Prez). But my point is that it seems like nobody can win, yet someone has to.  Do you know other examples where A OR B OR C has to be true, yet none of A,B,C look plausible?

I can phrase this another way:

novices can't win  (e.g., Ben Carson) AND first timers can't win (e.g., Mario Rubio) AND too moderate can't win (e.g., perception of Jeb) AND unknowns can't win (e.g., John Kasich).

They all can't be right, but looking at it   `first timers' is prob the weak link in my reasoning. I could replace it with `inexperienced politician'. Even so, sure looks like nobody can win.

Ted Cruz: Both of the quotes I attribute to him he really did say.
I don't know Ben Carson's original quote but he backtracked to Obama reminds you of a psychopath,
which is much better than saying Obama IS a psychopath . But he never said sociopath, so the quote I gave is NOT from Ben Carson.

On Obamacare he said its the worst thing to happen in America since slavery. But later
opposite-of-backtracking said it was in a way like slavery because it robs you of  your ability to control your own life.

Thursday, May 07, 2015

The Virtuous Cycle

Last week we have a celebration of the new CS graduates at Georgia Tech where each graduating student says what their plans are for next year. Other than the few going to grad school, they all have great jobs but the difference I see this year is how many are staying in Atlanta. Any CS student who wants to stay in Atlanta can find a great job in Atlanta.

A large number of companies are moving their headquarters or established large facilities in the Atlanta area and the reasons they give almost always include the students and researchers at Georgia Tech. We're starting to grow a good start-up culture here as well. 

Companies in Atlanta want our students and our research. That helps add to the prestige of Georgia Tech which in turn draws more companies. A virtuous cycle. A similar story is playing out in several other cities across the country and I even saw it in tiny but growing Bozeman where I visited Montana State earlier this week. Computer Science these days plays a major role if not the largest role in many of these industries.

All this growth leads to challenges such as finding the people and resources to meet the growing demands. All told though a good problem to have.

We don't want the success of the STEM fields to come at the expense of the rest of academics. It shouldn't have to be CS vs Classics, Physics or Philosophy. How we make it all successful will be the great challenge in higher education in the years to come.

Monday, May 04, 2015

Sizes of DFAs, NFAs, DPDAs, UCFG, CFGs, CSLs.

If A is a decider (e.g, DFA)  or generator (e.g., CFG) then L(A) is the language  that it decides or generates.

The following are well known:

L(DFA) = L(NDFA) ⊂ L(DPDA) ⊂ L(PDA) ⊂ L(CSL).

We are concerned with the  size of these devices. For a DFA and NDFA the size is
the number of states, for a DPDA, PDA the size is the sum of the stack alphabet and the number of states, and for CSL its the number of nonterminals.

If D and E are two classes of devices (e.g., DFAs and DPDAs) then a bounding function for (D,E)  is a function f such that if L is recognized by both a D-device and an E-device, and L is recognized by an E-device of size n, then it is recognized by a D-device of size ≤ f(n). We abbreviate b-fun

Readers of this column likely know that f(n)=2^n is a b-fun for (DFA,NFA) and prob know that this is tight. Below are some results that I suspect many readers don't know. Some of them  may be suitable for inclusion in an undergrad theory class. In what is below ≤ means Turing-Less-Than-Or-Equal.

  1.  Stearns showed that f(n) = n^{n^{n^{O(n)}}} is a b-fun for (DFA,DPDA).
  2.  Valiant improved this to double exp for a b-fun for (DFA,DPDA).
  3.  Meyer and Fischer showed the 2^n lower bound for (DFA,NDFA). They also showed a lower bound of 2^{2^{O(n)}} for (DFA,DPDA). I think the question of closing the gap between Valiant's result and the Meyer-Fischer result is still open; however, if you know a ref please leave a comment.
  4. Valiant showed that the if f is a b-fun for (DPDA,UCFG) then HALT ≤  f.
  5. Schmidt showed that if f is a b-fun for (UCFG,CFG) then HALT ≤  f. 
  6. Hartmanis showed that if f is a b-fun for (DPDA,PDA) then HALT ≤ f
  7. Hay  showed that if f is a b-fun for (DPDA,PDA) then f is NOT computable-in-HALT. 
  8. Beigel and Gasarch prove a general theorem from which they obtain the following: (a) if f is a b-fun  for (DPDA,PDA) then f ≤_INF. (It is easy to show that there exists a b-fun f ≤ INF for (DPDA,PDA) so the Turing degree is now precise), and (b) if f is a b-fun for (PDA,CSL) then SAME AS PART (a).
Results 3,4,5,6,7 can be restated in the following way--- we'll do result 6 as an example:

 If f  ≤  HALT then there exists infinitely many n such that there exists L_n such that  (a) L_n is DPDA, (b) there is a PDA for L_n of size n, (c) any DPDA for L_n is of size  at least f(n).

 Are there any ``for almost all n '' type bounds? There are but they are much weaker.  The following theorems are from the Beigel-Gasarch paper pointed to above.

  1. For almost all n there exists a  cofinite (hence DPDA) L_n that has a PDA of size O(n) but any DPDA for it has size 2^2^{n^{Ω(1)}}. 
  2. Same as point 1 but for PDA,CSL.

Both results 1,2 above use natural languages  in that they are not created for the sole purpose of proving the theorem, no diagonalization (my spell check says thats spelled wrong but I think its spelled right). Using a construction Beigel-Gasarch obtained (Meyer probably had the result 40 years earlier with a diff proof, see the Beigel-Gasarch paper  for historical details) that if f  ≤  HALT then for almost all n there is a lang L_n such that L_n has a CSL of size n but any PDA for it is of size at least f(n).