A Metric Group Product

A guest post by Dylan McKay, recently graduated from Georgia Tech and soon to be PhD student at Stanford.

[Editor's Note: Turns out the given solution doesn't work and whether a metric group product over the non-negative reals exists remains open. Update 7/10/18: A non-constructive solution was posted on Mathoverflow back in 2010]

Here a cute puzzle motivated by a pair of undergrads and their poor understanding of what the phrase “Algebraic Geometry” really should mean:

Find a function f from the nonnegative reals to the nonnegative reals that satisfies the group axioms and the metric axioms, or prove that there is no such function. That is, find an f:RxR→R such that (R,f) is a group and a metric space. (I am using R to refer to the set of nonnegative real numbers).

As a quick reminder, the group axioms are:

- Closure: f(a,b) must be in R

- Identity: There must exist an element e such that for all a, f(e,a)=f(a,e)=a

- Associative: for all a,b,c, f(f(a,b),c) = f(a,f(b,c))

- Inverse: for all a, there must exist a b such that f(a,b)=e

And the metric axioms are:

- f(a,b) = 0 iff a=b

- f(a,b) <= f(a,c) + f(c,b)

- f(a,b) = f(b,a)

One really bizarre thing about this supposed function is that, what this metric does is essentially guarantee that, given some number x, every other number has a distance from x that is unique -- no other number is exactly that distance away! This is quite counterintuitive to how we think about distance. Each number is its own distance from 0, though, which is very much in line with intuition.

(If you want to solve the puzzle yourself, don't read any further until you do! Unless you want some hints, in which case, look at the next paragraph.)

We can, however, make some other observations that point us in the right direction. For instance, f(e,e) = 0 from the metric axioms and f(e,e) = e from the group axioms, so you get that e = 0. From this and the metric axioms, you get that every element must be it’s own inverse!

Here, we are reminded of a pretty familiar and trusty function, this exclusive-or (XOR) function! And indeed, this will lead us a to a solution.

Here is our candidate function:

Consider x and y in their binary representations. If they do not have the same number of bits to the left of the decimal point (or should it be called the binary point?), pad the one with fewer such bits with 0’s so they have the same number of such bits. Then f(x,y) will be the number represented in binary by the bit-wise exclusive or of these two strings (maintaining the same number of bits to the left of the decimal point, of course). And there we have it: our function!

Now of course, we still need to prove that it satisfies the axioms. But if you do not believe that it does, work it out for yourself! Each axiom is pretty simple to check, especially as we are (for the most part) familiar with this as an extension of an idea for some smaller groups. Well, all of them except our good friend the triangle inequality. This one actually isn’t too bad, but if you have trouble, I will include a hint at the bottom of the post.

Anyway, I hope you liked our puzzle!

Thanks!

-Dylan

Hint for triangle inequality: XOR and addition are really similar, but there is a key difference between them that make a XOR b <= a + b.

The city where the book publishers resides is Funkytown!

A while back  when I got back the galleys for a paper the publisher wanted to know the complete postal address of one of the co-authors and also the city where the publisher of a book in he bibliography was printed. The publisher didn't really have a city, so I wrote in FUNKYTOWN.

Is it important in a paper to have the city that the publisher is in? I am assuming that at one time it was (though I am not sure why even that is) but now it is
completely irrelevant.

Is it important to have an authors postal address? I can't imagine why nowadays in 2015, though this I can imagine as being important in an earlier era.


Question: What information do publishers ask for in galleys that are no longer relevant? Why do they? When will they stop?

Langs with provably bigger CFG's then CSG's

In a prior blog entry HERE I discussed very large differences in the size of machines. I didn't discuss CFG vs CSG so I'll do that now. We assume that the CFGs and CSGs are in Chomsky Normal Form (for CSG this means that RHS is any rule is of length 1 or 2).

The following are known:

1)  Let PERMn be the set of permutations of {1,...,n} (so  PERM3 is {123,132,213,231,312,321}). Then (1) there exists a CSG for PERMn of size O(n²), but (2) every CFG for PERMn requires size 2^{Ω(n)  .} The upper bound is easy, the lower bound is by Ellul, Krawetz, Shallit, Wang Here.

2) Let Wn = { ww : |w|=n}.  Then (1) there exists a CSG for Wn of size O(n), but (2) every CFG for Wn requires size  2^Ω(n). The upper bound we leave to the reader.    Filmus HERE for the lower bound.(ADDED LATER: In Comment Below Filmus (yes, the same one!) claims Wn has a CSG of size O(log n)).

3) Let Sn = { w : |w|=3n and the numbers of a's, b's, c's in w are all the same}. Then (1) there exists a CSG for Sn of size O(log n) but (2) every CFG for Wn requires size 2^Ω(n). The upper bound is  from Beigel and Gasarch (If you know of an earlier source leave a polite comment.)  The lower bound is from Filmus (the ref above).(ADDED LATER- In Comment Below Filmus (yes, the same one!) corrects me, his paper did not show Sn has exp lower bound, and in fact Sn DOES have a CFG of size O(n^3)).

4)  The languages above are (informally) natural. If one goes to unnatural languages then there is a result where the CFG is GINORMOUS:  For all f  ≤_T HALT, for all n, there is a lang Ln such that (1) there exists a CSG for Ln of size n, but (2) every CFG for Ln has  size ≥  f(n). First proven by Meyer, but a new proof by Beigel and Gasarch is in the paper pointed to above. (See that paper for the history.)

The results stated above are suitable for an ugrad formal lang theory course.

Are there natural langs with triple-exp or larger gap between CFG's and CSG's? There are very few techniques to get lower bounds for CFG's (the papers of Ellul et al, and Filmus, are the only ones I know) so  new techniques may be needed. Are  there even any good candidates for natural languages  with small CSG's and really really large CFG's?

Award Season

László Babai will receive the 2015 Knuth Prize and Daniel Spielman and Shang-Hua Teng will receive the 2015 Gödel Prize. ACM issued a press release for both awards which will be presented at the upcoming STOC at FCRC.

Babai did seminal research on interactive proofs, communication complexity, group algorithms and much more. One cannot count the number of PhD theses in mathematics and computer science that can trace themselves back to some initial work by Babai. I was lucky to have Laci Babai as a colleague, mentor and friend during my years at the University of Chicago.

Spielman and Teng, who received the 2008 Gödel Prize for smooth analysis, won again for three papers using nearly linear time Laplacian solvers for a series of graph problems.

• Spectral sparsification of graphs. SIAM J. Computing 40:981-1025, 2011.
• A local clustering algorithm for massive graphs and its application to nearly linear time graph partitioning. SIAM J. Computing 42:1-26, 2013.
• Nearly linear time algorithms for preconditioning and solving symmetric, diagonally dominant linear systems. SIAM J. Matrix Anal. Appl. 35:835-885, 2014. 

The ACM Awards ceremony later this month will have a number of theory related prizes.

• Dan Boneh is the recipient of the ACM-Infosys Foundation Award in the Computing Sciences for the development of pairing-based cryptography and its application in identity-based encryption.
• James Demmel is the recipient of the Paris Kanellakis Theory and Practice Award for his work on numerical linear algebra libraries, including LAPACK.
• Charles Leiserson will receive the ACM-IEEE CS Ken Kennedy award for his work on parallel computing.
• Jon Kleinberg is the 2014 recipient of the ACM – AAAI Allen Newell Award for social and information networks, information retrieval, and data science, and for bridging computing, economics and the social sciences.
• New ACM Fellows include Al Borodin, Faith Ellen, Michael Kearns, Valerie King, Yishay Mansour, Michael Mitzenmacher, Omer Reingold, Ronitt Rubinfeld and Aravind Srinivasan.

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.

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.

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.

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.

Edit

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.

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.

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.

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).

Is Logarithmic Space Closed Under Kleene Star?

A Georgia Tech student asked the title question in an introductory theory course. The instructor asked his TA, the TA asked me and I asked the oracle of all things log space, Eric Allender. Eric didn't disappoint and pointed me to Burkhard Monien’s 1975 theorem

L is closed under Kleene star if and only if L = NL.

L here is the set of problems solved in deterministic O(log n) space and NL is the nondeterministic counterpart. For a set of strings A, Kleene star, denoted A^* is the set of all finite concatenations of strings of A. For example if A = {00,1} then A^* = {ε, 1, 00, 11, 001, 100, 111, 0000, 0011, 1100, …} where ε is the zero-length string.

Kleene star comes up in regular expressions but also makes for many a good homework problem.

1. Show that if A is in NL then A^* is also in NL.
2. Show that if A is in P then A^* is also in P.
3. Show that if A is c.e. (recognizable) then A^* is also c.e.

Problem 1 above is equivalent to saying NL is closed under Kleene star and implies the “if” part of Monien’s result. Here is a simple proof of the other direction that L closed under Kleene star implies L = NL.

Consider the following NL-complete problem: The set of triples (G,s,t) such that G is a directed graph with a restriction that all edges (i,j) have i<j and there is a path from node s to node t.

Define the following language

B = {G#i+1#G#i+2#...#G#j# | there is an edge (i,j) in G}

B is computable in log space and the string G#s+1#G#s+2#…#G#t# is in B^* if and only if there is a path from node s to node t. QED

Allender, Arvind and Mahajan give some generalizations to log-space counting classes and also notes that there are languages computable in AC⁰ (constant-depth circuits) whose Kleene star is NL-complete. B above is one such set.

Advice on running a theory day

Last semester I ran a THEORY DAY AT UMCP. Below I have ADVICE for people running theory days. Some I did, some I didn't do but wish I did, and some are just questions you need to ask yourself.

1) Picking the day- I had two external speakers (Avrim Blum and Sanjeev Arrora) so I was able to ask THEM what day was good for THEM. Another way is to pick the DAY first and then asking for speakers.

2) Number of external speakers- We had two, and the rest were internal. The external speakers had hour-long talks, the internal had 20-minute talks. This worked well; however, one can have more or even all external speakers.

3) Whoever is paying for it should be told of it towards the  beginning of the process.

4) Lunch- catered or out? I recommend catered if you can afford it  since good time for people to all talk. See next point.

5) If its catered you need a head count so you need people to register. The number you get may be off- you may want to ask when they register if they want lunch. Then add ten percent.

6) Tell the guest speakers what time is good for them to arrive before they make plans so you can coordinate their dinner the previous night.

7) If have the money and energy do name tags ahead of time. If not then just have some sticky tags and a magic marker.

8) Guest speakers- getting them FROM Amtrak or Airport to dinner/hotel --- give them a personal lift (they may be new to the city and a bit lost). Getting them from the event back TO the airport or amtrak, can call airline limo and taxi. (though if can give a ride, that's of course good.)

9) Pick a day early and stick with it. NO day is perfect, so if someone can't make it, or there is a faculty meeting that day, then don't worry about it.

10) Have website, speakers, all set at least a month ahead of time. Advertise on theory-local email lists, blogs you have access to, and analogs of theory-local for other places (I did NY, NJ, PA). Also email people  to spread the word.

11) Advertise to ugrads. Students are the future!

12) If you are the organizer you might not want to give a talk since you'll be too busy doing other things.

13) Well established regular theory days (e.g., NY theory day) can ignore some of the above as they already have things running pretty well.

Fifty Years of Moore's Law

Gordon Moore formulated his famous law in a paper dated fifty years and five days ago. We all have seen how Moore's law has changed real-world computing, but how does it relate to computational complexity?

In complexity we typically focus on running times but we really care about how large a problem we can solve in current technology. In one of my early posts I showed how this view can change how we judge running time improvements from faster algorithms. Improved technology also allows us to solve bigger problems. This is one justification for asymptotic analysis. For polynomial-time algorithms a doubling of processor speed gives a constant multiplicative factor increase in the size of the problem we can solve. We only get an additive factor for an exponential-time algorithm.

Although Moore's law continues, computers have stopped getting faster ten years ago. Instead we've seen the rise of new technologies: GPUs and other specialized processors, multicore, cloud computing and more on the horizon.

The complexity and algorithmic communities are slow to catch up. With some exceptions, we still focus on single-core single-thread algorithms. Rather we need to find good models for these new technologies and develop algorithms and complexity bounds that map nicely into our current computing reality.

The New Oracle Result! The new circuit result! which do you care about?

You have likely heard of the new result by Ben Roco, and Li-Yang on random oracles (see here for preprint) from either Lance or Scott or some other source:

Lance's headline was PH infinite under random oracle

Scott's headline was Two papers but when he stated the result he also stated it as a random oracle result.

The paper itself has the title

An average case depth hierarchy theorem for Boolean circuits

and the abstract is:

We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy theorem says that for every d≥2, there is an explicit n-variable Boolean function f, computed by a linear-size depth-d formula, which is such that any depth-(d−1) circuit that agrees with f on (1/2+on(1)) fraction of all inputs must have size exp(nΩ(1/d)). This answers an open question posed by Hastad in his Ph.D. thesis.
Our average-case depth hierarchy theorem implies that the polynomial hierarchy is infinite relative to a random oracle with probability 1, confirming a conjecture of Hastad, Cai, and Babai. We also use our result to show that there is no "approximate converse" to the results of Linial, Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus answering a question posed by O'Donnell, Kalai, and Hatami.
A key ingredient in our proof is a notion of \emph{random projections} which generalize random restrictions.

Note that they emphasize the circuit aspect.

In Yao's paper where he showed PARITY in constant dept requires exp size the title was

Separating the polynomial hiearchy by oracles

Hastad's paper and book had titles about circuits, not oracles.

When Scott showed that for a random oracle P^NP is properly in Sigma_2^p the title was

Counterexample to the general Linial-Nissan Conjecture

However the abstarct begins with a statement of the oracle result.

SO, here is the real question: What is more interesting, the circuit lower bounds or the oracle results that follow? The authors titles and abstracts might tell you what they are thinking, then again they might not. For  example, I can't really claim to know that Yao cared about oracles more than circuits.

Roughly speaking the Circuit results are interesting since they are actual lower bounds, often on reasonable models for natural problems (both of these statements can be counter-argued), oracle results are interesting since they give us a sense that certain proof teachniques are not going to work. Random oracle results are interesting since for classes like these (that is not well defined) things true for random oracles tend to be things we think are true.

But I want to hear from you, the reader: For example which of PARITY NOT IN AC_0 and THERE IS AN ORACLE SEP PH FROM PSPACE do you find more interesting? Is easier to motivate to other theorists? To non-theorists (for non-theorists I think PARITY).

PH Infinite Under a Random Oracle

Benjamin Rossman, Rocco Servedio and Li-Yang Tan show new circuit lower bounds that imply, among other things, that the polynomial-time hierarchy is infinite relative to a random oracle. What does that mean, and why is it important?

The polynomial-time hierarchy can be defined inductively as follows: Σ^P₀ = P, the set of problems solvable in polynomial-time. Σ^P_i+1 = NP^ΣP_i, the set of problems computable in nondeterministic polynomial-time that can ask arbitrary questions to the previous level. We say the polynomial-time hierarchy is infinite if Σ^P_i+1 ≠ Σ^P_i for all i and it collapses otherwise.

Whether the polynomial-time hierarchy is infinite is one of the major assumptions in computational complexity and would imply a large number of statements we believe to be true including that NP-complete problems do not have small circuits and that Graph Isomorphism is not co-NP-complete.

We don't have the techniques to settle whether or not the polynomial-time hierarchy is infinite so we can look at relativized worlds, where all machines have access to the same oracle. The Baker-Gill-Solovay oracle that makes P = NP also collapses the hierarchy. Finding an oracle that makes the hierarchy infinite was a larger challenge and required new results in circuit complexity.

In 1985, Yao in his paper Separating the polynomial-time hierarchy by oracles showed that there were functions that had small depth d+1 circuits but large depth d circuits which was needed for the oracle. Håstad gave a simplified proof. Cai proved that PSPACE ≠ Σ^P_i for all i even if we choose the oracle at random (with probability one). Babai later and independently gave a simpler proof.

Whether a randomly chosen oracle would make the hierarchy infinite required showing the depth separation of circuits in the average case which remained open for three decades. Rossman, Servedio and Tan solved that circuit problem and get the random oracle result as a consequence. They build on Håstad's proof technique of randomly restricting variables to true and false. Rossman et al. generalize to a random projection method that projects onto a new set of variables. Read their paper to see all the details.

In 1994, Ron Book showed that if the polynomial-time hierarchy was infinite that it remained infinite relativized to a random oracle. Rossman et al. thus gives even more evidence to believe that the hierarchy is indeed infinite, in the sense that if they had proven the opposite result than the hierarchy would have collapsed.

I used Book's paper to show that a number of complexity hypothesis held simultaneously with the hierarchy being infinite, now a trivial consequence of the Rossman et al. result. I can live with that.

Baseball is More Than Data

As baseball starts its second week, lets reflect a bit on how data analytics has changed the game. Not just the Moneyball phenomenon of ranking players but also the extensive use of defensive shifts (repositioning the infielders and outfielders for each batter) and other maneuvers. We're not quite to the point that technology can replace managers and umpires but give it another decade or two.

We've seen a huge increase in data analysis in sports. ESPN ranked teams based on their use of analytics and it correlates well with how those teams are faring. Eventually everyone will use the same learning algorithms and games will just be a random coin toss with coins weighted by how much each team can spend.

Steve Kettmann wrote an NYT op-ed piece Don't Let StatisticsRuin Baseball. At first I thought this was just another luddite who will be left behind but he makes a salient point. We don’t go to baseball to watch the stats. We go to see people play. We enjoy the suspense of every pitch, the one-on-one battle between pitcher and batter and the great defensive moves. Maybe statistics can tell which players a team should acquire and where the fielders should stand but it still is people that play the game.

Kettmann worries about the obsession of baseball writers with statistics. Those who write based on stats can be replaced by machines. Baseball is a great game to listen on the radio for the best broadcasters don't talk about the numbers, they talk about the people. Otherwise you might as well listen to competitive tic-tac-toe.

FCRC 2015

Every four years the Association for Computing Machinery organizes a Federated Computing Research Conference consisting of several co-located conferences and some joint events. This year's event will be held June 13-20 in Portland, Oregon and includes Michael Stonebraker's Turing award lecture. There is a single registration site for all conferences (early deadline May 18th) and I recommend booking hotels early and definitely before the May 16th cutoff.

Theoretical computer science is well represented.

• 47th ACM Symposium on the Theory of Computing. Apply for student travel support by May 9th. 
• 30th Computational Complexity Conference, now an independent conference. Student travel support deadline of May 9th. CCC is looking for a new logo, if you have ideas send them to Dieter van Melkebeek.
• 16th ACM Conference on Economics and Computation and its associated workshops and tutorials
• 27th ACM Symposium on Parallelism in Algorithms and Architectures
• A plenary lecture by Andy Yao

The CRA-W is organizing mentoring workshops for early career and mid-career faculty and faculty supervising undergraduate research.

A number of other major conferences will also be part of FCRC including HPDC, ISCA, PLDI and SIGMETRICS. There are many algorithmic challenges in all these areas and FCRC really gives you an opportunity to sit in talks outside your comfort zone. You might be surprised in what you see.

See you in Portland!

Two cutoffs about Warings problem for cubes

Known:

1. All numbers except 23 can be written as the sum of 8 cubes
2. All but a finite number of numbers can be written as the sum of 7 cubes 
3. There are an infinite number of numbers that cannot be written as the sum of 3 cubes(this you can prove yourself, the other two are hard, deep theorems).

Open: Find x such that:

1. All but a finite number of numbers can be written as the sum of x cubes.
2. There exists an infinite number of numbers that cannot be written as the sum of x-1 cubes.

It is known that 4 ≤ x ≤ 7

Lets say you didn't know any of this and were looking at empirical data.

1. If you find that every number ≤ 10 can be written as the sum of 7 cubes this is NOT interesting because 10 is too small.
2.  If you find that every number ≤ 1,000,000 except 23 can be written as the sum of 8 cubes this IS interesting since 1,000,000 is big enough that one thinks this is telling us something (though we could be wrong). What if you find all but 10 numbers (I do not know if that is true) ≤ 1,000,000 are the sum of seven cubes?

Open but too informal to be a real question: Find x such that

1. Information about sums-of-cubes for all numbers ≤ x-1 is NOT interesting
2. Information about sums-of-cubes for all numbers ≤ x IS interesting.

By the intermediate value theorem such an x exists. But of course this is silly. The fallacy probably relies on the informal notion `interesting'. But a serious question: How big does x have to be before data about this would be considered interesting? (NO- I won't come back with `what about x-1').

More advanced form: Find a function f(x,y) and constants c1 and c2 such that

1. If f(x,y) ≥ c1  then the statement all but y numbers ≤ x are the sum of 7 cubes is interesting.
2. If f(x,y) ≤  c2 then the statement all but y numbers ≤ x are the sum of 7 cubes is not interesting. 

To end with a more concrete question: Show that there are an infinite number of numbers that cannot be written as the sum of 14 4th powers.