Thursday, October 16, 2014

The Curious Case of NP and NEXP

NP (nondeterministic polynomial time) and NEXP (nondeterministic exponential time) are provably different classes by the nondeterministic time hierarchy. No surprise, given exponentially more time we expect to solve more problems. But the proof requires collapses at many input lengths and odd things happen when we look at the infinitely-often question.

We say a language L is in i.o.-C for a complexity class C if there is an A in C such that for infinitely many n, A and L agree on strings of length n (for all x of length n, x is in A if and only if x is in L). Straightforward diagonalization shows that EXP is not in i.o.-P.

However we showed a relativized world where NEXP is in i.o.-NP in a recent paper (Theorem 18).
The construction is not particularly difficult but rather surprising: There is a possibility that one can get exponential improvement for nondeterministic computation for infinitely many input lengths.

Also consider the following facts: NEXP is not in i.o.-co-NP by straight diagonalization. If NEXP is is in i.o.-NP then
  • NEXP is in i.o.-EXP
  • EXP is in i.o.-NP and thus EXP is in i.o.-co-NP (since EXP is closed under complement).
This is not an immediate contradiction since i.o. inclusion is not transitive, even though those i.o.'s happen at about the same length. You can't combine these relativizable statements to prove NEXP is not in i.o.-NP.

NEXP in i.o.-NP is not one of my most complicated relativization results, but definitely one of the strangest.

Monday, October 13, 2014

Luddite or not?

My first ever guest post for Lance was on Are you a luddite. I certainly am to some extent a luddite, but there are some things where it not clear if they are luddite-ish or not.

  1. I prefer reading books to blogs. This came up when I reviewed both Lipton and Lipton-Regan blog-books, and I am now reading some of Terry Tao's Blog book.  l look forward to reading Scott's Blog book. At first I thought that preferring books was luddite-ish. But some high tech people and some young people who I've asked AGREE with me. Why is this?
  •  when reading a blog (or doing anything on line) its so easy to get distracted, e.g. OH, I WONDER IF WHITEY FORD IS STILL ALIVE SO I"LL PUT HIM ON MY LIST OF LIVING FAMOUS PEOPLE OVER 80 (he is, he's 85, and has the same birthday (though not year) as Martin Gardner), OH, I wonder if the word Buypartisan (that is NOT misspelled) is on my list-of-cool-new-words that I keep, OH I wonder how many people have registered for Theory Day. OH, Lipton just posted about Definitions not being needed and used that quote from The Treasure of Sierra Madre (see here) that was satirized in the movie UHF, I wonder if that clip is on You-Tube (It is here). OH, I can write a blog about Math in Weird-Al songs, for example Polka Patterns.
  • If I read a blog with a proof in it I tend to say  I'll read that later.
  •  I work better with pen and paper on hand. This may change if the way to mark up pdf and other documents gets better.
  1. (I do not why it restarted at number 1. I don't care to fix it- is that Luddite or not wanting to waste time on something unimportant?)
  2. Of course, the blog reading issue  is MY fault for being distracted.
  3. I don't pay my bills on line. There have been many data breaches and that gets darling and I nervous. Is this Luddite? Not sure--- is banking off-line any safer? I ask non-rhetorically.
  4. In a small class I use the blackboard. Some of my systems faculty have gone from board to slides and then back to board.  For a big class I have to use slides, though that may be an argument for small classes.

BILL: When I goto that conference I am going to bring some math to read during the talks I don't understand.

DARLING: Isn't that rude?

BILL: Many people in the audience will have their laptops out, reading email, managing their facebook page, etc.

DARLING: But the speaker can at least imagine they are taking notes

BILL: Unlikely. In the next talk the speaker will become the laptop person.

The fact that I don't look at a laptop during a talk is probably a plus- and not a Luddite thing.
  1. We still don't have Netflix. We watch less TV this way? Worse TV this way? Not clear how this one goes.
  2. I used to write things out before typing them in, now I type them in directly. I wonder if that's good or bad.
  3. I used to have notebooks of random math stuff in them. Now I can't get myself to write things out by hand. That's probably bad.
  4. If someone asks me a question I am too quick to goto the web rather than try to answer it myself. This is mixed--- I don't waste time on problems I can't solve, but I also don't have the joy of solving them. I think of myself as not being a good problems-solver, but this could be a self-fulfilling prophecy that the web makes easier to indulge in.
  5. This is a DUH-comment- I hate technology that does not work. One of worst episodes of Star Trek was The Ultimate Computer which showed that a good human is better than a MALFUNCTIONING computer. Well DUH. I had a rant about electronic refereeing refereeing- and not a single comment accused me of being a Luddite. In short- I hate technology that doesn't work. Duh.
So- your thoughts? Some Luddite things may not be Luddite, but just better. And technology will change yet again, making us all Luddites.

Thursday, October 09, 2014

2014 Fall Jobs Post

Tis the season for the fall jobs post. Please list any jobs, academic or industrial, in theoretical computer science broadly construed in the comments to this post. If you are a job seeker check this page often as new jobs get added over time.

As always the best places to look for academic CS positions are the job sites at the CRA and the ACM. Also check out postdoc and other opportunities on Theory Announcements. It never hurts to check out the webpages of departments or to contact people to see if positions are available.

I expect the computer science market to be quite robust again. CS enrollments continue to explode putting pressure to hire more faculty. Several good places last year had open positions unfilled.

We should also see a growth in instructor positions, as deans and provosts need to meet enrollment demands but aren't ready to commit faculty positions until they are sure CS is not in another bubble. Don't ignore these positions, think of an instructor as a teaching postdoc.

The market in theoretical computer science is a harder call. What will be the effect of Microsoft adding fifteen theorists to the market? That also suggests fewer industrial research and postdoc opportunities in theory.

Try to make yourself more versatile. Perhaps you could teach machine learning or computer security, areas of strong need closely related to theory.

Good luck to everyone on the market. I look forward to seeing your names in the 2015 spring jobs post.

Monday, October 06, 2014

The Complexity of NIM. Open?

Recall 1-pile NIM:

Let A be a finite set of Naturals. NIM(A) is the following game: There are n stones on the board. Players I and II alternate removing a\in A stones. The first player who can't win loses. Note that if 1\in A then `can't move' means that the other player took the last stone. If (say) 2 is the min elt of A then its possible there is 1 stone on the board and a player can't move.


The following  are known and easy to prove:
  1. If A={1,L} and L is even then II wins iff n\equiv 0,2,4,...,L-2 mod L+1
  2. If A={1,L,L+1} and L is odd then II wins iff n\equiv 0,2,4,...L-1 mod 2L+1
  3. If  A={1,L,L+1} and L is even then II wins iff n\equiv o,2,4,...,L-2 mod 2L
  4. If A= {L,...,M} then II wins iff n\equiv 0,2,4,...,L-2 mod L+1
  5. For ANY set A there will be a mod pattern, after a certain point.
(I think that if 1\in A then the mod pattern goes from the beginning, but if 1\notin A then its possible that it does not start for a while.)

This raises the following computational problem: How hard is the problem of, given finite set A, find the mod pattern. I would want to know the complexity as a function of the size of the representation of A, or possibly just |A|log_2(max elt of A).  Has this been looked at? Some Google searches and asking around did not yield anything. I'm hoping that asking my readers may yield something.

Thursday, October 02, 2014

Favorite Theorems: Multilinear Circuits

In the past decade we have seen a strong program in algebraic circuit complexity. If you just define circuits using multiplication and addition gates, sometimes you can use the algebraic structure to get lower bounds that prove much more difficult in the Boolean case. For October's favorite theorem we highlight a couple of Ran Raz's results.

The main result of the first paper is basically in the title. Raz creates a polynomial function f that can be computed by a polynomial multilinear circuit of depth O(log2 n) but cannot be computed by any multilinear formula. (A formula, unlike a circuit, cannot use the output of a gate more than once). His proof works by examining the rank of the partial derivative matrix of a function chosen in a specified random way.

Ran Raz followed up with Amir Shpilka and Amir Yehudayoff to show lower bounds for syntactically multilinear circuits and exponential bounds for constant-depth multilinear circuits.

The second paper showed the most well-known of the algebraic functions (permanent and determinant) do not have poly-size multilinear formulas. Raz again starts with the partial derivative matrix, this time combined with random restrictions.

More recent work in algebraic circuit complexity focuses on the VP versus VNP problem, the algebraic version of P v NP. VP = VNP if we can solve the permanent on poly-size algebraic circuits. Proving VP ≠ VNP is a goal of the Geometric Complexity Theory program as well as a consequence of tighter bounds on constant-depth algebraic circuits

Tuesday, September 30, 2014

Dagstuhl on Algebra in Computational Complexity

(Reminder- Theory day at UMCP: here is the link. )

There was a Dagstuhl on Algebra in Computational Complexity Sept 22-26.
I learned stuff in the talks, over meals, and even in my room alone at night.

1) Recall that a while back Ryan Williams (the theorist, not the American-Football player) showed that NEXP is not in ACC. His proof involved MANY things but one of the core things was an ALGORITHM for a version of  SAT (I think Succinct-SAT) that was ever-so-slightly better than brute force. So one lesson is that people in complexity theory should know some algorithms. At Dagstuhl Ryan presented work that shows that people in algorithms should know complexity. He used some old results about circuits to obtain  algorithm for all-pairs shortest path that has complexity n^3/X where X=2^{\Omega(log n)^{1/2}. The ultimate goal is to either prove or disproof that all-pairs... has n^{3-ep} algorithms or not. On the NOT side he has (with other people, including Virginia Williams) a large class of problems , including APSP, that either all have n^{3=ep} or none of them do.

2)  There were two (possibly three) talks on VP and VNP. Both are circuit classes defined by Valiant. Meena Mahajan has some natural problems that are complete for VNP (if you consider looking at Homomorphic polynomials natural) and Eric Allennder had a dual notion to VP and VNP. A sequence of polynomials is in VP  if there is an arithmetic circuit  family of polynomial bounded size and degree that computes the sequence. (Circuit C_n computes poly f_n). VNP is if there is a sequence C_n of poly bounded size and degree such that f_n(x) = sum as y\in {0,1}^p(n) C_n(x,y).

This is usually discussed over a finite field. Eric's result depended on which field it was.

3)  Stephen Fenner talked about some combinatorial games that were PSPACE complete. The black-white-poset-game is as follows: there is a poset where every node is colored white or black. One player is called black, the other is called white. Players alternate removing nodes of their color, and if they remove a node they remove all nodes above it. Either player can go first, so you may have a game where if B goes first he wins, but if he goes second he does not. Fenner and his co-authors have shown that the general problem of, given a Black-whie Poset and who goes first, determining who wins, is PSPACE complete. They showed that other versions are in P.

4)  In the 1990's Karchmar and Wigderson had an approach to NC^1 vs NC^2 and P vs NC^1 that looked promising--- they defined problems in communication complexity (where we actually DO have results!) that would imply some separations. This lead to monotone circuit results, but not to real circuit results. Or Meir spoke on some problems in that realm which can now be approaced with information complexity. Are we closer to a true separation? Next Dagstuhl!

5)  David Zuckerman gave a nice talk on Malleable codes. I was intrigued by some of the math that he did. The Sum-Product theorems are along the lines of: If A, B are large sets of reals (or other domains) then either A+A or AA is large. Or one could say that if A,B,C are all large than AB+C is large. David used an entropy version of this--- if A,B,C have large min-entropy, then so does AB+C.

6)  Kopparty showed that Polynomial Id Testing and Polynomail Factoring are related and may be equivalent.

7)  Over Lunch Jacobo Toran told me the following rather odd result.

a) Imagine the following GI algorithm: given two graphs look at the degree sequence, then the degrees of the degress, then... (this goes on n times). Two graphs are isom if they are never found to be non-isom. DOESN"T WORK-  Cai-First-Immerman showed that. Even so, we'll call that FOCSI-isom. Note that FOCS-isom is in P.

b) Recall that G (as a matrix) and H (as a matrix) are isom if there exists a Perm Matrix P such that GP = PH. We can expand what P can be- say to doubly-stocastic (every row and every column adds to 1) We call two graphs G,H STOC-isom if there exists a Double stocastic matrix P such that GP=PH. This is in Poly Time by Linera Programming.

c) FOCS-isom and STOC-isom are the same! Too bad, I thought that STOC-isom might be a way to get GI in P.

8) Sometimes at a conference I find a book in the library and read it at night and learn something out of nowhere. This happened with Mitzenmacher-Upfal book on Prob. and Computing (It could be called ``The Alice book'' as there is a picture of Alice from Alice in Wonderland on the cover. For the longest time a famous compiler book was called The Dragon Book). I read parts of it and even made up notes that I may use in an ugrad course:  the coupon collector problem: A cereal company puts coupons labeled 1,2,...,n at random in boxes. If you mail in one of each you get a free box of cereal. You are DETERMINED to get that box of cereal. What is the expected number of boxes of cereal you must buy? It turns out to be nln(n), and is very tight around that.

9) I learned more from the talks, more from the meals, and more from my own alone time, but what is above is a good sampling.

10) More chalk-talks then I would have thought. Either chalk or slides can be done well or poorly.

11) Looking forward to the next Dagstuhl!




Saturday, September 27, 2014

MikeFest


I rarely highlight individual events on the blog, but one's advisor only turns sixty once. We will honor Michael Sipser at MIT on Sunday, October 26 with a one-day symposium full of great speakers in complexity. As one of Sipser's PhD students, I'm helping to organize the event and serving as emcee for the dinner speeches. Please send me any funny or embarrassing stories or pictures of Sipser through the decades.

Many of you know Sipser from his popular textbook Introduction to the Theory of Computation. Sipser was one of the driving forces in complexity in the 70's and 80's. Sipser initiated the program of using circuit complexity to separate complexity classes and, with Merrick Furst and James Saxe, showed parity could not be computed by poly-size constant-depth circuits. His research includes how to simulate randomness by alternation, was the first to explore hardness vs randomness, made great advances in the complexity of interactive proofs, and much more. Sipser led the MIT Mathematics department for the last ten years and was recently appointed Dean of Science.

Join us in Cambridge for this great day to celebrate an extraordinary man.

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

P.S. STOC call posted, Deadline November 4

Wednesday, September 24, 2014

Typecasting in Dagstuhl

After this pre-recorded typecast, we learned of the tragic death of Alexey Chervonenkis, the C of VC dimension, a huge loss to the learning community. We’ll have a proper obit soon. Now onto the typecast.

Lance: Hello and welcome to Dagstuhl for our first typecast since the 2014 Complexity Conference. Howdy Bill.

Bill: Hi Lance. Are you enjoying Dagstuhl?

Lance: I always have fun at Dagstuhl especially when you are here Bill.

Bill: I have not seen you at many talks.

Lance: So maybe you should go to more talks Bill.

Bill: Never mind. As you told me Scott is writing a blog book. Should we too?

Lance: Something we discussed many times.

Bill: How about a slightly different idea? At the end of this year you will have had FIVE lists of TEN best theorems. (Doing math in his head) That’s FIFTY theorems. There’s a book with a unified theme.

Lance: And I’m glad you’re going to write it.

Bill: That’s not exactly what I had in mind. But I’m happy to help you write it?

Lance: Do you think there are people who would want to buy this book?

Bill: I need your help BLOG AUDIENCE. Leave a comment to say if you would read this book. Would you read the book if you have to pay for it?

Lance: I certainly wouldn’t.

Bill: You don’t count. But they (points to the audience) do. [Bill leaves to get ice cream and comes back] I’m sure it will sell well in Silicon Valley.

Lance: Speaking of Silicon Valley, that was one tough post to write on MSR-SVC, basically an obituary post for a research lab.

Bill: Isn’t rather grim calling it an obituary?

Lance: Exactly.

Bill: Do you always give one word answers?

Lance: No.

Bill: You are man of few words.

Lance: You are a man of a few words too many.

Bill: Yes, I like to keep conversations flowing.

Lance: Indeed you are of the few extroverts in complexity. Introverts like me think deeply of what to say before we say it.

Bill: Did you just insult me? How did an introvert like you become a department chair?

Lance: I fake it well. [Quickly changing topic] I hear there’s exciting news out of Maryland. And I’m not talking about the Orioles.

Bill: We’re getting a new building, The Brendan Iribe Center for Computer Science and Innovation.

Lance: Because there’s no innovation in computer science. Brendan who?

Bill: He co-founded Oculus which was sold to Facebook for Ackerman of O(1) dollars.

Lance: Sounds exciting. It is one pretty ugly building you are in now.

Bill: Moving on, how the Complexity Conference in Vancouver?

Lance: You didn’t read my blog post?

Bill: [Reading blog post] Wow, no best paper and only 66 participants. Seems a bit lower than last year.

Lance: We were correlated with STOC last year and next year at FCRC as well. Though not with the IEEE anymore.

Bill: Is complexity theory dying?

Lance: The talks at this Dagstuhl alone prove otherwise.

Bill: I particularly liked David Zuckerman’s talk about using statistical sum-product theorems to create non-malleable codes. Why is it so empty in here?

Lance: It’s a rare sunny day at Dagstuhl and we’re inside doing this typecast. What other topics are exciting you at Dagstuhl?

Bill: There’s a resurgence of interest in VP and VNP, Valiant’s algebraic analogues of P and NP and genuine optimism that VP <> VNP might be provable in the near future.

Lance: There is some great work there but let’s wrap this up while we have still have some daylight.

Bill: You know what to say Lance.

Lance: In a complex world, best to keep it simple.