Favorite Theorems: Connecting in Log Space

We start the favorite theorems with a result that might surprise many is still less than ten years old.

Undirected Connectivity in Log-Space by Omer Reingold (PDF)

Intuitively, this result says you can tell if two points are connected in a complex maze by only having to remember the equivalent of a constant number of locations in the maze. Reingold's algorithm builds an expander graph based on the zig-zag construction in a very clever way that uses very little space to construct and to check that two points connect.

In 1979, Aleliunas, Karp, Lipton, Lovász and Rackoff showed that one can solve s-t connectivity in randomized logarithmic space by taking a random walk on the graph. My last favorite theorem from 2004 talked about derandomizing space algorithms and before Reingold the best algorithm for s-t connectivity required log^4/3 space. Indepently of Reingold, Vladimir Trifonov gave a O(log n log log n) space algorithm for s-t connectivity, a victim of bad timing.

One neat implication of Reingold's result is a new and simpler characterization of log-space as the set of problems expressible in first-order logic with ordering and symmetric transitive closure.

After Reingold's result we might have expected solutions to a number of related problems but we didn't see much progress.

• Can every randomized log-space algorithm be derandomized in log space?
• Do there exist log-space computable universal traversal sequences? 
• Can we solve directed s-t connectivity better than Savitch? 
• Can we modify Reingold's algorithm to bring log space into NC¹?

Contribute to the Martin Gardner Centennial

Dana Richards emailed us about a place to write how Martin Gardner influenced you. You can leave such comments here.  I left a comment there, but I expand it for this blog entry.

When I got interested in mathematics in high school I went to the public library looking for math books (this was before Al Gore invented the internet). I found some books by Martin Gardner and began reading them. They were just right for the level of math I was on at the time. My very first proof that I read on my own (outside of a class) was in those books- the proof that (in the terminology I use now) a graph is Eulerian iff every vertex has even degree.

I  learned about SOMA cubes (I bought a set and did every puzzle in the book in about 2 days.This is the only evidence that as a kid I was good at math). I learned the unexpected hanging paradox which confused me then (and still does). I learned the hercules-hydra game and other games that go on for a LOOOOOOOOOONG time. They are related to things in logic. I also learned about NIM games which I have used as a starting point for several student projects.

There have been some conferences in his honors, the Gathering-for-Gardner. I had the pleasure of reviewing some of the books from it. (My review is here.) These articles show that while his work was recreational this is not a well defined term- some if relates to very important and deep mathematics, and some deep math has arisen from such problems. The books also have articles about Gardner the Magician.

In  the 2000's some of his books were reprinted and I was asked to review them for my SIGACT News book review column.  I took this opp to do a joint review of several math recreational books. What a delight to reread his books and contrast them to those of his successors. And I STILL learned some math that I didn't know from them. (My review is here.)

Shortly before a column appears I always email the authors-of-books, authors-of-reviews, and publishers a first draft of my column. His publisher told me that he didn't use email (he was in his 90's!) so I postal mailed him my review. He read it, corrected some typos, but otherwise was quite happy with the review. He died a few months later. I was happy to have some contact, albeit short, with the man who helped keep me interested in math in high school and beyond.

Snow Days

An unexpected snowstorm hits the city in the middle of a workday. The roads get hopelessly clogged and I'm lucky to get home--many others just abandoned their cars, or slept in them. I'm talking about Valentine's Day, February 14, 1990 in Chicago. But the same story hit Atlanta yesterday. One big difference--Georgia Tech is closed today and tomorrow because the city can't handle the ice. The University of Chicago was open on February 15th. 

When these events happen, people wonder about the planning. Was it wise for all schools and businesses to shut down about the same time, early yesterday afternoon? Lots of blame to go around (and having CNN based in Atlanta guarantees coverage) but it is not clear that any plan would have done much better--how do you get millions of people safely home with dangerous roads and a limited public transit system? One of these times you wish P = NP and you can just find the right algorithm. One of the issues is that freak mid-day snowstorms don't happen that often, the last major one in Atlanta was 1982.

Meanwhile back in Chicago, schools were closed earlier this week, not for snow but for cold. But it was that cold on a regular basis back in the 90's. Global warming has changed expectations, as so brilliantly illustrated in this xkcd. 

Fermat's Last Theorem and Large Cardinals. Really!

A brilliant math ugrad at UMCP, Doug, is also a creative writer who
wants to work on large cardinals. His creative writing may help him there.
We had the following conversation:

DOUG: The proof of Fermat's last theorem depends on the existence
of certain large cardinals and hence is not in ZFC.

BILL: That is not true.
DOUG: Have you read the proof?
BILL: No, however, if that were true I would know it. See this blog entry.
DOUG: Why would you know it?
BILL: If FLT required LCs then

a) Number theorists would be nervous.
b) Logicians would be ecstatic
c) The math community would not have announced to the world that FLT was solved.
d) Wiles would not have collected his prize money for solving it.
e) Again, I would know it.

DOUG: All compelling arguments. Even so, FLT requires LCs.
BILL: I will bet you five dollars that the current proof of FLT does not depend on LCs.
DOUG: Uh. Your counter arguments are compelling.
BILL: So... no bet?
DOUG: Uh. No.

The next day I got an email from Doug with the subject heading

        I cheated myself out of five dollars.

 Doug found this article, What does it take to prove Fermat's Last Theorem? Grothendieck and the logic of number theory by Colin McLarty, from 2009.The article says that YES the  current proof of FLT DOES depend on LCs. Note that the proof is quite long and uses lots of other stuff that is sort of buried in it. So--- whats the catch?Why aren't number theorists nervous and logicians ecstatic? According to the article anyone who reads the proof of FLT and wanted to could unwind it  and get it down to ZFC (and likely down to PA). But nobody has bothered yet.Hence nobody is nervous or ecstatic.

I will take their word for it, but it does make ME nervous. NOT about FLT which I am sure enough people have looked at (and looked at the background literature) that it really can be made to work in ZFC.I am more worried about papers that are not quite so looked at as having LC assumptions that are hidden from the reader that cannot be easily removed.

However KUDOS to Doug for telling me something in math that I did not know and should have.  I will treat him to a more-than-five-dollar-lunch.

Postscript: AH, the article was right: FLT was proven using ordinary set theory last year. (See here) by Colin McLarty (I assume its the same person). I will still take Doug to lunch- in a stupid, pedantic, technical sense I was right- the current (2014) proof of FLT did not use LC. But for the real issue of there being any problem at all, I was clearly wrong. Only a logician would say I was right. Hmm- Doug is a logician. Its up to him. (Hmm- my spell checker allows `Hmm' but not `Hmmm')

What will we wrought?

When I went to college in the early 80's, students protested against college endowments invested in companies that had business in apartheid South Africa. My mother worked as a statistician for one of those companies. An interesting dilemma, do I support a policy that hurts the company that is indirectly helping to put me through college?

Now my daughter is in college and worrying that the computing revolution will make it hard to find a job once she graduates and making her consider those job prospects in the major she chooses. And what am I? Chair of a computer science department that helps push that revolution forward.

Computing gets quite a bit of blame these days for the widening income gap between the have and the have nots, and jobs taken over by automation, but without causing a corresponding need for other types of jobs, other than those that serve computation itself. Are those fears real? We can't answer that question yet, positively or negatively. Time will tell.

For now, we just need to do our jobs, making computing better but also understanding and mitigating the negative effects of computing. We need to make sure that computing technology becomes a growing sea that raises all boats, and not just making the world better for the technological elite.

While I stand in awe in how computer science has changed the world, I hope we don't ever end up with CS leaders getting together and saying "What have we wrought?"

We don't care about Ballroom Dancing. Should we?

YOU got into your undergrad school because not only were you good at Math but you were on
the Fencing Team and in the Latin Club (so you could taunt your opponents in Latin: ouyah allcay athay an alestrabay!). Also you had a letter from your principal who never had you for a class but can comment on your leadership since you organized a pep rally for the football team. Why does UNDERGRAD admissions care about these things? Because, while they want good students, they also want to build a community of scholars of different interests and abilities.

YOU apply to grad school in Computer Science. Hey, it worked once maybe it will work again! You write about being in the ballroom dancing club and you have a letter from the Dean, who never had you in a class,
but you worked in his office and he can attest that you are a good leader and a hard worker.

Does the admissions committee care? NO. The only things we care about are CS, MATH, and RESEARCH. A letter from someone not in math or science is worthless. Some exceptions and thoughts:

1. If you recorded ballroom dancing and made a project out of how to teach it using some interesting new technology this IS good. This is likely an Human-computer-interaction project; however, I would care about this no matter what field you are going into.
2. If you have an interest in Nat Lang Proc and know Linguistics I would care.  I would think that knowing a foreign language would also be good.
3. If you are going to go into Human computer Interaction then Psychology helps.
4. If you are going to do Quantum Computing then Physics is good. However, whatever you do Physics is good as its more evidence of math ability.
5. For ugrad its been said that if your parents are powerful OR donors you may have an easier time getting into some UGRAD schools. What about Grad school? I've honestly never seen a case of this so I honestly don't know. 

I know a student who is an excellent math major but also a creative writer. I doubt this will help him.
but should it?

I once saw in a students application a letter from his preacher attesting to his fine moral character.
Do we care? should we? How about the other way around- if someone was an EXCELLENT programmer and math person but served 8 years for armed robbery would we care? This might not be fair since perhaps he reformed.

but my real question is- for grad admissions we don't care about Ballroom Dancing or other misc.
Is this a mistake? If someone was NOT as good at math BUT a better writer, should we take them?

Favorite Theorems: Introduction

I was invited to give a talk at the FST&TCS conference held in December 1994 in Madras (now Chennai). As I searched for a topic, I realized I was just finishing up my first decade as a computational complexity theorist so I decided to recap the past decade by listing My favorite ten complexity theorems of the past decade (PDF). Not so much to choose winners, but use the theorems to survey the great research during those past ten years.

In 2004, I repeated the exercise in my then young blog for the years 1995-2004. In 2005, I went back in time and chose my favorite theorems from the first decade of complexity (1965-1974) and in 2006 I covered 1975-1984, completing the backlog of the entire history of computational complexity.

Now in 2014 we start again, recapping my favorite theorems from 2005-2014, one a month from February through November with a recap in December. These theorems are chosen by a committee of one, a reward only worth the paper they are not written on. I choose theorems not primarily for technical depth, but because they change the way we think about complexity. I purposely choose theorems with breadth in mind, using each theorem to talk about the progress of a certain area in complexity. I hope you'll be presently surprised by progress we've made in complexity over the past decade.

A short History of Crypto

I taught a 3-week summer course to High School Students called

Computer Science: A Hands Off Approach

which did some theory. One thing I did was the following storyline:

1. Shift Cipher
2. Affine Cipher
3. Gen perm cipher (any perm of a,b,c,...,z
4. PROOF that perm is unbreakable: Eve has to go through all 26! possibilities
5. PROOF that perm IS breakable: Freq analysis. Moral of the story: Any proof that a system is unbreakable makes some assumptions that Eve might not agree to. Hence proving security is tricky.
6. Matrix Ciphers. PROOF that if you use a big enough matrix its unbreakable. Sort-of true for ciphertext only (though I doubt really proven). PROOF that requires going through all possibile nxn matrices that have det rel prime to 26 to crack it using plaintext only. PROOF that this is NOT true (I leave that to my reader).
7. Vig Cipher. Proof that its unbreakble, Proof that you can break it

I was very happy with this since it really instilled in them that proofs of security are nontrivial and always have assumptions. That does not mean they are not worth anything, but you want to get the assumptions explicit to if (or even WHEN) the system is broken, you can see what assumption needs to be attended to for the next iteration. One caution- these were VERY GOOD students so they GOT IT. They didn't mistake the false proofs for real ones.
(NOTE-- I never teach the cows paradox - all cows are the same color- when
doing induction since half the class will think induction can prove anything and the other half will think that all cows are the same color.)

I then encapsulated all of this with what I call A SHORT HISTORY OF CRYPTO:

 For i=1 to infinity
                Alice and Bob: We have a cipher that nobody can crack
                Alice and Bob: We have PROVEN that it can't be cracked
                Eve: I just cracked it
                Alice and Bob: Whoops.


(I later did Diffie-Hellman in the class which I will talk about in a later blog.)

Is Traveling Salesman NP-Complete?

[Nina Balcan asked me to mention that the COLT submission deadline is February 7]

Jean Francois Puget writes a controversial post No, The TSP Isn't NP Complete which I discovered during a lengthy twitter discussion with Puget and Peter Cacioppi.

There is a well-known technicality for the Euclidean Traveling Salesman problem but let's focus instead where we are given a complete graph weighted with positive integers. One version of TSP is truly NP-complete

TSP Decision: Given an integer B, is there a cycle through all the vertices such that the total weight of the edges used is at most B?

TSP Decision is in NP by guessing the cycle and hardness by a simple reduction from Hamiltonian Cycle.

Puget's makes the point that we normally think of the TSP problem as an optimization question

TSP Minimization: Find the cycle through all the vertices that minimizes the total weight used.

TSP Minimization is not even a decision problem. In the 80's, Mark Krentel created a complexity class OptP to capture optimization problems and showed that TSP Minimization is Opt-P-complete.

One can use TSP Decision to solve TSP minimization by doing binary search, so they have effectively have the same complexity.

Puget points out that even if we are given a tour, checking that it is the shortest tour is not believed to be in NP. That problem is in co-NP and I'm guessing co-NP-complete. [Update 3/20: I was right]

Puget doesn't like when people claim TSP is NP-complete when they are talking about the optimization problem. For example from my P v NP survey

The NP-complete traveling salesperson problem asks for the smallest distance tour through a set of specified cities. 

I'm far less bothered than Puget. Those who understand the technicalities of NP-completeness know that one has to convert the optimization problem to an appropriate decision problem to formally get an NP-complete set. Others aren't led too far astray, for we do have an equivalence that P = NP if and only if there is an efficient (polynomial-time) algorithm for TSP Minimization.

Tell me more about Alice and Bob

A while back my parents were in town on a weekend when I was scheduled to give a talk to HS students who had done well on the Maryland math competition. Logistics dictated that my parents goto the talk. (They were both English majors and wouldn't like me using the word `goto' since its not a word. Fortunately they don't read this blog and see what else I do to the English Lang.)
I gave a talk on Communication Complexity (slides are here) where I did the following:

1. I stated the problem: Alice has x, Bob has y, both strings of length n. They want to know if x=y without too much communication.
   
2. I noted that they can easily solve this with n+1 bits of communication and raised the question of Can They Do Better?
   
3. We discussed this. Someone mentioned average case (informally), which helped me clarify the problem. Someone else suggested sending the number-of-1's and if it didn't match they weren't equal, but also noted that if they did--- weren't sure. Most thought that one COULD do better or else I wouldn't be talking about it.
   
4. I tell them that NO you can't do better (I do not prove this).
   
5. I told them about mod arithmetic and how in mod p, p a prime, poly of degree d have at most d roots.
   
6. I presented the O(log n), error 1/n, randomized protocol for equality that uses polynomials mod p.
   
7. I briefly talked about comm complexity in general.
   

The talk went well. It is a good topic for good HS students, and if you want to borrow my slides you can (you may need to update the political reference). Having not understood ANY of the talk Mom had the following question:
MOM: Alice and Bob-- are they married?
BILL: Oh. I'll say no.
MOM: If they are not married then how come they have such a hard time communicating?
DAD: (he didn't say anything but I could tell he agreed).

Two cheers for the Pardon of Turing. But not three.

As I am sure readers of this blog know Alan Turing was prosecuted for homosexuality in 1952, forced into hormone treatment, and committed suicide in 1954 (I had always heard that that was WHY he committed suicide
though the dates don't quite line up--- at that time he seemed to be recovered form the ordeal. So there are some legit questions about this.)

He was recently given a Royal Pardon. While I am glad he was pardoned this does raise some questions. Lets me logical.

1. If we believe the law criminalizing homosexual acts was unjust (as I am sure that all of my readers do) then Turing is a red herring- they should pardon ALL people convicted. AND note that  according to this there are 15,000 men who were convicted of this crime who are still alive.
2. Pardon means that the person didn't do the crime. This is not the case here. However, laws are supposed to promote justice, not block it.

Here is hoping that the Pardon of Turing will lead to a general pardon.

(NOTE- the pointer in item 1 says more of what I wanted to say, but says it
more elegantly than I ever could.)

2013 Complexity Year in Review

The complexity result of the year goes to The Matching Polytope has Exponential Extension Complexity by Thomas Rothvoss. Last year's paper of the year showed that the Traveling Salesman Problem cannot have a subexponential-size linear program formulation. If one could show that every problem in P has a short polynomial-size LP formulation then we would have a separation of P and NP. Rothvoss' paper shoots down that approach by giving an exponential lower bound for the polynomial-time computable matching problem. This story is reminiscent of the exponential monotone circuit lower bounds first for clique then matching in the 1980's.

If you expand to all of mathematics, one cannot ignore Yitang Zhang's work showing the liminf of the difference between consecutive primes is a constant. Dick and Ken have other great results for the year.

A big year for theoretical computer science. Silvio Micali and Shafi Goldwasser received the ACM Turing Award. The P v NP problem makes a prominent appearance on a major US television series. We are seeing the rise of currencies based on complexity. Large-scale algorithms, cryptography and privacy play center stage in the Snowden revelations on the National Security Agency.

Generally good news on the funding and jobs front in the US. After a year of sequestration and a government shutdown, looks like some stability for science funding now that congress has actually passed a budget. Plenty of theorists got academic jobs last spring and given the number of ads, this year's CS job market should be quite robust as well.

A year for books. Of course my own Golden Ticket as well as Scott Aaronson's Democritus and Tom Cormen's Algorithms Unlocked.

An odd year for the blog in 2013 without a single obituary post. Nevertheless let us remember 4-Colorer Kenneth Appel, Georgia Tech Software Engineering Professor Mary Jean Harrold and Wenqi Huang who led the Institute of Theoretical Computer Sciences at the Huazhong University of Science and Technology. In 2013 we also said goodbye to Alta Vista, Intrade, Google Reader and the CS GRE.

In 2014 we'll have the next installment of My Favorite Ten Complexity Theorems of the Past Decade and the centenaries of George Dantzig and Martin Gardner. Enjoy New Years and keep reading.

To Write, or Not to Write, That Is the Question

Guest post by Vijay Vazirani

Our field has been blessed with some great books: classics such as Knuth's volumes and Garey & Johnson literally got the field going, and the books of Lovász brought much clarity in difficult, important areas. The new millennium brought a plethora of new books by famous TCS researchers including Arora, Barack, Goldreich, Kleinberg, Nisan and Tardos. With an astonishing 33 short titles in theory, Now Publishers has virtually created an assembly line for producing books.

Even so, when faced with the prospect of writing a new book, I am filled with trepidation and self-doubt: Am I up to the effort needed? Do I have a genuinely new point of view to expound? If so, have I figured out the "right'' format and style for expounding it in the simplest and clearest manner?

The issue arose when I decided to co-teach, with my able postdoc Ruta Mehta, the course Equilibrium Computation in Spring. With her characteristic enthusiasm, Ruta has practically forced me into a long-term commitment of writing a book on this topic. However, knowing what it takes, so far I have managed to resist an unequivocal "yes".

Most people would say it is a losing proposition -- the gains are typically minuscule compared to the toils. So why do so many people write books? I can imagine several reasons, but best to leave this discussion to my esteemed colleagues who are probably yearning for something to ponder on in this holiday season ...

That brings me to another question, "Are some of the books written in this millennium eventually going to be regarded as classics in the style of Knuth or Lovász's books?'' One could look at citation counts or average citations per year to get an "objective'' reading of the situation. However, when one is talking about true class, such crude estimators are clearly not the way to go.

Your thoughts are eagerly sought ...

Our Journal is 99 44/100 percent pure!!

Sometimes we are asked to evaluate how good a journal or conference  formally(Excellent, Very Good, Good, Fair, Better-than-being-poked-by-a-stick,pass the stick). Sometimes we talk about these things informally (ICALP is the European STOC! SODA is as hard to get into as FOCS!, CCC is a topTier topics-conference!)


But is there a way to judge these things formally? And should there be?Be aware of Goodhart's Law:

When a measure becomes a target, it ceases to be a good measure.

 One way to measure how good a journal or conference is  impact factor which is based on number-of-citations. Does this work well? Some issues:

1. Even with an honest effort these things are hard to do well. People may cite the conference version, the journal version, the arXiv version, or just give a website.
2. Even with an honest effort just a few articles can skew the results.
3. Its hard to compare cross-fields.  I suspect that pure math journals have lower citations rates than biology journals.
4. If an author cites himself, should that count?
5. The above points were all about HONEST efforts. Could a journal do things to boost its impact factor and then brag about how high their impact factor is? Would they? Alas yes, see this article and this article.

So what are we left with? We could just go on our gut, but that favors journals that used to be good and aren't any longer. But the problem is deeper than all that--- can't we just say Thats a good article  and not have to proof it by saying where it was published? Alas no- all fields have gotten specialized that we must use these proxies to inform us.

 I COULD rant that we should all be well rounded enough to read outside of our field, but I know how hard that is.

I COULD rant about the dishonesty pointed out in the above links, but that's only part of the problem.

I COULD say What do you think? so I will.

Security Changes

A couple of policy changes recently, one that supposedly enhances privacy and another that could reduce it.

Google has been implementing perfect forward secrecy since 2011 and other major Internet players, such as Facebook and Twitter, have started using perfect forward secrecy in the wake of the Snowden revelations that the NSA has been collecting Internet traffic to these companies.

So what is perfect forward secrecy? Not an easy question to find the answer to on the Internet. The wikipedia article says little. So I asked a couple of our security folks in the department.

The rough idea: We want to communicate several rounds of messages but if the current keys are compromised they can't be used to decrypt earlier messages. A couple of immediate thoughts: This isn't "perfect", you can still discover the earlier messages by breaking the encryption (say if P = NP). Also this isn't that exciting a problem from a theoretical perspective, you can just use a standard public-key protocol and start with fresh private and public keys each round and deleting the old ones. But that isn't very efficient.

One approach to PFS: Have a standard public/private key scheme to set up a session key (used in an AES or similar private key protocol) then run separate Diffie-Hellman schemes for each message. In RSA if you have the factors for N you can decrypt, where in Diffie-Hellman you can keep the same group without compromising security.

Chris Peikert calls this a poor-man's perfect forward security and there are better schemes though a bit more complicated.

On a different front, Google recently announced that images by default would be displayed in gmail messages. The images would not come directly from the sender, which could contain malware that avoids Google's filters, but rather from Google's servers after being downloaded and cleansed by Google.

Downloading an image often tells the sender that the image was read, typically with some id encoded in the filename. So once again we give up privacy for convenience. At least Google gives us the option to turn off the automated displaying.

Analogs between Quantum Computing and Parallelism

(Jon Katz wanted me to mention this:  A wise man once noted that there are fewer quantum algorithms than thereare quantum-algorithms textbooks! But there is still a lot of interest inquantum computation from cademia,government, industry, and the broader public. Univ of MD and NIST have recently formed a
center devoted to quantum computing and involving faculty and researchers from both physics and computer science communities. As part of this they are advertising a posdoctoral fellowship.)

A long time ago people in theory did  a lot of work on parallel computing before
there were many parallel computers build. Today people are doing a lot of work on quantum computing before quantum computers are build. What is similar and different here?

1.  When parallel computers were actually built they were not like PRAM's. Many of the models assumed shared memory which wasn't true. Even so, did the work on PRAMS and other models help the practioners? Directly? Indirectly?
2. Are the models of quantum computing studied now helping the practioners? Is the development of quantum computers at too early a stage to even ask this question?
3. Even if no quantum computers are ever built the study has been a sucess since some classical problems have been solved using quantum techniques (and I think this will happen more and more). And some interesting math has come out of it. And physicists and others have learned more about quantum mechanics from quantum computing. Could the same thing have been said for parallelism- even if parallel computers had not been built would the study of them have still be useful?
4. One big difference- many problems can be parallelized in some form and solved that way (and some cannot be solved any other way). A wise man named Jon Katz referred above to a wise man named Ronald de Wolf  who wrote, in a review of 3 books on quanum computing:

 A quick survey on amazon.com shows that the number of books on quantum computing (at least 20) is more than 10 times as high as the number of quantum algorithms (2: Shor's and Grover's). (Footnote: Note that this review was written in 2003 so this statement is no longer true.)

While  I think he meant there are more quantum algorithms (quantum random walks, quantum simulations, quantum selection-type problems?, quantum number-theory-type-problems?) now than in 2003, I will note that there are also more books on quantum computing now than then- on a cursory look at amazon I think I counted 30, but with the uncertainly principle, its hard to tell.  The point is, aside from factoring and of course quantum simulation I wonder if when  quantum computers, if they are build, will be able to do much more
BUT SEE NEXT PARAGRAPH. (Every year my students are surprised to find out that quantum computers probably CANNOT solve SAT in poly time.)

ADDED LATER: Comment 6 has a pointer to a survey of MANY quantum algorithms for MANY algebraic problems, and also a pointer to a more recent article on quantum algorithms. I will be delighted if the number of quantum algorithms now exceeds the number of books on quantum computing.

Approximate Computing

Hadi Esmaeilzadeh is the newest professor in the School of Computer Science at Georgia Tech. Hadi works in computer architecture and did some great work on dark silicon. (Just noticed this is starting to look like a Lipton-style blog post.)

I had Hadi give a lecture in my theory class (the dreaded day-before-Thanksgiving lecture). Hadi talked about his new research directions in approximate computing. Approximate computing is a new paradigm in the architecture community, doesn't even have its own wikipedia entry yet.

When you do arithmetic operations, say addition and multiplication on real numbers, you typically want full precision up to the limits of the bits we use to store those numbers. Suppose you allow some error, say 5%. For logical operations, this would be a disaster giving a very wrong answer. Running various optimization algorithms, like simplex, these error might compound leading to very suboptimal results.

But there are several scenarios where approximate computing might not hurt that much. Processing media, like pictures, sound and video, are not exact anyway and a small error might not degrade the quality successfully. Statistical methods that sample a large space, such as when we analyze big data, still could yield reasonable results using approximate computing.

Why do approximate computing? Because of the architecture--approximate computing can be done often faster and with less power consumption. The tradeoffs may allow approximate computing to handle tasks beyond what we can do with traditional computing.

Approximate computing needs good theoretical models and that's where our community can come it. What's the right way to model the power-speed-accuracy tradeoffs and how can we determine the right computational problems that can take advantage of these tradeoffs. Might have nice connections to learning theory and property testing.

Inventions are Non-Commutative: Amazon vs Amazon

(Tal Rabin, Shubhangi Saraf and Lisa Zhang asked me to remind you to publicize this: the bi-annual Women in theory (WIT workshop), NYC, May 28-30, 2014. Apps due Jan 20, 2014. Go here for all relevant information)

If FAX machines had come out 20 years earlier they would have had far MORE impact.
If FAX machines had come out 20 years later they would have had NO impact since by then
we all had email and scanners and what not. So when an invention comes out matters.

Ask your grandparents what white-out was for corrections on a typewriter (while you are at it ask your grandparents what a typewriter is). If it came out 20 years later then Bette Nesmith  would have not made 50 million dollars, her son Michael Nesmith would not have had the spare time to become a musician and there might not be a group  called THE MONKEES. Gee, we would have no LAST TRAIN TO CLARKSVILLE. But I digress (from what?).

If Lasik  eye surgery came first and glasses later, glasses may have been seen as a great way to avoid surgery!

Amazon is working on a technology that may become obsolete before it really takes off. I am referring to the Amazon Drone Project. The idea is that when your order an item from Amazon a drone will get it to your house (or perhaps wherever you are) VERY FAST - maybe within 30 minutes. This works well for objects under 5 pounds. Say like what Amazon is best know for BOOKS.

But there is an a technology out there that may kill this idea before it gets off the ground. There is a competing company called Amazon which has already developed ways for people to get books on what they call a Kindle, which is like emailing them a book. So there is NO Physical object to be delivered.
Hence the Amazon Drone project may, like the 8-track tape (ask your great grandparents) become obsolete due to technology being developed by Amazon.

I wonder- had Amazon Drone come out 20 years ago would it have had more impact? Would it have made less of a demand for Amazon Kindle?

Amazon Drone may still have an impact by delivering other things. But I wonder how long it will take before 3-d printers make MOST things that Amazon delivers not need to be physical objects.

Bitcoins Revisited

Two years ago I gave a lecture and posted about bitcoin. Of course what I didn't do was buy a bitcoin whose value back then was about $3 and today runs in the $1000 range.

Bitcoins have received quite a bit of press, particularly with the FBI shutting down Silk Road, the drug trafficking site which used bitcoins for their transactions. Then people realized that bitcoins are starting to become a real currency, with a market cap of about US$11 billion not far now from the money supply of the Costa Rica Colones (US$13 billion). Now governments are deciding on how to deal with bitcoins as a currency, one which they really can't regulate or control.

The Economist has one of the better articles on Bitcoins, talking about some of the technical issues involved.

The Bitcoin system is designed to cope with the fact that improvements in computer hardware make it cheaper and faster to perform the mathematical operations, known as hashes, involved in mining. Every 2,016 blocks, or roughly every two weeks, the system calculates how long it would take for blocks to be created at precisely 10-minute intervals, and resets a difficulty factor in the calculation accordingly. As equipment gets faster, in short, mining gets harder. But faster equipment is constantly coming online, reducing the potential rewards for other miners unless they, too, buy more kit. Miners have formed groups that pool processing power and parcel out the ensuing rewards. Once done with ordinary computers, mining shifted to graphics-processing units, which can perform some calculations more efficiently. Miners then moved on to flexible chips that can be configured for particular tasks, called field-programmable gate arrays. In the past year, bespoke chips called ASICs (application-specific integrated circuits) have appeared on the scene.

Then there was the paper Dorit Ron and Adi Shamir wrote that explored the bitcoin transaction graph and suggested a (now supposedly debunked) connection between the mysterious creator of bitcoins, "Santoshi Nakamoto", and Ross William Ulbricht aka Dread Pirate Roberts, the founder of Silk Road.

Bitcoins even make it to my Thanksgiving table. My brother thought they were a scam even though I pointed out the systems has no scammers. He remains unconvinced though he invests heavily in gold, which has the same property of the value mostly being there because people believe it has value.

I taught a lecture on bitcoins again in my intro theory course. We all generally agreed that a few years from now we'll all remember the days when bitcoins were worth $1000. Not sure we'll remember those days because bitcoins will be worth millions or because they'll be worth pennies.

Global Warming and the Axiom of Choice

Who was the first scientist to warn of Global Warning? These questions are complicated, but I would say it was Bing Crosby in a paper called White Christmas. Here are the first two lines:

I'm dreaming of a white christmas
Just like the ones I used to know

Why are there no more white christmas's? Because global warming made it stop snowing!

Why do otherwise intelligent people refuse to believe that Global Warming is real and is caused by humans and we we need to do something about it? I have a conjecture and an analog. Here is what I think the reasoning is

1. Republicans have the following AXIOM (until they are in office): government IS the problem, not the solution. More than this, they think that there is NO problem that requires government action.
   
2. Consequence: Government should do NOTHING about Global Warming.
   
3. Since Government shouldn't do anything about Global Warming, it is not a problem.
   

Rather than rethink their AXIOM they accept the conclusion that Global Warming is either not a problem or not caused by humans. The shame of it is that there ARE economically viable ways, perhaps moderate republican ways, to fight global warming- some version of Cap-and-trade, or pay-to-pollute. And getting off of Fossil Fuels would be good for other reasons. I can picture history going a different way so that Republicans want more fuel-eff cars to get us off of Mideast Oil. I can picture a history where the insurance companies are more powerful than the oil companies for lobbying and hence Government takes LOTS of action against global warming.

Are their things in math where people accept an axiom despite its absurd consequences?Yes:

1. Most math people believe the Axiom of Choice.
   
2. Consequence: the Banach-Tarski Paradox
   

Rather than rethink their AXIOM they accept the absurd conclusion that you can break a ball into 5 pieces, reassemble, and get twice the volume. Fortunately, believing this does not endanger the planet.