Shtetl-Optimized

A few weeks ago, I had the pleasure of attending FQXi’s Setting Time Aright conference, part of which took place on a cruise from Bergen, Norway to Copenhagen, Denmark.  (Why aren’t theoretical computer science conferences ever held on cruises?  If nothing else, it certainly cuts down on attendees sneaking away from the conference venue.)  This conference brought together physicists, cosmologists, philosophers, biologists, psychologists, and (for some strange reason) one quantum complexity blogger to pontificate about the existence, directionality, and nature of time.  If you want to know more about the conference, check out Sean Carroll’s Cosmic Variance posts here and here.

Sean also delivered the opening talk of the conference, during which (among other things) he asked a beautiful question: why does “complexity” or “interestingness” of physical systems seem to increase with time and then hit a maximum and decrease, in contrast to the entropy, which of course increases monotonically?

My purpose, in this post, is to sketch a possible answer to Sean’s question, drawing on concepts from Kolmogorov complexity.  If this answer has been suggested before, I’m sure someone will let me know in the comments section.

First, some background: we all know the Second Law, which says that the entropy of any closed system tends to increase with time until it reaches a maximum value.  Here “entropy” is slippery to define—we’ll come back to that later—but somehow measures how “random” or “generic” or “disordered” a system is.  As Sean points out in his wonderful book From Eternity to Here, the Second Law is almost a tautology: how could a system not tend to evolve to more “generic” configurations?  if it didn’t, those configurations wouldn’t be generic!  So the real question is not why the entropy is increasing, but why it was ever low to begin with.  In other words, why did the universe’s initial state at the big bang contain so much order for the universe’s subsequent evolution to destroy?  I won’t address that celebrated mystery in this post, but will simply take the low entropy of the initial state as given.

The point that interests us is this: even though isolated physical systems get monotonically more entropic, they don’t get monotonically more “complicated” or “interesting.”  Sean didn’t define what he meant by “complicated” or “interesting” here—indeed, defining those concepts was part of his challenge—but he illustrated what he had in mind with the example of a coffee cup.  Shamelessly ripping off his slides:

Entropy increases monotonically from left to right, but intuitively, the “complexity” seems highest in the middle picture: the one with all the tendrils of milk.  And same is true for the whole universe: shortly after the big bang, the universe was basically just a low-entropy soup of high-energy particles.  A googol years from now, after the last black holes have sputtered away in bursts of Hawking radiation, the universe will basically be just a high-entropy soup of low-energy particles.  But today, in between, the universe contains interesting structures such as galaxies and brains and hot-dog-shaped novelty vehicles.  We see the pattern:

In answering Sean’s provocative question (whether there’s some “law of complexodynamics” that would explain his graph), it seems to me that the challenge is twofold:

1. Come up with a plausible formal definition of “complexity.”
2. Prove that the “complexity,” so defined, is large at intermediate times in natural model systems, despite being close to zero at the initial time and close to zero at late times.

To clarify: it’s not hard to explain, at least at a handwaving level, why the complexity should be close to zero at the initial time.  It’s because we assumed the entropy is close to zero, and entropy plausibly gives an upper bound on complexity.  Nor is it hard to explain why the complexity should be close to zero at late times: it’s because the system reaches equilibrium (i.e., something resembling the uniform distribution over all possible states), which we’re essentially defining to be simple.  At intermediate times, neither of those constraints is operative, and therefore the complexity could become large.  But does it become large?  How large?  How could we predict?  And what kind of “complexity” are we talking about, anyway?

After thinking on and off about these questions, I now conjecture that they can be answered using a notion called sophistication from the theory of Kolmogorov complexity.  Recall that the Kolmogorov complexity of a string x is the length of the shortest computer program that outputs x (in some Turing-universal programming language—the exact choice can be shown not to matter much).  Sophistication is a more … well, sophisticated concept, but we’ll get to that later.

As a first step, let’s use Kolmogorov complexity to define entropy.  Already it’s not quite obvious how to do that.  If you start, say, a cellular automaton, or a system of billiard balls, in some simple initial configuration, and then let it evolve for a while according to dynamical laws, visually it will look like the entropy is going up.  But if the system happens to be deterministic, then mathematically, its state can always be specified by giving (1) the initial state, and (2) the number of steps t it’s been run for.  The former takes a constant number of bits to specify (independent of t), while the latter takes log(t) bits.  It follows that, if we use Kolmogorov complexity as our stand-in for entropy, then the entropy can increase at most logarithmically with t—much slower than the linear or polynomial increase that we’d intuitively expect.

There are at least two ways to solve this problem.  The first is to consider probabilistic systems, rather than deterministic ones.  In the probabilistic case, the Kolmogorov complexity really does increase at a polynomial rate, as you’d expect.  The second solution is to replace the Kolmogorov complexity by the resource-bounded Kolmogorov complexity: the length of the shortest computer program that outputs the state in a short amount of time (or the size of the smallest, say, depth-3 circuit that outputs the state—for present purposes, it doesn’t even matter much what kind of resource bound we impose, as long as the bound is severe enough).  Even though there’s a computer program only log(t) bits long to compute the state of the system after t time steps, that program will typically use an amount of time that grows with t (or even faster), so if we rule out sufficiently complex programs, we can again get our program size to increase with t at a polynomial rate.

OK, that was entropy.  What about the thing Sean was calling “complexity”—which, to avoid confusion with other kinds of complexity, from now on I’m going to call “complextropy”?  For this, we’re going to need a cluster of related ideas that go under names like sophistication, Kolmogorov structure functions, and algorithmic statistics.  The backstory is that, in the 1970s (after introducing Kolmogorov complexity), Kolmogorov made an observation that was closely related to Sean’s observation above.  A uniformly random string, he said, has close-to-maximal Kolmogorov complexity, but it’s also one of the least “complex” or “interesting” strings imaginable.  After all, we can describe essentially everything you’d ever want to know about the string by saying “it’s random”!  But is there a way to formalize that intuition?  Indeed there is.

First, given a set S of n-bit strings, let K(S) be the number of bits in the shortest computer program that outputs the elements of S and then halts.  Also, given such a set S and an element x of S, let K(x|S) be the length of the shortest program that outputs x, given an oracle for testing membership in S.  Then we can let the sophistication of x, or Soph(x), be the smallest possible value of K(S), over all sets S such that

1. x∈S and
2. K(x|S) ≥ log₂(|S|) – c, for some constant c.  (In other words, one can distill all the “nonrandom” information in x just by saying that x belongs that S.)

Intuitively, Soph(x) is the length of the shortest computer program that describes, not necessarily x itself, but a set S of which x is a “random” or “generic” member.  To illustrate, any string x with small Kolmogorov complexity has small sophistication, since we can let S be the singleton set {x}.  However, a uniformly-random string also has small sophistication, since we can let S be the set {0,1}ⁿ of all n-bit strings.  In fact, the question arises of whether there are any sophisticated strings!  Apparently, after Kolmogorov raised this question in the early 1980s, it was answered in the affirmative by Alexander Shen (for more, see this paper by Gács, Tromp, and Vitányi).  The construction is via a diagonalization argument that’s a bit too complicated to fit in this blog post.

But what does any of this have to do with coffee cups?  Well, at first glance, sophistication seems to have exactly the properties that we were looking for in a “complextropy” measure: it’s small for both simple strings and uniformly random strings, but large for strings in a weird third category of “neither simple nor random.”  Unfortunately, as we defined it above, sophistication still doesn’t do the job.  For deterministic systems, the problem is the same as the one pointed out earlier for Kolmogorov complexity: we can always describe the system’s state after t time steps by specifying the initial state, the transition rule, and t.  Therefore the sophistication can never exceed log(t)+c.  Even for probabilistic systems, though, we can specify the set S(t) of all possible states after t time steps by specifying the initial state, the probabilistic transition rule, and t.  And, at least assuming that the probability distribution over S(t) is uniform, by a simple counting argument the state after t steps will almost always be a “generic” element of S(t).  So again, the sophistication will almost never exceed log(t)+c.  (If the distribution over S(t) is nonuniform, then some technical further arguments are needed, which I omit.)

How can we fix this problem?  I think the key is to bring computational resource bounds into the picture.  (We already saw a hint of this in the discussion of entropy.)  In particular, suppose we define the complextropy of an n-bit string x to be something like the following:

the number of bits in the shortest computer program that runs in n log(n) time, and that outputs a nearly-uniform sample from a set S such that (i) x∈S, and (ii) any computer program that outputs x in n log(n) time, given an oracle that provides independent, uniform samples from S, has at least log₂(|S|)-c bits, for some constant c.

Here n log(n) is just intended as a concrete example of a complexity bound: one could replace it with some other time bound, or a restriction to (say) constant-depth circuits or some other weak model of computation.  The motivation for the definition is that we want some “complextropy” measure that will assign a value close to 0 to the first and third coffee cups in the picture, but a large value to the second coffee cup.  And thus we consider the length of the shortest efficient computer program that outputs, not necessarily the target string x itself, but a sample from a probability distribution D such that x is not efficiently compressible with respect to D.  (In other words, x looks to any efficient algorithm like a “random” or “generic” sample from D.)

Note that it’s essential for this definition that we imposed a computational efficiency requirement in two places: on the sampling algorithm, and also on the algorithm that reconstructs x given the sampling oracle.  Without the first efficiency constraint, the complextropy could never exceed log(t)+c by the previous argument.  Meanwhile, without the second efficiency constraint, the complextropy would increase, but then it would probably keep right on increasing, for the following reason: a time-bounded sampling algorithm wouldn’t be able to sample from exactly the right set S, only a reasonable facsimile thereof, and a reconstruction algorithm with unlimited time could probably then use special properties of the target string x to reconstruct x with fewer than log₂(|S|)-c bits.

But as long as we remember to put computational efficiency requirements on both algorithms, I conjecture that the complextropy will satisfy the “First Law of Complexodynamics,” exhibiting exactly the behavior that Sean wants: small for the initial state, large for intermediate states, then small again once the mixing has finished.  I don’t yet know how to prove this conjecture.  But crucially, it’s not a hopelessly open-ended question that one tosses out just to show how wide-ranging one’s thoughts are, but a relatively-bounded question about which actual theorems could be proved and actual papers published.

If you want to do so, the first step will be to “instantiate” everything I said above with a particular model system and particular resource constraints.  One good choice could be a discretized “coffee cup,” consisting of a 2D array of black and white pixels (the “coffee” and “milk”), which are initially in separated components and then subject to random nearest-neighbor mixing dynamics.  (E.g., at each time step, we pick an adjacent coffee pixel and milk pixel uniformly at random, and swap the two.)  Can we show that for such a system, the complextropy becomes large at intermediate times (intuitively, because of the need to specify the irregular boundaries between the regions of all-black pixels, all-white pixels, and mixed black-and-white pixels)?

One could try to show such a statement either theoretically or empirically.  Theoretically, I have no idea where to begin in proving it, despite a clear intuition that such a statement should hold: let me toss it out as a wonderful (I think) open problem!  At an empirical level, one could simply try to plot the complextropy in some simulated system, like the discrete coffee cup, and show that it has the predicted small-large-small behavior.   One obvious difficulty here is that the complextropy, under any definition like the one I gave, is almost certainly going to be intractable to compute or even approximate.  However, one could try to get around that problem the same way many others have, in empirical research inspired by Kolmogorov complexity: namely, by using something you can compute (e.g., the size of a gzip compressed file) as a rough-and-ready substitute for something you can’t compute (e.g., the Kolmogorov complexity K(x)).  In the interest of a full disclosure, a wonderful MIT undergrad, Lauren Oullette, recently started a research project with me where she’s trying to do exactly that.  So hopefully, by the end of the semester, we’ll be able to answer Sean’s question at least at a physics level of rigor!  Answering the question at a math/CS level of rigor could take a while longer.

PS (unrelated). Are neutrinos traveling faster than light?  See this xkcd strip (which does what I was trying to do in the Deolalikar affair, but better).

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I thought I’d let Shtetl-Optimized readers know about an experimental new course I’m teaching this fall (starting tomorrow): 6.893 Philosophy and Theoretical Computer Science.  The course was directly inspired by my Why Philosophers Should Care About Computational Complexity essay, and will cover many of the same topics.  Here’s the description:

This new offering will examine the relevance of modern theoretical computer science to traditional questions in philosophy, and conversely, what philosophy can contribute to theoretical computer science.  Topics include: the status of the Church-Turing Thesis and its modern polynomial-time variants; quantum computing and the interpretation of quantum mechanics; complexity aspects of the strong-AI and free-will debates; complexity aspects of Darwinian evolution; the claim that “computation is physical”; the analog/digital distinction in computer science and physics; Kolmogorov complexity and the foundations of probability; computational learning theory and the problem of induction; bounded rationality and common knowledge; new notions of proof (probabilistic, interactive, zero-knowledge, quantum) and the nature of mathematical knowledge.  Intended for graduate students and advanced undergraduates in computer science, philosophy, mathematics, and physics.  Participation and discussion are an essential part of the course.

If you’d like to follow remotely, the course homepage has links to lots of interesting readings, and students will also be posting their personal reactions to the class discussions as the semester progresses.

Update (Sept. 7): By overwhelming request not only from readers but from students in the class, and with several of those students’ extremely kind assistance, we will be making audio recordings—although the audio quality probably won’t be great.

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Update (August 11, 2011): Thanks to everyone who offered useful feedback!  I uploaded a slightly-revised version, adding a “note of humility” to the introduction, correcting the footnote about Cramer’s Conjecture, incorporating Gil Kalai’s point that an efficient program to pass the Turing Test could exist but be computationally intractable to find, adding some more references, and starting the statement of Valiant’s sample-size theorem with the word “Consider…” instead of “Fix…”

I just posted a 53-page essay of that name to ECCC; it’s what I was writing pretty much nonstop for the last two months.  The essay will appear in a volume entitled “Computability: Gödel, Turing, Church, and beyond,” which MIT Press will be publishing next year (to coincide with Alan T.’s hundredth birthday).

Note that, to explain why philosophers should care about computational complexity, I also had to touch on the related questions of why anyone should care about computational complexity, and why computational complexity theorists should care about philosophy.  Anyway, here’s the abstract:

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance.  In this essay, I offer a detailed case that one would be wrong.  In particular, I argue that computational complexity theory—the field that studies the resources (such as time, space, and randomness) needed to solve computational problems—leads to new perspectives on the nature of mathematical knowledge, the strong AI debate, computationalism, the problem of logical omniscience, Hume’s problem of induction and Goodman’s grue riddle, the foundations of quantum mechanics, economic rationality, closed timelike curves, and several other topics of philosophical interest.  I end by discussing aspects of complexity theory itself that could benefit from philosophical analysis.

Weighing in with 70 footnotes and 126 references, the essay is basically a huge, sprawling mess; I hope that at least some of you will enjoy getting lost in it.  I’d like to thank my editor, Oron Shagrir, for kicking me for more than a year until I finally wrote this thing.

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We live in perilous times.  Within a few days, the United States might default on its debt, plunging the country into an unprecedented catastrophe.  Meanwhile, the tragedy in Norway (a country I’ll visit for the first time next month) reminds us that the civilized world faces threats from extremists of every ideology.  All this news, of course, occurs against the backdrop of record-breaking heatwaves, the decimation of worldwide fish stocks, the dwindling supply of accessible oil, and the failure of the Large Hadron Collider to find supersymmetry.

But although the future may have seldom seemed bleaker, I want people to know that we in MIT’s complexity theory group are doing everything we can to respond to the most pressing global challenges.  And nothing illustrates that commitment better than a beautiful recent paper by my PhD student Andy Drucker (who many of you will recognize from his years of insightful contributions to Shtetl-Optimized: most recently, solving an open problem raised by my previous post).

Briefly, what Andy has done is to invent—and demonstrate—a breakthrough method by which anyone, including you, can easily learn to multiply ten-digit numbers in your head, using only a collection of stock photos from Flickr to jog your memory.

Now, you might object: “but isn’t it cheating to use a collection of photos to help you do mental math—just like it would be cheating to use pencil and paper?”  However, the crucial point is that you’re not allowed to modify or rearrange the photos, or otherwise use them to record any information about the computation while you’re performing it.  You can only use the photos as aids to your own memory.

By using his method, Andy—who has no special mental-math training or experience whatsoever—was able to calculate 9883603368 x 4288997768 = 42390752785149282624 in his head in a mere seven hours.  I haven’t tried the method myself yet, but hope to do so on my next long plane flight.

Crucially, the “Flickr method” isn’t limited to multiplication.  It works for any mental memorization or calculation task—in other words, for simulating an arbitrary Boolean circuit or Turing machine.  As I see it, this method provides probably the most convincing demonstration so far that the human brain, unaided by pencil and paper, can indeed solve arbitrary problems in the class P (albeit thousands of times more slowly than a pocket calculator).  In his paper, Andy discusses possible applications of the method for cognitive science: most notably, using it to test conjectures about the working of human memory.  If that or other applications pan out, then—like many other research projects that seem explicitly designed to be as useless as possible—Andy’s might end up failing at that goal.

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Michael Mitzenmacher asked me to post the following announcement:

On August 29-30, 2011, there will be a conference in celebration of Michael Rabin‘s 80th birthday at the Harvard School of Engineering & Applied Sciences.    The speakers include Yonatan Aumann, Michael Ben-Or, Richard Karp, Dick Lipton, Silvio Micali, Michael Mitzenmacher, David Parkes, Tal Rabin, Ron Rivest, Dana Scott, Madhu Sudan, Salil Vadhan, Moshe Vardi, and Avi Wigderson.  The conference is open to the public, but registration is required by August 25.  For more information, see the conference website at https://www.events.harvard.edu/web/4352.

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The poll is here; my (slightly-edited) responses are below.  It took heroic self-restraint, but I tried to answer with straightforward statements of what I actually think, rather than ironic humor.

1. Do you think P=NP or not? You may give other answers as well.

I think P≠NP (on more-or-less the same grounds that I think I won’t be devoured tomorrow by a 500-foot-tall salsa-dancing marmoset from Venus, despite my lack of proof in both cases).

2. When do you think it will be resolved?

In his recent book The Beginning of Infinity, David Deutsch argues that we can’t even make decent probabilistic predictions about a future event, to whatever extent that event depends on new knowledge being created.  I agree with him on this: a proof of P≠NP, like other major mathematical advances, would depend almost entirely on new knowledge, and because of that, my uncertainty applies not only to the approximate number of years but to the approximate log of that number: decades, centuries, millennia, who knows?  Maybe the question should be rephrased: “will humans manage to prove P≠NP before they either kill themselves out or are transcended by superintelligent cyborgs?  And if the latter, will the cyborgs be able to prove P≠NP?”

3. What kinds of techniques do you think will be used?

Obviously I don’t know—but if we look at the techniques used in (say) Ryan Williams’ recent result, and then remember that that proof only separates NEXP from ACC⁰, we can get a weak hint about the scale of the techniques that would be needed for problems like P vs. NP.  Right now, Mulmuley’s GCT is the only approach out there that even tries to grapple with the biggest barrier we know, beyond even relativization, natural proofs, and algebrization: the barrier that many nontrivial problems (including matching and linear programming) are in P!  That’s not to say Mulmuley’s specific program will succeed: indeed, I suspect that the right chain of reasoning might diverge from Mulmuley’s at an earlier rather than later point.  But even for the seemingly-easier permanent versus determinant problem, I fear Mulmuley is basically right that the key insights lie in yellow books yet to be written.

4. Will the problem still be relevant given advances in algorithms and in SAT Solvers?

Yes, in the same way the Second Law of Thermodynamics is still relevant given advances in hybrid cars.

5. Feel free to comment on anything else: Graph Isomorphism, Factoring, Derandomization, Quantum computers, and/or your own favorite problem.

Graph Isomorphism: Probably in P.

Factoring: Probably hard for classical computers, but unlike with NP-complete problems, if it isn’t then we’re still living on Earth.

Derandomization: I think P=BPP (with essentially the same strength of conviction as P≠NP), and likewise L=RL, etc.

Quantum computing: I think BPP≠BQP (though not with the same strength of conviction as P≠NP), and also predict that no bizarre changes to quantum mechanics will be discovered of the sort needed to make scalable quantum computing impossible.

For those who are still reading, as a special bonus I present my answers to the large and interesting questions asked by a commenter on my last post named Mike S.

One thing I’ve heard before about NP(-hard) problems is that often certain instances are much harder than others. What are your feelings on the physical practicality of a computer that solves only most cases of NP(-hard) problems quickly? Also, is determining the ‘difficulty’ of particular instances of NP-complete problems NP(-hard)?

It depends what you mean by “most”! I think it’s almost certainly possible to generate a probability distribution over 3SAT instances almost all of which are hard (indeed, that assumption is central to modern cryptography). As one example, the approximate shortest vector problem is known to be just as hard on average as it is in the worst case, and it can easily be reduced to 3SAT. Another candidate is random k-SAT instances at the “critical ratio” of clauses to variables, for k≥4.

But maybe what you meant was those instances of NP-hard problems that “typically arise in real life.” Here all sorts of issues come into play: for example, often the instances that arise in practice have symmetries or other structure that makes them easy. And often your goal is not to find the best solution, but just a better solution than your competitors. And often we terminate trains of thought long before they lead to hard instances of NP-complete problems—we’re usually not even conscious that that’s what we’re doing; we just have an intuition that “such-and-such would require a hopeless search.”

But at the same time, when we do ask explicitly for optimal solutions, that request for optimality often has a way of finding the hard instances for us.

Less seriously, you said something along the lines of ‘P!=NP keeps mathematicians in business’. If math is so hard computationally, how do WE do it? Or on the other hand, if the computational complexity of certain problems is a fundamental property of the universe, and we are part of the universe, doesn’t it follow that we could make computers that are as good or better at doing math than we are?

The short answer is that math (as practiced by humans) is an extremely hit-or-miss business!  A billion years of evolution have equipped us with a lot of useful heuristics, as has the much faster evolution of mathematical ideas over the last few thousand years.

Probably even more important, we normally don’t care about arbitrary mathematical questions (does this random Turing machine halt?), but only questions that arise in some explanatory framework. And that criterion tends to select extremely strongly for questions that we can answer! Why it does so is a profound question itself, but whatever the answer, the history of math provides overwhelming evidence that it does. Goldbach’s Conjecture and the Collatz 3x+1 Conjecture are more-or-less “arbitrary” questions (at least in our present state of knowledge), and indeed they haven’t been answered yet. Fermat’s Last Theorem might have seemed pretty arbitrary at first (Gauss regarded it as such), but it wasn’t.  Indeed, in the 1980s it was embedded into the deep explanatory framework of elliptic curves and modularity, and a decade later it was solved.

Of course, despite these factors in mathematicians’ favor, they’re very far from having a general-purpose method to solve all the problems they want solved.

Incidentally, “P≠NP means computers can never replace human mathematicians” is a forehead-bangingly common misunderstanding. Personally, I see no reason why the brain couldn’t be simulated by computer (neuron-by-neuron if necessary), and P≠NP does nothing to challenge that belief.  All P≠NP suggests is that, once the robots do overtake us, they won’t have a general-purpose way to automate mathematical discovery any more than we do today.

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For those who are interested, video of my Buhl Lecture in Physics at Carnegie Mellon is now available on YouTube.  (The lecture was on April 29; the topic was “Quantum Computing and the Limits of the Efficiently Computable.”)  Thanks to everyone at CMU for their amazing hospitality.

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Shafi Goldwasser has asked me to announce that the next Innovations in Theoretical Computer Science (ITCS) conference—previously called Innovations in Computer Science (ICS)—will be held January 8-10, 2012 in Cambridge, MA, the first I(T)CS to be held outside of China.   The submission deadline is August 7.  The call for papers is here, and the conference website is here.

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You’re deep in the Congo.  You’ve got an iPhone with some charge left, but there’s no cell tower for hundreds of miles.  With life-or-death urgency, you need to know the definition of the complexity class SBP and its relationship to BPP_path.  What do you do?

Not to worry: Satoshi Hada has created a free Complexity Zoo app for the iPad and iPhone.  I tried it out and it works great!

You get a cold call from yet another solitary genius who’s discovered a simple linear-time 3SAT algorithm.  You tell him to implement the algorithm and test it out, and then you’ll talk.  Half an hour later, he tells you he’s done so and it works perfectly.  So you tell him to go factor the 617-digit RSA challenge number.  But being an iconoclastic genius, he never learned how to reduce factoring to 3SAT.  What do you do?

Relax: answering challenge #2 from this blog post even before the post went up, USC students Henry Yuen and Joe Bebel have created a great web application called ToughSAT, which generates hard SAT instances on demand, based on factoring, subset sum, or even a “hard problem cocktail.”  As a commenter helpfully alerted us, a few years ago Paul Purdom and Amr Sabry of Indiana University already created a similar site that generates hard SAT instances based on factoring.

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Here’s an interview with me by math grad student Samuel Hansen, as part of a podcast he runs called Strongly Connected Components.  (Also check out the interviews with Steven Rudich, Steven Rudich a second time, Lance Fortnow, Doron Zeilberger, and your other favorite stars of the nerdosphere!)  In the interview, I talk about my passion for baseball stats, what you don’t know about llama-breeding, the use of color in Matisse’s later works … oh all right, it’s mostly about quantum computing and P vs. NP.

Here’s a story I told for an event called Story Collider, which was back-to-back with a superb production of Breaking the Code (Hugh Whitemore’s acclaimed play about the life of Alan Turing) in Cambridge’s Central Square Theater.  I was honored to serve as a “scientific consultant” to the Breaking the Code production, and to do audience Q&A before and after a couple performances.  In the Story Collider, I talk about the “Turing phase” I went through as a teenager and Alan T.’s impact on my life.

(Note: For the past couple years, I’ve avoided talking much about my personal life on this blog, since I pride myself on being someone who learns from experience and adjusts his behavior accordingly.  But two months ago, something truly happy occurred in my life, and if you listen to the end of the Story Collider, you’ll find out what it was…)

One last personal note: I’m at the Federated Computing Research Conference in San Jose all week.  If you read Shtetl-Optimized, are here at FCRC, see me, and wouldn’t do so otherwise, come and say hi!

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