Deborah Belle is a psychology professor at Boston University (BU) who is interested in gender differences in social behavior. She has reported a shocking result about bias.
Today I thought I would discuss the issue of gender bias and also the related issue of the advantages of diversity.
Lately at Tech we have had a long email discussion on implicit bias and how we might do a better job of avoiding it in the future. My usual inclination is to think about such issues and see if there is some science behind our assumptions. One colleague stated:
The importance of diversity is beyond reasonable doubt, isn’t it?
I agree. But I am always looking for “proof.”
Do not get me wrong. I have always been for diversity. I helped hire the first female assistant professor to engineering at Princeton decades ago. And I have always felt that it is important to have more diversity in all aspects of computer science. But is there some science behind this belief? Or is it just axiomatic—something that we believe and needs no argument—that it is “beyond reasonable doubt?”
This is how I found Deborah Belle, while looking on the web for “proof.” I will just quote the BU Today article on her work:
Here’s an old riddle. If you haven’t heard it, give yourself time to answer before reading past this paragraph: a father and son are in a horrible car crash that kills the dad. The son is rushed to the hospital; just as he’s about to go under the knife, the surgeon says, “I can’t operate—that boy is my son!” Explain …
If you guessed that the surgeon is the boy’s gay, second father, you get a point for enlightenment… But did you also guess the surgeon could be the boy’s mother? If not, you’re part of a surprising majority.
In research conducted by Mikaela Wapman […] and Deborah Belle […], even young people and self-described feminists tended to overlook the possibility that the surgeon in the riddle was a she. The researchers ran the riddle by two groups: 197 BU psychology students and 103 children, ages 7 to 17, from Brookline summer camps.
In both groups, only a small minority of subjects—15 percent of the children and 14 percent of the BU students—came up with the mom’s-the-surgeon answer. Curiously, life experiences that might [prompt] the ‘mom’ answer “had no association with how one performed on the riddle,” Wapman says. For example, the BU student cohort, where women outnumbered men two-to-one, typically had mothers who were employed or were doctors—“and yet they had so much difficulty with this riddle,” says Belle. Self-described feminists did better, she says, but even so, 78 percent did not say the surgeon was the mother.
This shocked me. I knew this riddle forever it seems. But was surprised to see that the riddle is still an issue. Ken recalls from his time in England in the 1980s that surgeons were elevated from being addressed as “Doctor X” to the title “Mister X.” No mention of any “Miss/Mrs/Ms” possibility then, but this is now. I think this demonstrates in a pretty stark manner how important it is to be aware of implicit bias. My word, things are worse than I ever thought.
I looked some more and discovered that there was, I believe, bias in even studies of bias. This may be even more shocking: top researchers into the importance of diversity have made implicit bias errors of their own. At least that is how I view their research.
Again I will quote an article, this time from Stanford:
In 2006 Margaret Neale of Stanford University, Gregory Northcraft of the University of Illinois at Urbana-Champaign and I set out to examine the impact of racial diversity on small decision-making groups in an experiment where sharing information was a requirement for success. Our subjects were undergraduate students taking business courses at the University of Illinois. We put together three-person groups—some consisting of all white members, others with two whites and one nonwhite member—and had them perform a murder mystery exercise. We made sure that all group members shared a common set of information, but we also gave each member important clues that only he or she knew. To find out who committed the murder, the group members would have to share all the information they collectively possessed during discussion. The groups with racial diversity significantly outperformed the groups with no racial diversity. Being with similar others leads us to think we all hold the same information and share the same perspective. This perspective, which stopped the all-white groups from effectively processing the information, is what hinders creativity and innovation.
Nice study. But why only choose to study all-white groups and groups of two whites and one black? What about the other two possibilities: all black and two blacks and one white? Did this not even occur to the researchers? I could imagine that all-black do the best, or that two black and one white do the worst. Who knows. The sin here seems to be not even considering all the four combinations.
Tolga Bolukbasi, Kai-Wei Chang, James Zou, Venkatesh Saligrama, Adam Kalai have a recent paper in NIPS with the wonderful title, “Man is to Computer Programmer as Woman is to Homemaker? Debiasing Word Embeddings.”
Again we will simply quote the paper:
The blind application of machine learning runs the risk of amplifying biases present in data. Such a danger is facing us with word embedding, a popular framework to represent text data as vectors, which has been used in many machine learning and natural language processing tasks. We show that even word embeddings trained on Google News articles exhibit female/male gender stereotypes to a disturbing extent. This raises concerns because their widespread use, as we describe, often tends to amplify these biases. Geometrically, gender bias is first shown to be captured by a direction in the word embedding. Second, gender neutral words are shown to be linearly separable from gender definition words in the word embedding. Using these properties, we provide a methodology for modifying an embedding to remove gender stereotypes, such as the association between the words receptionist and female, while maintaining desired associations such as between the words queen and female. Using crowd-worker evaluation as well as standard benchmarks, we empirically demonstrate that our algorithms significantly reduce gender bias in embeddings while preserving the its useful properties such as the ability to cluster related concepts and to solve analogy tasks. The resulting embeddings can be used in applications without amplifying gender bias.
Here is one of their examples. Suppose we want to fill X in the analogy, “he is to doctor as she is to X.” A typical embedding prior to their algorithm may return X = nurse. Their hard-debiasing algorithm finds X = physician. Yet it recognizes cases where gender distinctions should be preserved, e.g., given “she is to ovarian cancer as he is to Y,” it fills in Y = prostate cancer. Their results show that their hard-debiasing algorithm performs significantly better than a “soft-debiasing” approach and performs as well or nearly as well on benchmarks apart from gender bias.
Overall, however, many have noted that machine learning algorithms are inhaling the bias that exists in lexical sources they data-mine. ProPublica has a whole series on this, including the article, “Breaking the Black Box: How Machines Learn to be Racist.” And sexist, we can add. The examples are not just linguistic—they include real policy decisions and actions that are biased.
Ken wonders whether aiming for parity in language will ever be effective in offsetting bias. Putting more weight in the center doesn’t achieve balance when all the other weight is on one side.
The e-mail thread among my colleagues centered on the recent magazine cover story in The Atlantic, “Why is Silicon Valley so Awful to Women?” The story includes this anecdote:
When [Tracy] Chou discovered a significant flaw in [her] company’s code and pointed it out, her engineering team dismissed her concerns, saying that they had been using the code for long enough that any problems would have been discovered. Chou persisted, saying she could demonstrate the conditions under which the bug was triggered. Finally, a male co-worker saw that she was right and raised the alarm, whereupon people in the office began to listen. Chou told her team that she knew how to fix the flaw; skeptical, they told her to have two other engineers review the changes and sign off on them, an unusual precaution.
One of my colleagues went on to ascribe the ‘horribleness’ of many computer systems in everyday use to the “brusque masculinism” of their creation. This leads me to wonder: can we find the “proof” I want by making a study of the possibility that “men are buggier”—or more solidly put, that gender diversity improves software development?
Recall Ken wrote a post on themes connected to his department’s Distinguished Speaker series for attracting women into computing. The series includes our own Ellen Zegura on April 22. The post includes Margaret Hamilton and her work for NASA’s Apollo missions, including the iconic photo of the stack of her code being taller than she. Arguments over the extent of Hamilton’s role can perhaps be resolved from sources listed here and here, but there is primary confirmation of her strong hand in code that had to be bug-free before deployment.
We recently posted our amazement of large-scale consequences of bugs in code at underclass college level, such as overflowing a buffer. Perhaps one can do a study of gender and project bugs from college or business applications where large data could be made available. The closest large-scale study we’ve found analyzed acceptance rates of coding suggestions (“pull requests”) from over 1.4 million users of GitHub (news summary) but this is not the same thing as analyzing design thoroughness and bug rates. Nor is anything like this getting at the benefits of having men and women teamed together on projects, or at least in a mutual consulting capacity.
It is easy to find sources a year ago hailing that study in terms like “Women are Better Coders Than Men…” Ordinarily that kind of “hype” repulses Ken and me, but Ken says maybe this lever has a rock to stand on. What if we ‘think different’ and embrace gender bias by positing that women approach software in significantly different ways—?—where having such differences is demonstrably helpful.
What would constitute “proof” that gender diversity is concretely helpful?
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Littlewood’s Law and Big Data
“Leprechaun-proofing” data source |
Neil L. is a leprechaun. He has visited Dick on St. Patrick’s Day or the evening before many times. Up until this night I had never seen him.
Today, Neil’s message is more important than ever.
With over a foot of snow in Buffalo this week and the wind still howling, I was not expecting anything green. Long after Debbie had gone to bed, I was enmeshed in the “big–data blues” that have haunted me since summer and before. I was so fixated it took me more than a few seconds to realize that wisps of green smoke floating between me and the computer screen were something I should investigate.
There on our kitchen-study divider sat Neil. He looked like the pictures Dick had posted of him, but frazzled. He cleaned his pipe into a big Notre Dame coffee mug I got as a gift. I’d had it out since Princeton went up against Notre Dame in “March Madness”—my Tigers missed a chance for a big upset in the closing five seconds. As if reading my mind, he remarked how the tournament always produces upsets in the first round:
“If there be no unusual results, ‘twould be most unusual.”
The Neil whom Dick described would have said this with wry mirth, but he sounded weary as if he had a dozen mouths to feed. I fired up the kettle and brought out the matching mug to offer tea or coffee, but he pointed to his hip flask and said “it’s better against the cold.”
That prompted me to ask, “Why didn’t you visit Dick? He and Kathryn have been enjoying sun at this week’s Bellairs workshop on Barbados.” I had been there two years ago when Neil had taken great pains to track Dick down. Neil puffed and replied, “Same reason I didn’t try finding you there back then—too far afield for a big family man.” The word “family” struck me as our dog Zoey, who had stayed sleeping in her computer-side bed at my feet, woke up to give Neil a barkless greeting. Of course, even leprechauns have relations…
Nodding to pictures of our children on the wall, I asked Neil how many he had. He took a long puff and replied:
“Several thousand. It’s too hard to keep count nowadays.”
Now Zoey barked, and this covered my gasp. Knowing that Neil was several centuries old, I did some mental arithmetic, but concluded he would still need a sizable harem. Reading my mind again, Neil cut in:
“Not as ye mortals do. What d’ye think we’re made of?”
I reached out to touch him, but Neil leaned away and vanished. A moment later he popped back and folded his arms, waiting for me to reply. I realized, ah, he is made of spirit matter. What can that be? Only one thing in this world it could be: information.
“Tá tú ceart” he whistled. “Right. And some o’ yer sages wit ye mortals have some o’ the same stuff. Max Tegmark, for one, wrote:
“… consciousness is the way information feels when being processed.”
And Roger Penrose has just founded a new institute on similar premises—up front he says chess holds a key to human consciousness so you of all people should know whereof I speak.”
Indeed, I had to nod. He continued, “And information has been growing faster than Moore’s Law. Hard to keep up…” The last words came with a puff of manly pride.
“Information is leprechauns??,” I blurted out. The propagation of “fake news” and outright falsehoods in recent months has been hard enough to take, but this boiled me over. I wanted to challenge Neil—and I recalled the protocol followed by William Hamilton’s wife: glove and shamrock at his feet. Well, I don’t wear gloves even in zero-degree weather, and good luck my finding a shamrock under two feet of snow. So I asked in a level voice, “can you give me some examples?”
Neil puffed and replied, “Not that information be us, but it bears us. And more and more ye can get to know us by reading your information carefully. But alas, more and more ye are confusing us with aliens.”
“Aliens?” This was all too much, and the dog wanted out. But Neil was happy to flit alongside me as I opened the door to the yard for her. He explained in simple tones:
“Ye have been reading the sky for many decades listening for alien intelligence. Up to last year ye had maybe one possible instance in 1977—apart from Nikola Tesla, who knew us well. But now reports are coming fast and furious. Not just fleeting sequences but recurrent ‘fast radio bursts’ observed in papers and discussed even this week by scientists from Harvard. Why so many now?”
I was quick to answer: “Because we are reading so much more data now.” Neil clapped his hands—I expected something to materialize by magic but he was just affirming my reply. I hedged, “But surely we understand the natural variation?” Neil retorted:
“Such understanding didn’t prevent over 500 physics papers being written on a months-long fluctuation at the Large Hadron Collider before it dissolved last summer.”
Indeed, the so-called diphoton anomaly had seemed on its way to confirmation because two separate experiments at the LHC were seeing it. An earlier LHC anomaly about so-called “penguin decays” has persisted since 2013 with seemingly no conclusion.
As I let the dog back in and toweled snow off her, I reflected: what was wrong with those 500 physics papers? A particle beyond the Standard Model would be the pot of gold at the end of a rainbow not only for many researchers but human knowledge on the whole. Then I remembered whom I was speaking with. Once free of the towel, Zoey scooted away, and I regrouped. I turned to Neil and said, “There is huge work on anomaly detection and data cleansing to identify and remove spurious data. Surely we are scaling that up as needed…”
Neil took a long drag on his pipe and arched up:
“I be not talking o’ bad data points but whole data sets, me lad.”
I sank into an armchair and an electrical voltage drop dimmed the lights as Neil took over, perched again on the divider. “Ye know John Littlewood’s law of a miracle per month, indeed you wrote a post on it. If ye do a million things or observe a million things, one o’ them is bound to be a million-to-one shot.”
I nodded, already aware of his point.
“No different ’tis with data sets. One in a million be one-in-a-million bad. A thousand in a million be—begorra—one-in-a-thousand bad. Or too good. If ye ha’e 50,000 companies and agencies and research groups doing upwards of 20 data sets each, that’s wha’ ye have. Moreover—”
Neil leaned forward enough to fall off the counter but of course he didn’t fall.
“All the cleansing, all the cross-validation ye do, all the confirmation ye believe, is merely brought inside this reckoning. All that also changes the community standards, and by those standards ye’re still one-in-a-million, one-in-a-thousand off. Now ye may say, 999-in-a-thousand are good, a fair run o’ the mill. But think of the impacts. Runs o’ the mill have run o’ the mill effects, but the stark ones, hoo–ee.”
He whistled. “The impacts of the ones we choose to reside in scale a thousand-to-one stronger, a million-to-one… An’ that is how we keep up a constant level of influence in affairs o’ the world. All o’ the world—yer hard science as well as social data.”
I thought of something important: “If you lot choose to commandeer one data set, does that give you free rein to infect another of the same kind?”
“Nae—ye know from Dick’s accounts, we must do our work within the bounds of probability. So if ye get a whiff of us or even espy us, ye can take double the data without fear of us. But—then ye be subject to the most subtle kind of sampling bias, which is the bias of deciding when to stop sampling.”
After the terrible anomaly I showed in December from four data points of chess players rated near 2200, 2250, 2300, and 2350 on the Elo scale, I had spent much of January filling in 2225, 2275, 2325, and 2375. Which improved the picture quite a lot. Of course I ran all the quarter-century marks from Elo 1025 to Elo 2775, over three million more moves in all. But instead of feeling pride, after Neil’s last point I looked down at the floor.
His final words were gentle:
“Cheer up lad, it not only could be worse, it would be worse. Another o’ your sages, Nassim Taleb, has pointed out what he calls the ‘tragedy of big data’: spurious correlations and falsity grow faster than information. See that article’s graphic, which looks quadratic or at any rate convex. Then be ye thankful, for we Leprechauns are hard at work keeping the troubles down to linear. But this needs many more of us, lad, so I must be parting anon.”
And with a pop he was gone.
Is Neil right? What examples might you know of big data sets suspected of being anomalous not for any known systematic reason but just the “luck of the draw”?
Happy St. Patrick’s Day anyway.
[some word changes]
Holly Dragoo, Yacin Nadji, Joel Odom, Chris Roberts, and Stone Tillotson are experts in computer security. They recently were featured in the GIT newsletter Cybersecurity Commentary.
Today, Ken and I consider how their comments raise a basic issue about cybersecurity. Simply put:
Is it possible?
In the column, they discuss various security breaks that recently happened to real systems. Here are some abstracts from their reports:
The last is an attempt to make attacks harder by using randomization to move around key pieces of systems data. It seems like a good idea, but Dan Boneh and his co-authors have shown that it can be broken. The group is Hovav Shacham, Eu-Jin Goh, Matthew Page, Nagendra Modadugu, and Ben Pfaff.
Here we talk about the first item at length, plus another item by Odom on the breaking of a famous hash function.
With all due respect to a famous song by Sonny Bono and Cherilyn Sarkisian, “The Beat Goes On“: I have changed it some, but I think it captures the current situation in cybersecurity.
The breaks go on, the breaks go on
Drums keep pounding
A rhythm to the brain
La de da de de, la de da de daLaptops was once the rage, uh huh
History has turned the page, uh huh
The iPhone’s the current thing, uh huh
Android is our newborn king, uh huh[Chorus]
A definition of insanity ascribed to Albert Einstein goes:
Insanity is doing the same thing over and over again and expecting different results.
I wonder lately whether we are all insane when it comes to security. Break-ins to systems continue; if anything they are increasing in frequency. Some of the attacks are simply so basic that it is incredible. One example is an attack on a company that is in the business of supplying security to their customers. Some of the attacks use methods that have been known for decades.
Ken especially joined me in being shocked about one low-level detail in the recent “Cloudbleed” bug. The company affected, Cloudflare, posted an article tracing the breach ultimately to these two lines of code that were auto-generated using a well-known parser-generator called Ragel:
if ( ++p == pe ) goto _test_eof;
The pointer p is in client hands, while pe is a system pointer marking the end of a buffer. It looks like p can only be incremented one memory unit at a time, so that it will eventually compare-equal to pe and cause control to jump out of the region where the client can govern HTML being processed. Wrong. Other parts of the code make it possible to enter this test with p > pe which allows undetected access to unprotected blocks of memory. Not only was it a memory leak but private information could be exposed.
The bug was avoidable by rewriting the code-generator so that it would give:
if ( ++p >= pe ) goto _test_eof;
But we have a more basic question:
Why are such low-level bits of 1960s-vintage code carrying such high-level responsibility for security?
There are oodles of such lines in deployed applications. They are not even up to the level of the standard C++ library which gives only == and != tests for basic iterators but at least enforces that the iterator must either be within the bounds of the data structure or must be on the end. Sophisticated analyzers help to find many bugs, but can they keep pace with the sheer volume of code?
Note: this code was auto-generated, so we not only have to debug actual code but potential code as well. The Cloudflare article makes clear that the bug turned from latent to actual only after a combination of other changes in code system patterns. It concludes with “Some Lessons”:
The engineers working on the new HTML parser had been so worried about bugs affecting our service that they had spent hours verifying that it did not contain security problems.
Unfortunately, it was the ancient piece of software that contained a latent security problem and that problem only showed up as we were in the process of migrating away from it. Our internal infosec team is now undertaking a project to fuzz older software looking for potential other security problems.
While admitting our lack of expertise in this area, we feel bound to query:
How do we know that today’s software won’t be tomorrow’s “older software” that will need to be “fuzzed” to look for potential security problems?
We are still writing in low-level code. That’s the “insanity” part.
My GIT colleagues also comment on Google’s recent announcement two weeks ago of feasible production of collisions in the SHA-1 hash function. Google fashioned two PDF files with identical hashes, meaning that once a system has accepted one the other can be maliciously substituted. They say:
It is now practically possible to craft two colliding PDF files and obtain a SHA-1 digital signature on the first PDF file which can also be abused as a valid signature on the second PDF file… [so that e.g.] it is possible to trick someone to create a valid signature for a high-rent contract by having him or her sign a low-rent contract.
Now SHA-1 had been under clouds for a dozen years already, since the first demonstration that collisions can found with expectation faster than brute force. It is, however, still being used. For instance, Microsoft’s sunset plan called for its phase 2-of-3 to be enacted in mid-2017. Google, Mozilla, and Apple have been doing similarly with their browser certificates. Perhaps the new exploit will force the sunsets into a total eclipse.
Besides SHA-2 there is SHA-3 which is the current gold standard. As with SHA-2 it comes in different block sizes: 224, 256, 384, or 512 bits, whereas SHA-1 gives only 160 bits. Doubling the block size does ramp up the time for attacks that have been conceived exponentially. Still, the exploit shows what theoretical advances plus unprecedented power of computation can do. Odom shows the big picture in a foreboding chart.
Is security really possible? Or are we all insane?
Ken thinks there are two classes of parallel universes. In one class, the sentient beings originally developed programming languages in which variables were mutable by default and one needed an extra fussy and forgettable keyword like const to make them constant. In the other class, they first thought of languages in which identifiers denoted ideal Platonic objects and the keyword mutable had to be added to make them changeable.
The latter enjoyed the advantage that safer and sleeker code became the lazy coder’s default. The mutable strain was treated as a logical subclass in accord with the substitution principle. Logical relations like Square “Is-A” Rectangle held without entailing that Square.Mutable be a subclass of Rectangle.Mutable, and this further permitted retroactive abstraction via “superclassing.” They developed safe structures for security and dominated their light cones. The first class was doomed.
[word changes in paragraph after pointer code: “end-user” –> “client”, HTML being “coded” –> “processed”.]
YouTube source |
Maurice Ashley is an American chess grandmaster. He played for the US Championship in 2003. He coached two youth teams from Harlem to national championships and played himself in one scene of the movie Brooklyn Castle. He created a TEDYouth video titled, “Working Backward to Solve Problems.”
Today we discuss retrograde analysis in chess and other problems, including one of my own.
Raymond Smullyan popularized retrograde chess puzzles in his 1979 book The Chess Mysteries of Sherlock Holmes and its 1981 sequel, The Chess Mysteries of the Arabian Knights. Here is the second example in the first book—except that I’ve added a white pawn on b4. What were the last three moves—two by White and one by Black—and can we tell the two moves before that?
Not only is Black checkmated, Black is outgunned on material. The puzzles do not try to be fair and many are “unnatural” as game positions. The point is that the positions are legal. There can occur in games from the starting position of chess—but only in certain ways that the solver must deduce.
White’s last move must have uncovered check from the white bishop on h1 because the bishop cannot have moved there. The uncovering could have come from White’s pawn on g2 capturing a black piece on h3 except that there is no way a bishop can get to h1 locked behind a white pawn on g2. So it must have been from the pawn on d6. If the last move were d5-d6 discovered check, then what was Black’s previous move? Black’s king cannot come from being adjacent to White’s king on b8 or b7, so it came from a7—but there it would have been in an impossible double check. The whole setup looks impossible, until we realize that White could have captured a black pawn en-passant after it moved from d7 to d5 to block the check. Play could have unfolded from this position:
The game could have gone 1. Bd6-c5+ Ka7-a8 2. e4-e5+ d7-d5 3. exd6 en-passant and checkmate. So the last three moves must have been 2. e4-e5+ d7-d5 3. exd6. But what about the first two? Suppose we were told that the checkmating move is the first time a white piece ever occupied the square d6. So the game didn’t go this way. Could it have gone another way that obeys this extra condition? The answer is at the end.
Ashley’s video raises the idea of retrograde analysis for planning and problem solving in life. If your goal is , then working backward from can tell you the subgoals needed to achieve it. The business executive Justin Bariso expanded on Ashley’s video even further in a neat post on his own blog. Bariso recommends to “plan your project backward,” opining that with the way things are usually done,
More time and money are scheduled for initial steps than are really needed.
Here is an example from Smullyan’s book which was also featured in a 2011 video by former world women’s champion Alexandra Kosteniuk:
How can this position come about—in particular, where was White’s queen captured? Focus on the main events—what was captured on b3, e6, h6, and in what order?—helps to plan the play. Incidentally, Kosteniuk was a hard-luck loser this past Friday in the semifinals of this year’s women’s world championship in Tehran.
What matters often in research, though, is finding the most propitious initial steps—and the time budget is open-ended. Often we build a new tool and set out to prove theorems whose statements we might not know in advance. Yet for statements like “Is ?” the question, “Is a theorem?” is a classic retrograde problem: Proof steps are the moves. A legal move either instantiates an axiom or follows by a rule of deduction from one or more earlier steps. Undecidability for the underlying formal system means there is no procedure to tell whether any given position is legal.
How might retrograde analysis help in complexity theory? Dick and I once ventured a notion of “progressive” algorithms. Maybe it can be supplemented by analyzing some necessary “regressive” behavior of algorithms. Or more simply put, using retrograde analysis to show that a statement is impossible may be what’s needed to prove .
I taught a hybrid algorithms-and-complexity short-course at the University of Calcutta last August. One of my points was to present breadth-first search (BFS) and depth-first search (DFS) as paradigms that correspond to the complexity classes and in particular. I illustrated more complicated algorithms that run in deterministic or nondeterministic logspace and explained how in principle they could be reduced to one call to BFS.
I presented pebbling in the guise of Monty Python’s version of King Arthur and his fellow “riders.” They wish to conquer a network of towns connected by one-way roads. Each town has a defensive strength . To conquer , the riders need to occupy at least other towns with incoming roads, then any rider may occupy . A rider can leave a town and go “in-country” at any time, but to re-enter a previously conquered town, that town must be re-conquered. Source node(s) can be freely (re-)occupied, and multiple riders can occupy the same node. Given a graph with source(s) s and goal node f, and an integer , the question is:
Can f be conquered by k riders starting on s, and if not, what is the minimum k?
The case of pebbling, strictly speaking, is when always equals the node’s in-degree, while is basically BFS. Various forms of pebbling have been studied and all were instrumental to various complexity results. Here is a simple example:
The answer is : Three riders can conquer in the order , then moves to and enters This gives the strength needed for to conquer , but needs to be re-conquered. This is done by starting again from to , and finally falls.
Now picture the following graph in which every OR gate has hit strength and every AND gate has hit strength . The gate labeled b is undetermined. If it is OR, then gate f can be conquered by riders as follows: Two riders staring at and first conquer gate a, and then using the free entry into gate b, have a conquer d. Then the rider on b rides back to and helps the third rider go from to e in like manner. Finally the riders on d and e conquer f. That was easy—but what if b is an AND gate? Can riders still do it? You may wish to ponder before looking below.
I gave this as part of a take-home final to over 30 students in the course, saying to argue that when b is an AND gate, riders are not enough. Almost all tried various forms of forward reasoning in their proofs. Many such proofs were incomplete, for instance not considering that riders could rewind to start.
Only a few found the neatest proof I know, which is retrograde: Before f is conquered, there must be riders on d and e. One of those must have been the last to arrive, say e. This means the immediately previous step had riders on d, b, and c. The only move before that must have been from a conquering d, so we have proved the necessity of the configuration (a,b,c). And this configuration has no predecessor. So three riders cannot conquer f.
Can a more-general, more-powerful lower bound technique be built from this kind of retrograde reasoning?
Chess answers: In the first puzzle, if no White piece had previously occupied d6, there is still a way the game could go. Look at the second diagram and picture White’s bishop on c5 with a knight on b6. White can play knight to the corner discovering check and Black’s king can take it, giving the overall moves 1. Nb6-a8+ Ka7xa8 2. e4-e5+ d7-d5 3. exd6 en-passant and checkmate.
In the second chess puzzle, the only piece White could have captured on b3 was Black’s queenside bishop. In order for it to leave its initial square c8, however, Black needed to capture on e6. The only White piece able to give itself up there was the missing knight, because White’s queen could not escape until the move a2xb3 happened. So Black’s capture on h6 was of White’s queen. I can find a legal game reaching this position (with White to play) in 18 moves; is that the minimum?
Retrograde chess puzzles become far more intricate than the examples in this column suggest. Besides Smullyan’s books, the great trove for them is maintained by Angela and Otto Janko here. Joe Kisenwether has some examples from games other than chess.
[fixed that exam problem was “to argue that…”]
Amazon source |
When Raymond Smullyan was born, Emanuel Lasker was still the world chess champion. Indeed, of the 16 universally recognized champions, only the first, Wilhelm Steinitz, lived outside Smullyan’s lifetime. Smullyan passed away a week ago Monday at age 97.
Today, Dick and I wish to add some thoughts to the many comments and tributes about Smullyan.
He was known for many things, but his best-known contributions were books with titles like: “What Is the Name of This Book?” Besides their obit in Sunday’s paper, the New York Times ran a sample of puzzles from these books. No doubt many enjoyed the books, and many may have been moved to study “real” logic and mathematics. His book To Mock a Mockingbird dressed up a serious introduction to combinatory logic. This logic is as powerful as normal predicate calculus but has no quantified variables. So making it readable to non-experts is a tribute to Smullyan’s ability to express deep ideas in ways that were so clear.
Smullyan earned his PhD under the guidance of Alonzo Church in 1959 at age 40. When I (Ken) was an undergrad at Princeton, I remember thinking, “gee, that’s old.” Well there’s old and there’s old… As we noted in our profile of him 19 months ago, he was still writing books at a splendid pace at age 95, including a textbook on logic.
Neither of us met him, so we never experienced his tricks and riddles firsthand, but we had impressions on the serious side. Here are some of Dick’s, first.
Smullyan and Melvin Fitting, who was one of his nine PhD students, wrote a wonderful book on set theory: Set Theory and the Continuum Problem (Oxford Logic Guides), 1996.
The blurb at Amazon says:
Set Theory and the Continuum Problem is a novel introduction to set theory … Part I introduces set theory, including basic axioms, development of the natural number system, Zorn’s Lemma and other maximal principles. Part II proves the consistency of the continuum hypothesis and the axiom of choice, with material on collapsing mappings, model-theoretic results, and constructible sets. Part III presents a version of Cohen’s proofs of the independence of the continuum hypothesis and the axiom of choice. It also presents, for the first time in a textbook, the double induction and superinduction principles, and Cowen’s theorem.
The blurb at Amazon says:
A lucid, elegant, and complete survey of set theory [in] three parts… Part One’s focus on axiomatic set theory features nine chapters that examine problems related to size comparisons between infinite sets, basics of class theory, and natural numbers. Additional topics include author Raymond Smullyan’s double induction principle, super induction, ordinal numbers, order isomorphism and transfinite recursion, and the axiom of foundation and cardinals. The six chapters of Part Two address Mostowski-Shepherdson mappings, reflection principles, constructible sets and constructibility, and the continuum hypothesis. The text concludes with a seven-chapter exploration of forcing and independence results.
Wait—are these the same book? Yes they are, and this is one way of saying the book is chock full of content while being self-contained. Neither blurb mentions the part that most grabbed me (Dick). This is their use of modal logic to explain forcing.
Modal logic has extra operators which is usually interpreted as “Necessarily ” and meaning “Possibly “; like the quantifiers and they obey the relation
Saul Kripke codified models as directed graphs whose nodes each have an interpretation of . Then holds at a node if all nodes reachable from satisfy (and hence ), while holds at if holds at some node reachable from . The nodes are “possible worlds.”
What Fitting and Smullyan do is define a translation from set theory to their modal logic such that is valid if and only if . Then the game is to build a node such that every Zermelo-Fraenkel axiom gets a but the translated continuum hypothesis fails in some world reachable from .
One reprinting of the book posed an inadvertent puzzle: many mathematical symbols were omitted. Symbols for set membership, subset, quantifiers, and so on were missing. As one online reviewer noted, “it really does make the book useless.” My copy at least was unaffected.
A week after our 7/28/15 Smullyan post mentioned above, I (Ken again) went with my family to Oregon for vacation. This included a trip to Powell’s Books in Portland, which may be the largest independent bookstore in the world. The math and science sections were larger and more eclectic than any Barnes or college bookstores I’ve seen. There were several copies of the 1961 Princeton Annals paperback edition of Smullyan’s PhD thesis, Theory of Formal Systems, on sale for $20. I felt spurred to buy one and felt it could be useful because of Smullyan’s penchant for combinatorial concreteness.
Sure enough, section B of Chapter IV formulates the rudimentary relations crisply and clearly. Here are Smullyan’s words atop page 78 on their motivation:
As remarked in the Preface, our proof follows novel lines in that all appeal to the traditional number theory devices … in the past—e.g., prime factorization, congruences and the Chinese remainder theorem—are avoided. Thus Gödel’s program of establishing incompleteness, even of first-order theories involving plus and times as their some arithmetical primitives, can, by the methods of this section, be carried out without appeal to number theory.
Simply said: Smullyan avoids all the complicated numerical machinery needed in the usual treatments and makes them—like magic—disappear. The main predicate needed by Smullyan is is a power of , provided is prime. From that he defined a predicate meaning that the binary representation of is the concatenation of those of and . The formal language is still that of logic and numbers but the operations are really manipulating strings. His predicates were able to fulfill all roles for which the class of primitive recursive relations and subclasses involving and had previously been employed.
Smullyan was writing in 1959. Turing machine complexity had not even been defined yet. It transpired later that Smullyan’s class contains nondeterministic logspace and equals the alternating linear time hierarchy. Linear time by itself is annoyingly dependent on machine details, but once you have a couple levels of quantifier alternation the class becomes very robust. Dick employed tricks with alternating linear time in some papers, and such alternation is used to amplify the time hierarchy theorem so that for multitape Turing machines leads to a contradiction higher up. It is also intriguing to see Smullyan write on page 81:
We do not know whether or not all constructive arithmetic attributes are rudimentary. Quine […] has shown that plus and times are first order definable from [concatenation] … but this leaves unanswered the question as to whether plus and times are themselves rudimentary.
The thesis has a footnote saying this had been done for plus, and times follows from remarks above, but whether the predicate is in deterministic linear time remains open. It is likely that Smullyan went through similar concrete thinking as Juris Hartmanis and Richard Stearns when they conjectured no. We wonder if anyone thought to ask Smullyan about this and wish we had.
Our condolences go to his family along with our appreciation for his writings.
Emanuel Lasker philosophized in his 1906 book Struggle about perfect strategists in any walk of life, calling them Macheeides after Greek for “battle species.” An improved edition of Google DeepMind’s AlphaGo probably joined their ranks by beating top human players 60-0 in games played via online servers, not counting one game disrupted by connection failure. The top ranked player, Ke Jie, lost 3 games and landed in the hospital, but desires a 4th try. Where will computer Macheeides strike next?
William Agnew is the chairperson of the Georgia Tech Theoretical Computer Science Club. He is, of course, an undergraduate at Tech with a multitude of interests—all related to computer science.
Today I want to report on a panel that we had the other night on the famous P vs. NP question.
The panel consisted of two local people, one semi-local person, and two remote people—the latter were virtually present thanks to Skype. The local people were Lance Fortnow and myself, and the semi-local one was Dylan McKay. He was present at the panel, and was an undergraduate a bit ago at Tech. He is now a graduate student working with Ryan Williams, who both are moving from Stanford to MIT. The last was the Scott Aaronson who is not only an expert on P vs. NP, but also all things related to quantum computation.
An excellent panel, which I was honored to be part of. We had a large audience of students, who were no doubt there because of the quality of the panelists: although—sandwiches, drinks, and brownies—may have had some effect. They listened and asked some really good questions—some of which we could even answer.
The panel, like many panels, was fun to be on; and hopefully was informative to the audience. I believe the one point that all on the panel agreed with is: we do not know very much about the nature of computation, and there remains many many interesting things to learn about algorithms. I like the way Ryan put it:
We are like cave men and women banging rocks together and trying to see what happens.
This is not an exact quote, but you get the point: we are in the dark about what computers can and cannot do.
I thought I would summarize the panel by listing just a few questions that were discussed.
Scott recently released a 121-page survey on P versus NP. He did not read all of it during the panel. In fact he did not read any of it. It is chock full of content—for instance, the story about the Traveling Salesman Problem and Extended Formulations is told in a long footnote. It was partly supported by NSSEFF, which is not a phonetic spelling of NSF but stands for the National Security Science and Engineering Faculty Fellowship, soon to be renamed for Vannevar Bush.
It takes the stand that . Over half of the non-bibliography pages are in the section 6 titled “Progress.” This is great and completely up to date—not only through Ryan’s circuit lower bounds but also last year’s rebuff to the simplest attack in Ketan Mulmuley’s Geometric Complexity Theory paradigm. It details the three major barriers—relativization, “Natural Proofs,” and “Algebrization”—right in the context of where they impeached progress.
The climax in sections 6.4 and 6.6 is what Scott calls “ironic complexity” and Mulmuley calls the “flip”: the principle that to prove a problem X is harder to compute than we know, one may need to prove that another problem Y is easier to compute than we know. This gets dicey when the problems X and Y flow together. For instance, a natural proof that the discrete logarithm is nonuniformly hard to compute makes it nonuniformly easier to compute. Hence such a proof cannot give any more than a “half-exponential” lower bound (see this for definition). Ryan’s result, which originally gave a “third-exponential” lower bound on circuits for NEXP, proves lower bounds on a exponential scaling of SAT via upper bounds on an -like version; the two are brought a little closer by the former needing only succinct instances. Scott’s survey also emphasizes the fine line between “in-P” and “NP-hard” within cases of computational problems, arguing that if P=NP then we’d have found these lines fuzzed up long ago.
For my part—Ken writing this section—I’ve experienced a phenomenon that calls to mind our old post on “self-defeating sentences.” To evade the natural-proofs barriers, I’ve tried to base hardness predicates on problems that are hard for exponential space in terms of the number of variables in . The idea is to prove that circuits computing need size where is a counting function that scales with the complexity of , in analogy to the Baur-Strassen bounds where is the “geometric degree” of a variety associated to .
The Baur-Strassen tops out at when is a polynomial of degree , and since the low-degree polynomials we care about have , this accounts for why the best known arithmetical circuit lower bounds for natural functions are only . But extending the Baur-Strassen mechanism to double-exponentially growing would yield the exponential lower bounds we strive for. Candidates with names like “arithmetical degree” and (Castelnuovo-Mumford-)“regularity” abound, giving double-exponential growth and -hardness, but the latter sows the self-defeat: The hardness means there is a reduction to from length- instances of problems but the shortness of can make fail. I’ve described a based on counting “minimal monomials” in an ideal associated to , which although not necessarily complete still met the same defeat.
So maybe the constructive fact behind a problem’s NP-completeness also embodies a mirror image of a problem in P, so that we cannot easily tell them apart. NP-complete problems may “masquerade” as being in P—since the known ones are all isomorphic, if one does they all do. This may explain the success of SAT-solvers and suspicion about P=NP being independent as voiced during the panel. It also suggests that intermediate problems may bear attacking first.
At the conclusion of the panel Agnew, who moderated it skillfully, asked the question:
If you had to state what you believe “at gunpoint” what do you believe about P vs. NP?
He was holding a Nerf gun, but we still all seemed to take the threat seriously. Not surprisingly, all but one “fool” said that they believed that P is not equal to NP. The sole fool, me, said that they felt that P=NP. I have stated this and argued it many times before: see this for more details on why.
Of course P vs. NP remains open, and again as the panel all agreed—including the fool—we need new ideas to resolve it.
[deleted sentence about “not considering P=NP”]
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Shinichi Mochizuki has claimed the famous ABC conjecture since 2012. It is still unclear whether or not the claimed proof is correct. We covered it then and have mentioned it a few times since, but have not delved in to check it. Anyway its probably way above our ability to understand in some finite time.
Today I want to talk about how to check proofs like that of the ABC conjecture.
The issue is simple:
Someone writes up a paper that “proves” that X is true, where X is some hard open problem. How do we check that X is proved?
The proof in question is almost always long and complex. So the checking is not a simple matter. In some cases the proof might even use nonstandard methods and make it even harder to understand. That is exactly the case with Mochizuki’s proof—see here for some comments.
Let’s further assume that the claimed proof resolves X which is the P vs. NP problem. What should we do? There are some possible answers:
Every once in a while he would get up and join our table to gossip or kibitz Then he would add, “The bigger my proof, the smaller the hole. The longer and larger the proof, the smaller the hole.”
…[N]o one wants to be the guy that spent years working to understand a proof, only to find that it was not really a proof after all.
P Does Not Equal NP: A Proof Via Non-Linear Fourier Methods
Alice Azure with Bob Blue
Here the “with” signals that Alice is the main author and Bob was simply a helper. Recall a maxim sometimes credited to President Harry Truman: “It is amazing what you can accomplish if you do not care who gets the credit.”
What do you think about ways to check proofs? Any better ideas?
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Impetus to study a new reducibility relation
See Mike’s other projects too |
Michael Wehar has just earned his PhD degree in near-record time in my department. He has posted the final version of his dissertation titled On the Complexity of Intersection Non-Emptiness Problems which he defended last month. The dissertation expands on his paper at ICALP 2014, joint paper at ICALP 2015 with Joseph Swernofsky, and joint paper at FoSSaCS 2016.
Today, Dick and I congratulate Mike on his accomplishment and wish him all the best in his upcoming plans, which center on his new job with CapSen Robotics near Pitt and CMU in Pittsburgh.
Mike’s thesis features a definition that arguably has been thought about by researchers in parameterized complexity but passed over as too particular. If I were classifying his definition like salsa or cheddar cheese,
then Mike’s definition would rate as
Too sharp, that is, for -notation, thus putting it beyond our usual complexity theory diet. But the problem it addresses has also seemed beyond the reach of complexity theory while we can’t even get a clear handle on versus :
What are the relationships between the levels of deterministic versus nondeterministic time and/or space, and versus or levels of fairly-counted deterministic and nondeterministic space?
“Fairly counted” space means that we cast away the “linear space-compression” theorem by restricting Turing machines to be binary, that is to use work alphabet . The blank character can be read but not written. This doesn’t mean swearing off -notation for space, but does mean employing it more carefully.
Questions about particular polynomial-time exponents and space-usage factors animate fine-grained complexity, which has seen a surge of interest recently. Consider this problem which is central in Mike’s thesis:
The “naive” Cartesian product algorithm works in time basically . The fine-grained question is, can one do better than naive? As we covered before, Mike’s early work showed that getting time would have the sharp consequence , which improved the conclusion in a 2003 paper by Dick with George Karakostas and Anastasios Viglas.
Mike’s thesis expands the mechanism that produces this and related results, with the aim of a two-pronged attack on complexity bounds. He has framed it in terms of parameterized complexity, which we’ll discuss before returning to the thesis.
Consider three classic -complete problems, each given an undirected graph and a number :
Parameterized complexity started with the question:
What happens when some value of the parameter is fixed?
For each problem and each , we can define the “slice” to be the language of strings—here denoting graphs —such that the answer for is “yes.” Each individual slice belongs to : just try all choices of , and when is fixed this gives polynomial time. The fine-grained question is:
Can we solve each slice faster? In particular, can we solve in time where the exponent is fixed independent of ?
The start of much combinatorial insight and fun is that the three problems appear to give three different answers:
The exact for Vertex Cover may be model-dependent; for RAMs we get and indeed time is known. But the fact of its being fixed holds in any reasonable model and classifies Vertex Cover as belonging to the class for fixed-parameter tractable. To address whether Clique and Dominating Set belong to , a novel reducibility and completeness theory was built.
To exemplify an FPT-reduction, consider the problem
When is variable this is a generic -complete problem, so of course we can reduce Clique to it, but consider the reduction done this way: Given and , make an alphabet with a character for each . Code with a table for to write down edges nondeterministically on its tape, then deterministically check whether only different nodes appear in these edges, accepting if so. The computation takes time that is and can be defined solely in terms of . The latter fact makes an FPT-eduction.
FPT-reductions work similarly to ordinary polynomial-time reductions. If our NTM problem belongs to then so does Clique. There is a trickier FPT-reduction the other way, so the problems are FPT-equivalent. Both are complete for the first level, called , of the -hierarchy. Dominating Set is hard for the second level, ; Clique FPT-reduces to it, but not vice-versa unless .
Just like all major -complete problems are related by logspace not just poly-time computable reductions, parameterized complexity has a logspace-founded version. The class consists of parameterized problems whose slices individually belong to —that is, to deterministic logspace—and and are defined similarly via the conditions and . Interestingly, the separation is known by standard diagonalization means, but is not known to be contained in for reasons similar to the vs. problem. A recent Master’s thesis by Jouke Witteveen has details on these classes.
All four of our problems belong to . Furthermore, the slices belong to “almost exactly ” space using binary TMs. The space is needed mainly to write down each candidate and can be quantified as . We could jump from this to define finer variations on and but Mike, joined by Swernofsky, chose to refine the reductions instead. Now we are primed to see how their ideas might impact separations.
Our reduction from Clique to Short NTM Acceptance bumped up to . Blowups in are important to a central concept called kernelization which deserves more space—see this survey. They can be exponential—the algorithm for Vertex Cover hints at how—or even worse. Hence people have refined parameterized reductions according to whether they blow up
But before now, we haven’t noted in this line a strong motivation to limit the blowup even further, such as to or for some fixed constant (however, see this).
Our reduction also bumped up the alphabet size. This appears necessary because for binary NTMs the problem is in : we can traverse the possible guesses made on the tape, and for each one, solve an instance of graph-reachability.
So what can we do if we insist on binary TMs? Mike and Joseph fastened on to the idea of making the in space become the parameter. We can define “Log NTM Acceptance” to be like “Short NTM Acceptance” except that the NTM is allowed space (where is the size of in states). We get a problem in that is in fact complete for under parameterized logspace reductions. Likewise “Log DTM Acceptance” is the case where is deterministic, which is complete for . Then we define “Medium DTM Acceptance” where the question asks about acceptance within time steps regardless of space. Mike’s thesis also covers “Long DTM Acceptance” where the time is .
The in allows a binary TM to write down the labels of nodes in a size- structure at a time. In that way it subsumes the use of in the above three graph problems but is more fluid—the space could be used for anything. Keying on drives the motivation for Mike’s definition of LBL (for “level-by-level”) reductions, whose uniform logspace version is as follows:
Definition 1 A parameterized problem LBL-reduces to a parameterized problem if there is a function computable in space such that for all and , , where is fixed independent of .
That is, in the general FPT-reduction form , we insist on exactly. This reduction notion turns out to be the neatest way to express how Mike’s thesis refines and extends previous work, even his own.
With our Chair, Chunming Qiao, and his UB CSE Graduate Leadership Award. |
The first reason to care about the sharper reduction is the following theorem which really summarizes known diagonalization facts. The subscript reminds us that the TMs have binary work alphabet size; it is not necessary on since it does not affect the exponent.
Theorem 2
- If a parameterized problem is LBL-hard for Log DTM Acceptance, then there is a constant and all , .
- If a parameterized problem is LBL-hard for Log NTM Acceptance, then there is a constant such that for all , .
- If is LBL-hard for Medium DTM Acceptance, then there is a constant such that for all , .
Moreover, if there is a constant such that each belongs to , , or , respectively, then these become LBL-equivalences.
The power, however, comes from populating these hardnesses and equivalences with natural problems. This is where the problem—as a parameterized family of problems asking about the intersection of the languages of -many -state DFAs—comes in. We can augment them by adding one pushdown store to just one of the DFAs, calling them where is the PDA. Then call the problem of whether
by the name . It curiously turns out not to matter whether or the are nondeterministic for the following results:
Theorem 3
- is LBL-equivalent to Log NTM Acceptance. Hence there is a such that for each , .
- is LBL-equivalent to Medium DTM Acceptance. Hence there is a such that for each , .
Between the ICALP 2015 paper and writing the thesis, Mike found that the consequence of (a) had been obtained in a 1985 paper by Takumi Kasai and Shigeki Iwata. Mike’s thesis has further results establishing a whole spectrum of variants that correspond to other major complexity classes and provide similar lower bounds on -levels within them. The question becomes how to leverage the greater level of detail to attack lower bound problems.
The way the attack on complexity questions is two-pronged is shown by considering the following pair of results.
Theorem 4 if and only if and are LBL-equivalent.
Theorem 5 If for every there is a such that , then .
It has been known going back to Steve Cook in 1971 that logspace machines with one auxiliary pushdown, whether deterministic or nondeterministic, capture exactly . Since pushdowns are restricted and concrete, this raised early hope of separating from —intuitively by “demoting” some aspect of a problem down to or which are known to be different from . There was also hope of separating and or at least finding more-critical conditions that pertain to their being equal.
What Mike’s theorem does is shift the combinatorial issue to what happens when the pushdown is added to a collection of DFAs. Unlike the case with an auxiliary pushdown to a space-bounded TM, there is a more-concrete sense in which the relative influence of the pushdown might “attenuate” as increases. Can this be leveraged for a finer analysis that unlocks some secrets of lower bounds?
At the very least, Mike’s LBL notion provides a succinct way to frame finer questions. For example, how does hang on the relation between individual levels of and levels of ? The simplest way to express this, using Mike’s notation for Medium DTM Acceptance and for the analogous parameterized problem for space-bounded acceptance, seems to be:
Theorem 6 if and only if and are LBL-equivalent.
Mike’s thesis itself is succinct, at 73 pages, yet contains a wealth of other variants for tree-shaped automata and other restricted models and connections to forms of the Exponential Time Hypothesis.
How can one appraise the benefit of expressing “polynomial” and “” complexity questions consciously in terms of individual and levels? Does the LBL reduction notion make this more attractive?
It has been a pleasure supervising Mike and seeing him blaze his way in the larger community, and Dick and I wish all the best in his upcoming endeavors.
[fixed first part of Theorem 2, added link to Cygan et al. 2011 paper, added award photo]
Cropped from src1 & src2 in gardens for karma |
Prasad Tetali and Robin Thomas are mathematicians at Georgia Tech who are organizing the Conference Celebrating the 25th Anniversary of the ACO Program. ACO stands for our multidisciplinary program in Algorithms, Combinatorics and Optimization. The conference is planned to be held starting this Monday, January 9–11, 2017.
Today I say “planned” because there is some chance that Mother Nature could mess up our plans.
Atlanta is expected to get a “major” snow storm this weekend. Tech was already closed this Friday. It could be that we will still be closed Monday. The storm is expected to drop 1-6 inches of snow and ice. That is not so much for cities like Buffalo in the north, but for us in Atlanta that is really a major issue. Ken once flew here to attend an AMS-sponsored workshop and play chess but the tournament was canceled by the snowfall described here. So we hope that the planned celebration really happens on time.
Attendance is free, so check here for how to register.
The program has a wide array of speakers. There are 25 talks in all including two by László Babai. I apologize for not listing every one. I’ve chosen to highlight the following for a variety of “random” reasons.
László Babai
Graph Isomorphism: The Emergence of the Johnson Graphs
Abstract: One of the fundamental computational problems in the complexity class NP on Karp’s 1973 list, the Graph Isomorphism problem asks to decide whether or not two given graphs are isomorphic. While program packages exist that solve this problem remarkably efficiently in practice (McKay, Piperno, and others), for complexity theorists the problem has been notorious for its unresolved asymptotic worst-case complexity.
In this talk we outline a key combinatorial ingredient of the speaker’s recent algorithm for the problem. A divide-and-conquer approach requires efficient canonical partitioning of graphs and higher-order relational structures. We shall indicate why Johnson graphs are the sole obstructions to this approach. This talk will be purely combinatorial, no familiarity with group theory will be required.
This talk is the keynote of the conference. Hopefully Babai will update us all on the state of this graph isomorphism result. We have discussed here his partial retraction. I am quite interested in seeing what he has to say about the role of Johnson graphs. These were discovered by Selmer Johnson. They are highly special: they are regular, vertex-transitive, distance-transitive, and Hamilton-connected. I find it very interesting that such special graphs seem to be the obstacle to progress on the isomorphism problem.
Petr Hliněný
A Short Proof of Euler-Poincaré Formula
Abstract: We provide a short self-contained inductive proof of famous Euler-Poincaré Formula for the numbers of faces of a convex polytope in every dimension. Our proof is elementary and it does not use shellability of polytopes.
The paper for this talk is remarkably short, only 3 pages. Of course the result has been around since the 1700s and 1800s, and David Eppstein already has a list of 20 proofs of it, so what is the point? It has to do with ways of proving things and the kind of dialogue we can have with ourselves and/or others about what is needed and what won’t work. Imre Lakatos famously codified this process, with this theorem as a running example conjuring up the so-called Lakatosian Monsters. Perhaps the talk will slay the monsters, but it will have to brave some snow and ice first.
Luke Postle
On the List Coloring Version of Reed’s Conjecture
Abstract: In 1998, Reed conjectured that chromatic number of a graph is at most halfway between its trivial lower bound, the clique number, and its trivial upper bound, the maximum degree plus one. Reed also proved that the chromatic number is at most some convex combination of the two bounds. In 2012, King and Reed gave a short proof of this fact. Last year, Bonamy, Perrett and I proved that a fraction of 1/26 away from the upper bound holds for large enough maximum degree. In this talk, we show using new techniques that the list-coloring versions of these results hold, namely that there is such a convex combination for which the statement holds for the list chromatic number. Furthermore, we show that for large enough maximum degree, a fraction of 1/13 suffices for the list chromatic number, improving also on the bound for ordinary chromatic number. This is joint work with Michelle Delcourt.
Mohit Singh
Nash Social Welfare, Permanents and Inequalities on Stable Polynomials
Abstract: Given a collection of items and agents, Nash social welfare problem aims to find a fair assignment of these items to agents. The Nash social welfare objective is to maximize the geometric mean of the valuation of the agents in the assignment. In this talk, we will give a new mathematical programming relaxation for the problem and give an approximation algorithm based on a simple randomized algorithm. To analyze the algorithm, we find new connections of the Nash social welfare problem to the problem of computation of permanent of a matrix. A crucial ingredient in this connection will be new inequalities on stable polynomials that generalize the work of Gurvits. Joint work with Nima Anari, Shayan Oveis-Gharan and Amin Saberi.
There are two. One is, will we be snowed in or snowed out this Monday? The other is, can some of the open problem raised by these talks be solved?
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Even after today’s retraction of quasi-polynomial time for graph isomorphism
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László Babai is famous for many things, and has made many seminal contributions to complexity theory. Last year he claimed that Graph Isomorphism (GI) is in quasi-polynomial time.
Today Laci posted a retraction of this claim, conceding that the proof has a flaw in the timing analysis, and Ken and I want to make a comment on what is up. Update 1/10: He has posted a 1/9 update reinstating the claim of quasi-polynomial time with a revised algorithm. As we’ve noted, he is currently speaking at Georgia Tech, and we hope to have more information soon.
Laci credits Harald Helfgott with finding the bug after “spending months studying the paper in full detail.” Helfgott’s effort and those by some others have also confirmed the mechanism of Laci’s algorithm and the group-theoretic analysis involved. Only the runtime analysis was wrong.
Helfgott is a number theorist whose 2003 thesis at Princeton was supervised by Henry Iwaniec with input by Peter Sarnak. Two years ago we discussed his claimed proof of the Weak Goldbach Conjecture, which is now widely accepted.
In December 2015, Laci posted to ArXiv an 89-page paper whose title claimed that GI can be solved in quasi-polynomial time. Recall that means that the algorithm runs in time for some constant . This an important time bound that is above polynomial time, but seems to be the right time bound for many problems. For example, group isomorphism has long been known to be in quasi polynomial time. But the case of graphs is much more complex, and this was reason that Babai’s claimed result was so exciting. We covered it here and here plus a followup about string isomorphism problems that were employed.
He also chose to give a series of talks on his result. Some details of the talks were reported by Jeremy Kun here.
Retracting a claim is one of the hardest things that any researcher can do. It is especially hard to say when to stop looking for a quick-fix and make an announcement. It may not help Laci feel any better, but we note that Andrew Wiles’s original proof of Fermat’s Last Theorem was also incorrect and took 15 months to fix. With help from Richard Taylor he repaired his proof and all is well. We wish Laci the same outcome—and we hope it takes less time.
In particular, his algorithm still runs faster than for any you care to name. For comparison, for more than three decades before this paper, the best worst-case time bound was essentially due to Eugene Luks in 1983. The new bound in full is
for some fixed that will emerge in the revised proof.
The important term is the . The function is exponential in . We previously encountered a recursion involving in the running time of space-conserving algorithms for undirected graph connectivity (see this paper) before Omer Reingold broke through by getting the space down to and (hence) the time down to polynomial. So there is some precedent for improving it.
As things stand, however, GI remains in the “extended neighborhood” of exponential time. Here is how to define that concept: Consider numerical functions given by formulas built using the operations and and . Assign each formula a level by the following rules:
Note that if has level then so does the power for any fixed because . The functions of level include not only all the polynomials but also all quasi-polynomial functions and ones such as , which is higher than quasi-polynomial when .
The amended bound on GI, however, belongs to level , which is what we mean by its staying in the extended neighborhood of exponential time. This is the limit on regarding the amended algorithm as “sub-exponential.”
It also makes us wonder about why it is so difficult to find natural problems with intermediate running times. We can define this notion by expanding the notion of “level” with a new rule for functions that are sufficiently well behaved:
Rule 5 subsumes rules 3 and 4 given that has level and has level . A special case is that when and has level , then has level .
We wonder when and where rule 5 might break down, but we note that careful application of rule 2 for multiplication when expanding a power makes it survive the fact that , , , and so on all have the same level. It enables defining functions of intermediate levels where .
Can the GI algorithm be improved to a level ?
We note one prominent instance of level in lower bounds: Alexander Razborov and Steven Rudich proved unconditionally in their famous “Natural Proofs” paper that no natural proof can show a level higher than for the discrete logarithm problem.
The obvious open problems are dual. Is the amended result fully correct? And can the original quasi-polynomial time be restored in the near future, or at least some intermediate level achieved? We hope so.
[fixed discussion of terms related to , added to the intro an update about the claim being reinstated]