Writer homepage source |
Claire Cameron is the senior science editor at an online publication called Inverse. She has written an incomplete article titled, “The 2020s: 20 science and tech predictions for the new decade.” She has posted 12 of the predictions, but the top 8 will be revealed on Wednesday, January 22. Update 1/22: The predictions will be revealed over the coming week.
Today we offer a game of predicting: what will she predict? We also make some predictions for this year, including some that have already come true—the best kind.
Here are her 12 predictions. See if you can guess her other 8:
There are already some clear big ones: yes to the Turing test and long lifespans and gene+brain editing but nix to self-driving cars. There may be some logical entailments, e.g., if 11 is false—so that self-driving cars are released—does that make 13 true?
None of these predictions is out-of-the-blue: they all connect to initiatives already underway. So there should be some basis for inferring the missing eight. Quantum computing is one obvious area. Reviewing last year’s predictions will bring us there.
We made 5 perennial predictions and 5 new ones:
The GORZ paper was #1 on this list of “The 10 Biggest Math Breakthroughs of 2019” by Popular Mechanics. We also covered the list’s #2, #3 (in one subsection), #4 (with a followup), and #10 items—not too bad.
The five non-perennial predictions from last year were:
Last year, if we had written “claimed” or even “achieved,” we would now say bingo. But writing “proved” sets a higher standard. To judge from this and this, it appears the Google-led team is upping their qubit count from 53 to 57 in order to shore up their quantum supremacy claim.
However, we look back at what we called the “three planks” of a quantum supremacy claim:
We don’t think IBM’s 2.5-day simulation that would run on the Summit supercomputer is “comparable hardware” enough to count as a “known classical approach.” So we claim part credit even with the “proved” wording based on plank 2 surviving IBM’s challenge and any other as far as we know. There is still our friend Gil Kalai’s now-detailed critique of the modeling that the Google team’s analysis leans on.
The level of plank 2 is what we meant a year ago anyway. See how we flip things in our last prediction below. Incidentally, in regard to the argument over the terms quantum “supremacy” and “advantage,” we stand by the different definitions toward the end of our predictions post a year ago.
For our five predictions—besides the five perennial ones—we start with something easy:
This isn’t a total gimme: the proof could be wrong. The idea and connections, however, already seem clear.
This may be more a stretch. We note this new paper by Anup Rao simplifying some aspects of the proof of last year’s advance on the problem by Ryan Alweiss, Shachar Lovett, Kewen Wu, and Jiapeng Zhang. Sunflower bounds are known to impact some complexity problems that seem more specialized, but neither paper mentions widening the contingencies. But talking about “mainstream” complexity:
We don’t have a concrete reason to think this, only the thought that if this were false in either direction then that might have been found already in the half-year since Huang’s paper. We note this survey from two weeks ago.
Now to get back to quantum. Instead of refining what we wrote a year ago, we do the opposite:
That is, the coming year will not see a result of the form: if polynomial-sized classical hardware can achieve the same statistical separation as modeled for the quantum hardware (under either the linear or logarithmic “cross-entropy benchmark”) then counting and hence the polynomial hierarchy collapses to . Thus we expect to still be talking about all this a year from now.
Talking about talking about quantum supremacy, I wrote a short poem on its larger significances:
‘It From Bit’ we once proclaimed,
but now the Bit has bit the dust
from whizzing quantum chips that gamed
coherence to evade the trust
that the Word drove creation’s hour:
Mother Nature fully lexical.
Why don’t our brains then have this power?
It is a status most perplexical.
The import of the polynomial-time Church-Turing thesis is that symbolic computation, our brains, our computing devices whose programs execute at the level of machine language, and natural processes on which any computing device can be based, are equivalent under broad notions of efficiency not just in computability. If this fails, then we can ask a question like the one about illusions quoted at the end of last week’s post on : why would so many smart people have had the illusion that the universe is based on classical information? This may have quick answers: the theory of quantum information wasn’t yet appreciated; it shares many similarities with classical information theory; classical information remains in the picture. Note that the Simons initiative It From Qubit has existed for some time.
But then comes a second question. Richard Dawkins speaks for many biologists in underscoring that the major open issues in the genesis of life concern the processing of information to build complex structures. If nature affords capability to this processing, one would expect complex biological structures—in particular, our brains—to evolve to avail it. Well, Roger Penrose spearheaded the effort to show that our brains employ distinctive quantum processes, but the physical side of his argument has been blunted by evidence. It could be that “quantum advantage” is not a life advantage. The classical falling-short of our grey matter would still be curious.
What are your predictions for 2020? What are your meta-predictions for the coming decade—that is, which 8 predictions do you think Claire Cameron will post on Wednesday? It may help you to know that last month she wrote a science article on the movie Cats.
[added note about It From Qubit, serial updates in intro]
Composite crop from homepages |
Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright, and Henry Yuen (JNVWY) have just posted a paper titled . The title means that multiple provers sharing quantum entanglement, given any Turing machine and string accepted by , can convince a polynomial-time bounded verifier with high probability that . The time is polynomial in regardless of how long takes to halt on .
Today we applaud this work and try to convey some short way of apprehending it.
Yoking a classic undecidable problem to a polynomial-time task is not the only surprise. The proof refutes a conjecture in classical functional analysis that had apparently been widely believed. Thus this story continues the theme of surprises and possibly working the wrong way on conjectures, as we also just mentioned in our previous post. The new work subsumes a major paper last year showing that contains nondeterministic double exponential time (), which proves it different from its classical counterpart , which László Babai, Lance Fortnow, and Carsten Lund proved equal to .
The developments have been covered by Scott Aaronson here, Lance here, Boaz Barak here, and in a personal way by Vidick here. The new paper weights in at 165 pages. We will give our own snap-summary and try to add a little from the side.
The refuted conjecture was made by the Fields Medalist Alain Connes in a context having no overt involvement of quantum mechanics. In these 2013 eight–lecture course notes on the conjecture, the word “quantum” appears only once, to say on page 2 of lecture 1:
Other very recent discoveries include the fact that Connes’ embedding conjecture is related to an important problem in Quantum Information Theory, the so-called Tsirelson’s problem…
The problem of Boris Tsirelson ultimately harks back to the theorem of John Bell about correlations that are physically realizable using quantum entanglement but not by any classical physical system. In the CHSH game form of Bell’s theorem, our old friends Alice and Bob can win the game over 85% of the time using quantum, only 75% otherwise. They can get this with just one pair of entangled qubits per trial. Tsirelson proved that the 85% (to wit, ) is optimal. In extensions of these games to larger-size cases, the question becomes: what are the gaps between quantum and classical?
Whether there is a gap of more than a fixed then feeds into interactive protocols. We can have parties trying to prove these gaps using their own entanglement. Where it gets tricky is when you allow Alice and Bob to use larger and larger quantum states and ask, can they achieve the gap with some large enough state? The limiting behavior of the gaps is complex. What JNVWY proved is that this becomes like a halting problem. Not just a halting problem, the Halting Problem. Yet two quantum provers, working for a given gap that is achievable, can prove this to a polynomial-time classical verifier. This is the magic of the theorem.
The reduction from halting to the problem about limits and gaps comes before introducing two-prover systems, as is reflected by JNVWY and also in the wonderful introduction of a 2017 paper by William Slofstra which they reference. In advance of saying more about it, we’ll remark that the new work may provide a new dictionary for translating between (i) issues of finite/infinite precision and other continuous matters, and (ii) possible evolutions of a system of finite size in discrete steps of time and size, where both are unbounded but (in positive cases) finite.
The results strikes Dick and me as shedding new light on a principle stated by David Deutsch in a 1985 paper:
Every finitely realisable physical system can be perfectly simulated by a universal model computing machine operating by finite means.
I was a student at Oxford alongside Deutsch in 1984–1985, and I remember more going on than the searchable record seems to reflect. Deutsch believed that his model of a quantum computer could solve the Halting problem in finite time. He gave at least one talk at the Oxford Mathematical Institute on that claim. As far as I know the claim stayed local to Oxford and generated intense discussion led by Robin Gandy, Roger Penrose, and (if I recall) Angus Macintyre and at least one other person who was versed in random physical processes.
My recollection is that the nub of the technical argument turned on a property of infinite random sequences that, when hashed out, made some associated predicates decidable, so that Deutsch’s functions were classically total computable after all. Thus the hypercomputation claim was withdrawn.
Now, however, I wonder whether the two-prover system constitutes the kind of “machine” that Deutsch was intuitively thinking of. As I recall, his claim was not deemed wrong from first principles but from how theorems about random sequences interacted with machine model definitions. The theory of interactive provers as computational systems was then in its infancy. Could Deutsch have had some inkling of it?
Again we congratulate JNVWY on this achievement of a long-term research goal. Looking at the past, does it relate to the discussion of hypercomputation stemming from the 1980s? We mean a stronger connection than treated here or in this 2018 paper. Is it much different from ones where “the mystery … vanishes when the level of precision is explicitly taken into account” (quoting this). Looking ahead, are there any connection to the physical issues of infinity in finite time that we recently discussed here?
Updates 1/17: Gil Kalai has a post with background on further conjectures impacted by (the failure of) Connes’s conjecture and on quantum prover systems, plus a plethora of links.
A new article in Nature includes the following quotations:
The article and comments on Scott’s blog include interpretations that seem to oppose rather than support Deutsch’s principle on the finiteness of nature. The tandem of two unlimited provers may not qualify as a “finite machine.”
There are comments below querying whether the theorem is in first-order arithmetic or how strong a choice axiom it may need.
[added to first paragraph in second section, added updates]
SME keynote lecture source |
Evelyn Lamb is a mathematician who is also a journalist. She has a blog called Roots of Unity on the Scientific American blog network. Also impressive is that she designed the AMS Page-a-Day Calendar on behalf of the American Mathematical Society. It is available from the AMS with member discounts and is filled with wonderful tidbits on math.
The other day she interviewed Ken and me on P=NP.
Ken just happened to be visiting me that afternoon in New York and we spoke to Evelyn about P=NP. The call was delayed because of equipment issues on both ends. Perhaps the fundamental problem is not P=NP after all, but making computerized equipment work. Oh well.
Evelyn is writing an article for the New Scientist magazine about P=NP. She said it was driven by the near 20th anniversary of the Clay Prizes. Recall there were seven of these, each with a million dollar prize. One, the Poincaré conjecture, was already solved. The others are still open—the million dollars is still there waiting for a brilliant researcher.
Here is our own rendition of some questions that came up. We did not keep a recording or notes on our end, and we have paraphrased and expanded some things that we said:
What is the update on P=NP?
Ken: The update is that here is no update. There is no recent progress on resolving P=NP—seemingly none this decade, I’d say. This is still light-years away from it and you could even say the difficulty needed for yea-much progress is discouraging. There are some conjectures that elaborate on P NP, including Unique Games and (S)ETH, but those two have gone less clear.
Does the Clay prize help researchers?
Dick: I do not see that the prize gets anyone to work on P=NP.
Ken: I disagree. The prize does help people explain quickly what they are working on to others and why. This is quite valuable.
Could P=NP be undecidable?
Dick: Who knows. I note that known proofs that some combinatorial problem is unprovable in Peano arithmetic somehow rely on a function that grows too fast. The famous Ramsey problem is a perfect example. I do not see any way to get such a function in the P=NP area. Of course I could easily be wrong.
Ken: This is the subject by which Dick and I first came to know each other in the 1980s, for instance this paper of Dick’s with Rich DeMillo vis-à-vis one of mine. I now believe it is resolvable but will need deep and complex techniques.
Here I, Ken, went into the topic of “Natural Proofs,” as Dick did again later.
When will P=NP be solved?
Ken: We just discussed this on our blog. The upshot is that the conjecture has been open long enough—coming to its 50th anniversary if you date it by Steve Cook’s famous 1970–71 paper, 64 years if you date it by Kurt Gödel’s 1956 letter to John von Neumann—that it is going outside the range where data on other conjectures has any predictive value.
Dick: There is a related issue with P=NP. Perhaps we have guessed the wrong direction. Most believe that P is not equal to NP. But many conjectures were finally resolved when someone worked on the right direction.
Who believes P=NP vs P NP?
Ken: Our friend Bill Gasarch has polled this three times since 2002. His article finds 80% support for P NP, which he says goes higher among those who have worked more on it. I believe unequal, but Dick’s opinion is fluid and Don Knuth recently said he takes the “equal” possibility seriously.
I (Ken) started hunting for how we’ve covered Knuth’s opinion on GLL—it seems only a brief mention here and in comments here—but Dick related hearing it in person from Knuth.
Why so Hard?
Ken: If you believe P NP, then it is hard because Nature—mathematical nature—does a bang-up job of making it seem like P NP. Most instances of NP-complete problems are easy; so called SAT-solvers have had much practical success. The larger issue is that nature has frustrated us from proving any nontrivial general lower bounds at all. You can allow a exponential-time algorithm to make exponentially-long queries to NP and yet no one has proved that the resulting language cannot be decided by linear-sized Boolean circuits. Ryan Williams needed a lot of effort to prove that this class does not have constant-depth poly-size modular-counting circuits, but those could be weaker in relevant ways than general linear-size circuits. But this class is a lot bigger than NP.
I then said another factor is that sometimes algorithms seem to make no visible progress until at the very end when they suddenly come up with a correct answer. Dick and I had tried to quantify a notion of progress. I then started talking about the “hardness versus randomness” phenomenon and the “Natural Proofs” barrier again (for which this 2016 paper by Ryan is a good reference) but Dick cut in with a nub of all these matters.
Dick: A key issue is what I call “Bait-and-Switch” (indeed, in a post on the first day of GLL). Suppose an output is believed to be hard. Add a random to it. The result is also random. One branch of an algorithm computing and another working on seem to have nothing to do with . Yet when you do bitwise you have . This destroys any lower-bound argument that would be based on metering progress toward .
Guessing wrong way?
Dick continued saying that this issue only affects the “not equal” position and maybe it’s a hint of people guessing the wrong way. This went back into some things that were said before, and then the call started winding up.
We had made mental notes while walking back from lunch across the street in time for the call, but forgot some of them. To recycle an old saying, a mental note isn’t worth the paper it’s written on.
One of them was to remark on Gerhard Woeginger’s P Versus NP claims page and the relative balance of claims of “Equal” and “Not equal.” As of its last update in September 2016, the 116 listed claims are (by Ken’s count) divided 62 for “Equal,” 49 for “Not equal,” 3 for unprovable/undecidable, 1 claiming no claim, and 1 for both “Equal” and “Not equal.” It may be that “Equal” predominates because its logical formula begins with and it seems easier to imagine one has found a single that works rather than to exclude all —an infinite number of them.
I (Ken) had intended to connect this and the P=NP poll results to our post two months ago about cases of open questions where one direction seems overwhelmingly supported both by evidence and sentiment. Whatever one thinks about the value of all the P-vs.-NP claims, they witness that P-vs.-NP is certainly not one of those cases.
Last, I had intended to mention the deepest evidence in favor of “not equal.” This is the sinuous thin line between hard and easy cases of problems. Going back at least to Thomas Schaefer’s great 1978 work classifying cases of SAT, we’ve been aware of a surprisingly sharp dichotomy between “hard” and “in P” with seemingly no buffer zone. And the line sometimes seems to fall capriciously. In the second half of this post we mentioned Jin-Yi Cai’s work on dichotomy and a case relevant also to quantum where counting solutions to a fixed family of quadratic mod-4 polynomials in is easy, but counting those in to the same polynomial is hard. For another instance of this widely-appreciated point, see Scott Aaronson’s discussion of 3SAT here.
The point is that if P=NP (or counting is in P) then all of this finely filigreed structure would vanish—poof—an illusion all along. Like if the Mandelbrot Set became a smeary blob. But then we’d be left with: why did we have this illusion in the first place?
We suggested that she speak to some others—people more expert than we are. For this and other reasons the article may be quite different. We hope giving our own summary here is a help. What points would you add to it?
We also forgot to ask her about her own work in Teichmüller theory. Here are a couple of her articles on the ABC Conjecture. But that is not a Clay problem and is a subject for another decade.
[Edited 50 to 20]
]]>Real Bernie Sanders reaction source |
Larry David is an American comedian. He was the lead writer and producer of the Seinfeld TV series. During the previous and current election cycles he has played Presidential candidate Bernie Sanders in skits on Saturday Night Live. His “Bernie” rails about issues with passwords.
Today I want to talk about reducing our dependence on passwords.
On SNL in October 2015, his “Bernie” branched off the topic of Hillary Clinton’s e-mails to famously rant:
What’s the deal with e-mails anyway? I forgot my password the other day, so they say “We’ll email you a new one.” But I can’t get into my email to get the password. I mean, talk about a ball-buster.
In the SNL cold open about the most recent debate three weeks ago, “Bernie” reprised the complaint:
Apple lies, Amazon lies, even my I-Phone lies. Every time it says it’s at 1 percent battery, it stays on for at least 20 minutes. The other times it’s at 7 percent—it shuts down immediately. Apple, what are you trying to hide? And what’s my password?!!
Passwords are still needed. But, and this is the key, we wish to reduce our dependency on them and still be safe. Also the method fits nicely with today’s need for access to encrypted information by law enforcement. We have already discussed this recently here. The trick is to make information safe enough to please us, but not have as many passwords that we need to remember. “Bernie” would be happy.
Let’s next review why we use passwords at all.
Passwords have been used forever. See our friends at Wikipedia for a story by Polybius, an ancient historian, on how eons ago the Roman military used watchwords. Passwords have been used since the beginning of computing, forever in our world, to safeguard computer systems. The MIT computer system, CTSS, used them starting in 1961 to protect users. This is the reason Fernando Corbató is credited with the invention of computer passwords.
The goal of passwords and watchwords, is simple: Control access. Both the real and fake Bernies use e-mail passwords to keep their communications private. We try to select good passwords, but the management of them can be challenging. It’s not just that people like “Bernie” forget theirs—it’s that our efforts to keep them in our heads often make them too easy to figure out. Jimmy Kimmel showed this in a 2017 segment on his own show.
Managing passwords has lead to the theme, “passwords are dead.” If they are dead, then we must have alternative methods. Some are based on biometrics such as fingerprints and eye scans; others are based on additional hardware. Fingerprints are one of the most popular, but have major drawbacks:
I can attest that fingerprints are unreliable. I use a Global Pass to speed my re-entry into the US. After my flight arrives at the airport, I use a Global Pass machine that checks my fingerprints. The machine fails to recognize my fingerprints, every time. This forces me to talk to a custom agent and convince them am who I claim to be. Thus defeating the reason for using the Global Pass.
A 2012 study by Joseph Bonneau, Cormac Herley, Paul Oorschot, and Frank Stajano tilted “The Quest to Replace Passwords” compared 35 alternatives to passwords on their security, usability, and deployability. They showed that most do better on security, many do better on usability. But all alternatives do worse on deployability. The authors conclude:
“Marginal gains are often not sufficient to reach the activation energy necessary to overcome significant transition costs, which may provide the best explanation of why we are likely to live considerably longer before seeing the funeral procession for passwords arrive at the cemetery.”
Colorful language, but the point is that passwords are hard to beat.
I have a proposal on how to avoid reliance on just passwords. At least for many applications. The idea is: You would use a password and add a secret password that you do not remember. You do not even know this password. You do not write it down. Clearly, this is additional security if you do not even know the secret password. No one can steal it from you or guess it.
So the issue is: How do you access your own stuff without accessing the secret password? The answer is simple: You run a program that tries all the possibilities. Let this take some time . This point is that for many situations you will not mind if is large. For example, when the access that you are protecting occurs rarely. The security that this affords is based on this: An attacker can and will definitely be able to get access by expending time, if they also know you standard password. But this protects against mass attacks. An attacker may be able to get your data, but will not be able to get millions of people’s data.
Let’s examine a few applications:
End-to-end Encryption: The secret password could be protected by a of order years or even decades of computer time. This would allow law enforcement to get information it needs, while stopping causal attackers.
Recovery of your data: The secret password could be protected by a of order days of computer time. Your computer backup data is secured by such a password. Then it crashes. You need to run such a computation to recover the data. In many situations this would be fine.
A banking site: You would still use a good password. The extra secret one could have a of an hour say. Then when you need to pay an on-line bill, you would have to wait a reasonable amount of time.
Not need to place passwords in your will: Omada King, in his 2013 book The Making Identity, wrote:
[A]ccording to a recent survey from the University of London, one in ten people are now leaving their passwords in their wills to pass on this important information when they die.
This could be avoided by using a secret password with of order months. When it is needed the heirs would run the recovery program and get the access they need.
Ken points out that merely trying more than a few possibilities will usually generate a suspicious-activity message and usually a shutdown. So a system would have to be set up to permit “self-hacking.” Ken has a small rotation of “password extenders” not written down and often has to try two or three before gaining access to non-financial sites where he is registered.
Ken is sometimes asked to monitor chess tournaments for possible cheating using the initial “screening” phase of his system. This phase requires minimal effort from Ken as it is supposed to be automated on a server that any chess tournament director would have quick access to, but for various reasons the International Chess Federation (FIDE) has not (yet) erected such a server. So Ken posts the auto-generated reports at a private location known only to him and the arbiter(s) of a particular tournament.
Ken does not maintain a password scheme for the reports. This would become a headache precisely because of the difficulties people have with passwords. Ken does not want to assume the responsibility for managing them. Instead he includes a “quasi-password” as part of the URL he creates. These are often multi-lingual puns or references to quirky artists or factoids in the home country of the tournament. Being memorable and unusual enables the arbiter’s browser to learn the word and link without collision.
This maximizes convenience: For Ken to e-mail reports or zipped folders as attachments would be cumbersome under daily updates. With Ken’s way, the arbiter can view updates even without having to pull up Ken’s previous e-mails with links, just by typing some letters of the weird word in the browser address bar. Hiding directory listings and a “no robots” directive completes minimal security for temporary use during the tournament.
We mentioned Fernando Corbató having originated passwords. But before his passing last July 12, he came to regret them. Here he is quoted:
“It’s become a kind of nightmare. I don’t think anybody can possibly remember all the passwords.”
How practical do you find the “no-password” idea? At least the above suggestion may save us from having to place passwords in our wills.
]]>
Using logic to try and understand the show Servant
[ M. Night ] |
M. Night Shyamalan is the creator of many wonderful horror movies, including The Village and The Sixth Sense. His films often have a twist ending.
Today I thought I would try to apply math and logic to his latest creation, the TV show called Servant.
I am not kidding. No number theory, no complexity theory, no strange theorems today. I thought I would try and apply some logic to attempt and understand M. Night’s story. The story was created by Tony Basgallop but M. Night’s direction modulates what we can see and how we feel it. Servant is on cable, on Apple TV, and is close to the end. It is a series of ten short episodes and the last two are due out in the next few days.
The story is strange, scary, and unusual. M. Night likes to play with his viewers and make stories that are difficult to understand. Like many of our problems: Think P=NP or the Riemann Hypothesis, or the function problem. Our problems are strange, scary, and unusual too. Our difficulty is that we do not have a date when all will be explained. As we discussed here, we might be waiting a long time before we get the “last” episode for P=NP, for example.
Take a look at the rest even if you have not watched the series. I would like to think that working out the “solution” to the Servant is possible, but probably hard. I am not likely to have the answer here, but I think I may be on the right track, at least.
Here is a brief summary:
Six weeks after the death of their 13-week-old son, Philadelphia couple Dorothy and Sean Turner hire a young nanny, Leanne, to move in and take care of Baby Jericho, a reborn doll. The doll, which Dorothy believes is her real child, was the only thing that brought her out of her catatonic state following Jericho’s death. While Sean deals with the grief on his own, he becomes deeply suspicious of Leanne.
The players are:
I like this statement about Servant:
Servant continues to frighten viewers while at the same time not providing audiences with answers. But have they been telling us more than we know?
Like P=NP there is much debate and discussion about Servant. Here is a typical comment:
Never mind the mystery of whether Jericho is truly alive again or not, there’s also the fact that we have no idea how he died in the first place. New theory is that Dorothy killed him, either on accident or due to a bad postpartum depression episode. This would mean Julian and Sean covered it up.
There are lots of details in the story that may or may not be important. There are many theories about what could be going on. One thing seems clear is that what is happening is mystery. Is it one that we can shed any light on?
The discussions on-line about what is happening seem to miss the key: Is the baby, Jericho, a plastic baby or a real baby? Most seem to accept that the baby was plastic and then was replaced by a real baby. This is where I can apply logical analysis. Let’s walk through the possible cases:
Perhaps Jericho is a magical object. Than the whole story is a dream. Or the players are ghosts. Or some other magical possibility. This reflects the fact that we do not have precise rules or axioms. M. Night could do anything he wishes. But I like to think that he plays by some rules. So let’s agree that this case is out.
Perhaps Jericho was a plastic baby at first and then became a real baby. This is the consensus of almost all viewers. At least if we can rely on their comments. The issue here is how does this work? How can a living baby suddenly appear? How can the mom and the nanny not notice the baby has become real?
Perhaps Jericho was always a plastic baby. This seems to not be considered as a plausible case. It does require that the father and the brother are crazy. I like this idea.
Perhaps Jericho was always a real baby. This seems to be the most unusual idea. I think M. Night might like this direction. It would be the biggest twist. The baby was always real. Here M. Night is laughing at us: he would get the biggest shock if Jericho is always real. It would explain much. Then the mom and the nanny are just fine, we do not need to assume that they are crazy at all. Then the father and the brother are nuts.
I think the last is my favorite. I wish I could “prove” that the last is the case. In a few days we should know what M. Night selected for the answer. I feel that our methods at least forced us to consider all the logical possibilities. Is this the right answer?
I hope you did not mind this diversion from hard core math. I think that applying logic may be useful here. Too bad we do not have a reveal date for our problems like P=NP.
]]>
Some fun about resolutions.
source |
Ben Orlin is a funny mathematician. His book title Change Is the Only Constant was selected by the blog Math-Frolic as the best mathematics book of 2019.
Today Ken and I want to try to get you to at least smile, if not laugh.
Orlin is a funny chap—okay I just got back from London—so forgive me for using “chap”. Check Orlin’s site out for proof that he is funny. Here are some of his examples of math types rewriting famous opening lines from books. We will make this into a kind of quiz. You must guess the book title from the modified quote:
The third is the first hard one—I did not get it. The last one (of three from Ken not Ben) is probably unfair. All six of them, however, are on the “First Lines Literature Coffee Mug” which Ken received a year ago as a Christmas present from his sister. I did not know this when I chose the first three.
Many of us make resolutions for the new year. Here are some examples from Orlin:
Be better at explaining what I do to family and friends. I, Dick, have trouble with this one.
Not to prove by contradiction what can be proved directly. Assume that and
Stop using the word “obviously.” Here at GLL we try to avoid this, at least when it is not obvious. We posted about phrases to avoid a year ago.
Here are some of ours:
Stop doubting quantum computer claims. Unless, adds Ken, you have a possible concrete way of challenging them…
Start trying to apply AI methods to complexity theory. Could there be a new learning approach to 3-SAT? Note that PAC learning kind-of came from there. See for instance the end of this.
Stop trying to understand proofs that Peano arithmetic is inconsistent. I still do not understand what logic they use to prove that Peano is inconsistent. What if that logic is inconsistent?
Start up some fundamental research ideas and attempts on hard problems again.
And try to make GLL better, including making it appeal to a wider community.
The answers are:
Have a happy new year, and make some fun resolutions. Please let us know some of them, or ideas for ours.
[added Update fixing last first-line answer.]
Kathryn Farley is my dear wife. We just celebrated Christmas together and then went off to London for a holiday.
Today I thought I would share a gift with you.
Kathryn and I exchange books for the holiday. We do not wrap them—our tradition, which she started years ago. I like doing this. Ken’s family re-uses wrapping paper year-to-year. I am pleased to see that there is a call to save wrapping on three gifts.
If every American family wrapped just 3 presents in reused materials, it would save enough paper to cover 45,000 football fields.
Kathryn gave me a wonderful collection of books. One of them is not the book: Computers, Rigidity, and Moduli: The Large-Scale Fractal Geometry of Riemannian Moduli Space. The author is Shmuel Weinberger a mathematician from the University of Chicago. I already had his book but finally took a look at it while browsing my other gifts. So it is a delayed gift, from a Christmas past.
The book is a combination of computer results, basic math results, and advanced math results. The last are a variety of topics that are well outside of my zone. I did like the book. Weinberger is a terrific writer, a terrific explainer, a gifted selector of topics. I was surprised to see that his book contained a puzzle that I had not thought about before. The puzzle is, like all good puzzles, easy to state—but hard to solve. At least for me. I did not see how to solve it. I hope you like it.
Imagine that there are distinct points in the plane. No three points are collinear. Half of them, are labeled red and the other half are labelled green. Festive colors. Your job is to connect each red point with a unique green point by a straight line. That is easy, of course. But we hate when lines cross. So your job is more: Please select the lines so that the fewest ones cross.
Thus the problem is. Find the fewest line crossings possible. As usual we wish to know the answer as a function of . Is it ? Or ? What is the best possible in the worst case?
The artwork on the wall behind Kathryn suggests a possible arrangement, likewise the floor. The art object directly behind her is in 3D space which is not allowed—but this is a photograph so its points are projected onto a plane anyway. Again, it’s important that no three points are collinear.
For start we note that there is always a best. Each of red points can match one of green points. So there are only a finite number of possible matchings, namely
This shows that there is always a best answer; moreover, it can be found. But can be huge, so another part of the question is: Can you find the best answer fast? How about finding the best answer in polynomial time?
The best answer is that the number of crossings is always at most
where is the constant
Here is a proof that this bound is correct. Suppose that you have selected the fewest number of line crossings. We will argue that there are at most line crossings. The plan is to show that we can descend—that is given a matching we can try and decrease the number of lines that cross.
Given any two red nodes and any two green nodes, we can define the operation swap by having them exchange partners. If their lines cross before the swap, then after the swap the lines will not cross. The following figure shows this:
Oops. I thought the idea was to define an operation so that the number of line crossing could always be decreased. Wait the argument does not work. In the above figure the switch increases the number of crossings.
There is a saving trick, however. Do not attempt to decrease the number of lines that cross. Instead decrease the total length of the lines selected. The above figure can be turned into a proof of a lemma that a swap of two crossing lines always decreases the sum of their lengths. The cool idea is that the swaps may actually increase the number of crossings, but they will always decrease the total length.
In the end the length is minimal and there are crossings. Thus
I cheated and used that
and so the constant is zero. Finally, no swap is ever undone—since the length always decreases—and so the process must finish within swaps.
Hope that you liked this puzzle. I think it shows that are sometimes methods that while simple may be hard to find. Is really the needed number of swaps? Are more than a linear number of swaps ever required?
Happy holidays to all.
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Has it outlasted the ability to estimate when?
Composite of src1, src2 |
Ryohei Hisano and Didier Sornette wrote in 2012 a paper titled, “On the distribution of time-to-proof of mathematical conjectures.”
Today Ken and I discuss predicting the end to mathematical conjectures. This is apart from considering odds on which way they will go, which we also talk about.
Nine years ago the Christmas issue of the New Scientist magazine analyzed a small set of solved mathematical conjectures and used it to forecast when the P vs. NP conjecture would be solved. The article estimated that the “probability for the P vs. NP problem to be solved by the year 2024 is roughly 50%”.
New Scientist 2019 “biggest problem” source |
This was done without any special insight into P vs. NP, just its prominence among the world’s best mathematicians and time since inception. How can one make the case that this argument is not silly, that it is not a wild guess?
The 2010 New Scientist article could be considered a bagatelle since they looked at only 18 conjectures, but Hisano and Sornette took the idea seriously. They widened the data field to Wikipedia’s list of mathematical conjectures. They removed a dozen-plus conjectures whose origin dates and/or solution dates are not firmly known, leaving 144 conjectures: 60 solved and 84 still open as of 2012.
It should be understood that their conclusions about the whole enterprise are negative. They say in their abstract:
We find however evidence that the mathematical process of creation is too much non-stationary, with too little data and constraints, to allow for a meaningful conclusion. … In conclusion we cannot really reject the simplest model of an exponential rate of conjecture proof with a rate of 0.01/year for the dataset that we have studied, translating into an average waiting time to proof of 100 years.
They found exponential growth in notable conjectures being formulated, which they ascribe to growth in the number of mathematicians. They try adjusting for this by invoking the overall growth curve of the world population.
Then they find a dearth of conjectures with midrange solution times, compared to the number needed to give strong fits to simple models. They try cutting quickly-resolved conjectures (ones solved in under 20 years) from the dataset to improve the fits, and obtain only an enhancement of evidence for a slow exponential curve of solution time. Here is a figure from their paper showing the exponential curves:
In the end they are not able to find many hard distinctions between possible models. Most in particular, they conclude by calling the simple -solution-chance-per-year model “our best model.”
But that model in turn, without memory of the age of the conjecture, struck me—Ken writing here—as running counter to a famous controversial argument of recent vintage. So Dick and I put our heads together again…
The Doomsday Argument can be phrased as a truism:
If your hearing about is a uniformly random event in the lifespan of , which began years ago, then the odds are 50-50 that the future lifetime of will be no less than years and no more than years.
This extends to say the probability is 90% that will be no less than and no more than . The numbers come from considering the middle half or middle 90% of the span.
The key word in our phrasing is if—if your cognizance is a uniformly random event of the lifespan. The argument’s force comes from taking your birthday as a random event in the span of humanity. We can adjust for the population growth curve and our ignorance of prehistoric times by taking to be your ordinal in human lives ordered by birthday and projecting in the same units. If there were 50 billion people before you, this speaks 95% confidence that the sum of human lives will not reach a trillion.
We can try the same reasoning on conjectures. Suppose you just learned about the P vs. NP problem from catching a reference to this blog. By Kurt Gödel’s letter to John von Neumann we date the problem to 1956, which makes years. Presuming your encounter is random, you can conclude a 50-50 chance of the conjecture being solved between the years 2040 (which is 21 years from now) and 2208 (which is still shorter than Fermat’s Last Theorem lasted). The less you knew about P vs. NP, the more random the encounter—and the stronger the inference.
Perhaps Wikipedia’s 1971 origin for P vs. NP from its precise statement by Steve Cook is more realistic. Was my learning about it in 1979 () a random event? If so, I should have been 75% confident of a solution by 2003. Its lasting 16 more years and counting busts the 83.3% confidence interval. Well, Dick learned about it within weeks, at most months, after inception. If that was random then its longevity counts as a 99.9% miss. Of course, Dick’s learning about it was far from random. But.
Our point, however, is this: Consider any conjecture that has lasted years. That’s the point at which the exponential and Doomsday models come into conflict:
The longer a conjecture lasts, the greater this conflict. We are not addressing the issue of uniform sampling over the lifespan of the conjecture, but this conflict applies to all observers past the 87-year mark. The honeycomb conjecture was documented by 36 BCE and finally solved by Thomas Hales in 1999.
Note that since , one cannot evade our model conflict by saying only a small fraction of conjectures last that long. Despite H-S noting that the honeycomb conjecture and Fermat and some others break their curve, their curve at left below shows excellent agreement up to and beyond the 100-year mark.
The curve at right shows that a rate fits even better when short-lived conjectures are deleted. Since , that makes the conflict set in at years old. Then it applies even for P vs. NP dated to Cook. At the conflict is considerable: Doomsday 66.7% of its lasting at least 50 more years, versus chance of its lasting that long by H-S.
This conflict says there must be a factor that invalidates either or both lines of reasoning. But the Hisano-Sornette estimates are supported by data, while our Doomsday representation is quite generic.
A further wrinkle is whether the collective balance of opinion on whether a conjecture is true or false has any bearing. If the field is evenly divided, does that argue that the conjecture will take longer to resolve?
The idea of “odds” is hard to quantify and has come into play only recently. Despite lots of chat about betting on P vs. NP, and Bill Gasarch’s wonderful polls on which side is true, data for other problems is scant.
This prompts us to mention that the Springer LNCS 10,000 anniversary volume came out in October. It includes a contribution by Ken titled, “Rating Computer Science Via Chess”—the title refers both to how the chess ratings of the best chess computers mirrored Moore’s Law and how they draw on many general CS achievements in both hardware and software. For now we note also the contribution by Ryan Williams titled, “Some Estimated Likelihoods for Computational Complexity.” Ryan places only 80% on and puts versus at 50-50.
There are also non-betting and non-opinion senses in which conjectures are regarded as “almost certainly true.” We recently posted about this. Perhaps something more can be gleaned from highlighting those conjectures in the H-S data set to which such criteria apply.
All of this speculation may be less idle and less isolated if it can be carried over to unsolved problems in other walks of life. Corporations and consortia doing open-ended research may already need to quantify the average time to achieve potential breakthroughs—or dismiss them—because they deal with many “feelers” at a time. The Doomsday assumption of random encounters probably doesn’t apply there. But there could be large-scale tests of it by “punters” seeking out random lifespans they can bet on, such as runs of shows or the time to make a movie of Cats, and profit by applying it.
What do you think about attempts to forecast the time to solve P vs. NP, or to forecast breakthroughs on larger scales?
There are many rebuttals to the Doomsday Argument, some general and some context-specific. Is our argument that it conflicts generally with memoryless exponential models subsumed by any of them?
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After 50 years we still are baffled by the primes.
Cornell faculty album source |
Juris Hartmanis needs no introduction. But we can note this long interview last February by the Heidelberg Laureate Foundation. His sister, Astrid Ivask, was a world–recognized poet.
Today Ken and I wish to discuss one of his old conjectures.
We have previously discussed one of his conjectures here. The conjecture we are interested in his one made in his joint paper with Herbert Shank titled, “On the Recognition of Primes by Automata.”
Shank once wrote a paper On the undecidability of finite planar cubic graphs with Solomon Garfunkel. I, Dick, liked this paper because it honestly pointed out that a predecessor paper by them was erroneous:
In the March, 1971 issue of this Journal (JSL) a paper of ours was published purporting to prove the hereditary undecidability of the first-order theory of finite planar graphs. The proof presented there contains an error which is unfortunately “unfixable” by the methods of that paper. The theorem however is true and we demonstrate here a generalization to finite cubic (exactly three edges at each vertex) planar graphs. The method involves coding the halting problem for a Turing machine into the theory of these graphs by considering special printouts of computations. Let us first consider a discussion of the aforementioned mistake and see what can be learned from it
Hartmanis and Shank (HS) asked the following conjecture—over fifty years ago—about testing whether an integer is a prime number:
We’ve written them in binary notation with the ones’ place first—just the way to give numbers on the input tape of a Turing machine. No error in their Turing representation here—just their brilliant conjecture connecting space to number theory: Suppose that primality is computed in space . Then
for some . That is, any primality test must take linear space.
As usual, space is the measure of how much storage the algorithm must use. The input tape is read-only and does count against the storage bound. The famous AKS primality test showed that whether a given -bit number is prime can be determined in time but did not improve the linear upper bound on space, which HS showed by trivial means: Try all possible divisors between and . The arithmetic and loop for this method can all be conducted in space.
What is some intuition for this conjecture? Why insist the space be linear when space is plenty—given that polynomial time could still be equal to logspace? Well, suppose and consider the integers from to , that is all having bits in binary notation. The number of machine configurations apart from the content of the input tape is at most:
This is . In fact, it is which is asymptotically the number of those that are prime according to the Prime Number Theorem. This means that the vast majority of prime numbers have no worktape configuration that is unique to their accepting computation, not even close. Thus each configuration would be shared with many other primes—and most would be shared with many non-primes as well. Whereas many non-primes are composite for the same reason—such as all strings beginning with being even—the intuition is that each prime has its distinct track of reasons that ought not to come together with other primes.
The HS paper notes that since the set of primes is not regular, it needs at least order-of space. This is a weak lower bound.
Both the bound and the above intuition become sharper if we require that the input tape be one-way as well as read-only: its head cannot go left to re-read lower-order bits. Every non-regular language requires order-of space by such 1-way machines.
If we write the prime numbers in unary notation—that is, —then we can prove they cannot be recognized by a machine in space. Having the input tape be unary makes the whole computation depend on the configuration of as described above. In consequence, has infinite arithmetic progressions of accepted strings. But by the celebrated theorem of Gustav Dirichlet, each such progression contains infinitely many primes as well as infinitely-many non-primes.
Scaling back from unary to binary seems to scale the space up to linear, so perhaps this result supports the HS conjecture. HS make related observations about log space and the density of the primes in the paper. On the other hand, polynomial time could still be equal to logspace (with two-way input tape), which would make their conjecture wrong.
We believe that the HS conjecture is hard, is related to some interesting questions, and could be false. That is, there might be some way to check that a number is prime in space. Proving lower bounds on space remains, after many decades, hopeless.
Here is a related conjecture. Using the same binary representation as above—where is 13 and is 17—Jeffrey Shallit has asked whether we can determine whether a given finite automaton accepts some input so that it is a prime number. We believe it is still open. Let’s assume for the moment that it is true. Then we can solve some open questions in number theory related to primes. This problem unlike the HS is worth some money. For example, the above would pay 50 British pounds—about $66.62.
Jeff points out that if the above is decidable then one can solve the famous problem: Are there any Fermat primes that are not known? Recall a Fermat prime is one of the form
The largest one known is
All such primes match the regular expression in binary notation, which is easy for a finite automaton to recognize. This expression matches numbers of the form , but as we discussed here, they can be prime only when is a power of .
Why are simple complexity questions about primes so hard? Why indeed?
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We all lost a great person
Cropped from Mexican NotiCyTI obit |
Héctor Garcia-Molina died just before Thanksgiving. He was a computer scientist and Professor in both the Departments of Computer Science and Electrical Engineering at Stanford University.
Today Ken and I write in sadness about his passing and to express some personal appreciations.
We at GLL have talked about him before. See this for his story about the fun of using an IBM mainframe for teaching. Or see this for a story about Héctor and meetings.
I, Dick, had the pleasure to have worked with him, while we were both faculty at Princeton in the 1980’s and beyond.
Héctor was the chair of the Stanford Computer Science Department from January 2001 to December 2004. Stanford then rotated the chair so all took their turn. I know that he “hated” being an administrator in general. But of course being a team player he took his turn.
One way to see his real feelings about being a chair was to look at the clock his students constructed for him. The clock was a digital timer that counted down the seconds that remained in his term as the chair. It started at roughly 126227808. I am sure he did fine, but the clock was a statement.
While Héctor was at Princeton we worked together on a project—the MMM project. It led to one of his least cited papers—a 1984 paper with me and Jacobo Valdes. OK, it has 48 citations according to IEEE. The idea of the project was to use memory rather than processors to speed up computations. He was a joy to work with: he was careful, and thoughtful, and just fun to work with on any project.
We did write a second paper with Richard Cullingford instead of Valdes, which Héctor presented at a meeting on knowledge-based management systems. The above Google Books link goes to the end with a discussion in which a simple issue was raised—we paraphrase:
Won’t it take forever just to write all zeros to the memory?
Héctor had a scientist’s answer: the project was still not at the prototype stage so he didn’t know. He said the project should be viewed as a scientific experiment to find out. Maybe he also had an inkling that what was coming was a decade of breakthroughs in CPU design and parallel/pipeline processing after all.
Héctor was unparalleled at seeing the future. I always thought that one of his abilities, one that set him apart, was his ability to predict directions of research. This allowed him, and his students, to write early papers in research areas before they became hot. This is one of the talents that made him so amazing.
I recall way before the world wide web was created he had students working on adding links to documents. I recall a talk by one of his students at Princeton that discussed what we now call URLs. One question that was raised during the talk was: How were the links going to be created? There was a lively discussion about this. Could they be created automatically? If not why would people take the time to create the links? Indeed.
Héctor saw that links would be created. That people would take the effort to create them. I must admit that he was right, and he saw the future better than most. I wish I had a fraction of his ability to see directions like he did.
Héctor told me that when he first got to Stanford the fund managers and investors roamed the halls. They would ask anyone they could if they had an idea for a company or a startup. It was a constant issue that Héctor had to deal with. They were continually trying to steal away students.
He told me he felt like he was the head of an abbey and was always having to protect his charges within the walls.
When the impetus came from within it was different. Of course, Héctor was the advisor of Sergey Brin at the time he and Larry Page conceived Google. Brin and Page found that their search engine prototypes were so good the dataflow was constantly straining Stanford’s machines. They needed to scrounge for more disks and processors to mount their servers. Héctor already oversaw the Stanford Digital Libraries Project and he arranged for funds to purchase spare parts for the data servers.
It is interesting that in this 2001 interview in the SIGMOD Record, Héctor did not have a high opinion of the industrial side of his area:
Again, I don’t think industry really does very much research. They come up with an idea and they try to sell it. If it was a good idea, maybe they will make money. Even if it was a bad idea, if they have good marketing people, they might still make money and we never know … I don’t think they have an advantage over [academics] in testing the ideas and evaluating them and performing measurements and really understanding what are the right techniques.
In the same interview, he had sage advice for students after completing their PhDs and during the tenure process, mainly on the side of not trying to play the system but focus on doing what you love.
Héctor had been a graduate student at Stanford. So his return there as a professor was a kind of homecoming. He was a home-team player in many senses. One of them is shown by this photo:
Stanford University obituary source |
He was a registered Stanford sports photographer. He also taught a course at Stanford on photography. We don’t know if he had special insights on detecting “deepfake” photos and videos.
Our condolences go to all his family. Héctor you will be missed. You are missed.
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