As we celebrate our successes in computational complexity here's a sobering fact: We have had no new non-relativizing techniques in the last 25 years.
A little background: In 1975, a few years after Cook established the P v NP problem, Baker, Gill and Solovay created two sets A and B such that every NP machine that could ask questions about A could be simulated by a P machine that could ask questions to A and there is some language accepted by an NP machine that could ask questions to B that cannot be solved by a P machine that could ask questions to B. In other words PA = NPA but PB ≠ NPB.
All the known proof techniques at the time relativized, i.e., if you proved two classes were the same or different, they would be the same of different relative to any set. BGS implied that these techniques could not settle the P v NP problem.
In the 80's and 90's we got very good at creating these relativized worlds and, with a couple of exceptions, we created relativized worlds that make nearly all the open questions of complexity classes between P and PSPACE both true and false.
There were some results in the that were non-relativizing for technical uninteresting reasons. In the late 80s/early 90s we had some progress, results like NP having zero-knowledge proofs, co-NP (and later PSPACE) having interactive proofs and NEXP having (exponential-sized) probabilistically checkable proofs, despite relativized worlds making those statements false. But then it stopped. We had a few more nonrelativizing results but those just used the results above, not any new techniques.
In 1994 I wrote on survey on what relativization meant post-interactive proofs, still mostly up to date. We have seen new barriers put up such as natural proofs and algebrization, but until we can at least get past traditional relativization we just cannot make much more progress with complexity classes.
Thursday, January 28, 2016
Sunday, January 24, 2016
When do we stop giving the original reference?
I was preparing a talk which included my result that there is a sane reduction 4-COL \le 3-COL (the paper is here) when I realized that if I am going to claim the reduction is by Gasarch then I should find out and credit the person who proved 3-COL NP-complete (I assume that the result 3-COL \le 4-COL is too trivial to have an author). I could not find the ref on the web (I am sure that one can find it on the web, but a cursory glance did not yield it). The reference was not in Goldreich's complexity book, nor the CLR Algorithms book. I suspect its not in most recent books.
The reference is Stockmeyer, SIGACT NEWS 1973,
Few modern paper references Cook or Levin's original papers for the Cook-Levin Theorem. I've even heard thats a sign its a crank paper.
Few modern papers reference Ramsey's original paper for Ramsey's Theorem.
But the questions arises--- when is a result such common knowledge that a ref is no longer needed?
A result can also go through a phase where the reference is to a book that contains it, rather than the original paper.
On the one hand, one SHOULD ref the original so that people know who did it and what year it was from. On the other hand there has to be a limit to this or else we would all be refering to Euclid and Pythagoras.
Where does Stockmeyer's result fall? I do not know; however, I've noticed that I didn't reference him, and I will correct that.
Thursday, January 21, 2016
The Growing Academic Divide
So reads the Report on the Modern Language Association Job Information List. The MLA is the main scholarly organization for the humanities in the US. The bright spot--we aren't Japan.The decreases of the past three years bring the number of advertised jobs to a new low, below the level reached after the severe drop between 2007–08 and 2009–10.
Meanwhile in computer science we continue to see large enrollment increases in major, minors and students just wanting to take computer science courses. In the upcoming CRA Snowbird Conference "a major focus of the conference will be booming enrollments, with a short plenary followed by parallel sessions devoted to the topic, its various ramifications, and ideas to help you deal with it, including best practices for managing growth."
The 2015 November CRA News had 83 pages of faculty job ads, up from 75 in 2014 and 34 in 2012. This doesn't even count that fact that many departments are looking to hire two, three, four or more positions in CS. It will be an interesting job market this spring.
All of this is driven by jobs. We can't produce enough strong computer scientists to fill industry demand. And it's becoming increasingly hard for a humanities major to get a good first job.
It's nice to be on the side of growth but it's a shame that faculty hiring seems to be a zero-sum game. We need poets as well as nerds. We've help create a world where we have made ourselves indispensable but is this a world we really want to live in?
Monday, January 18, 2016
Is it okay to praise an article or book in an article or book?
I recently had a paper accepted (YEAH!). The referees had some good corrections and one that puzzled me.
you wrote ``our proof is similar to the one in the wonderful book by Wilf on generationg functions [ref]''. You should not call a book wonderful as that is subjective. You can say it's well known.
I asked a pretension of professors about this. Is it okay to praise an article or book? Is it okay to state an opinon? Would the following be acceptable:
1) In Cook's groundbreaking paper SAT was shown to be NP-complete. It IS grounbreaking, so maybe thats okay.
2) Ramsey's paper, while brilliant, is hard for the modern reader to comprehend. Hence we give an exposition. If I am writing an exposition then I might need to say why the original is not good to read so this is informative.
3) Ryan Williams proved an important lower bound in [ref]. Is this okay to write? For most people yes, but NOT if you are Ryan Williams. (He never wrote such.)
4) William Gasarch proved an unimportant lower bound in [ref]. Is this okay to write ? Only if you ARE William Gasarch (He never wrote such).
The version that will be in a journal will indeed NOT call Wilf's book wonderful. The version on arXiv which will be far more read (not behind a paywall) will call Wilf's book wonderful.
you wrote ``our proof is similar to the one in the wonderful book by Wilf on generationg functions [ref]''. You should not call a book wonderful as that is subjective. You can say it's well known.
I asked a pretension of professors about this. Is it okay to praise an article or book? Is it okay to state an opinon? Would the following be acceptable:
1) In Cook's groundbreaking paper SAT was shown to be NP-complete. It IS grounbreaking, so maybe thats okay.
2) Ramsey's paper, while brilliant, is hard for the modern reader to comprehend. Hence we give an exposition. If I am writing an exposition then I might need to say why the original is not good to read so this is informative.
3) Ryan Williams proved an important lower bound in [ref]. Is this okay to write? For most people yes, but NOT if you are Ryan Williams. (He never wrote such.)
4) William Gasarch proved an unimportant lower bound in [ref]. Is this okay to write ? Only if you ARE William Gasarch (He never wrote such).
The version that will be in a journal will indeed NOT call Wilf's book wonderful. The version on arXiv which will be far more read (not behind a paywall) will call Wilf's book wonderful.
Thursday, January 14, 2016
A limit on the DFA-CFG divide for finite sets
It is easy to see that for Ln = {a,b}*a{a,b}2n
There IS a CFG of size O(n)
ANY DFA is of size double-exp-in-n
I was wondering if we can increase this gap. That is, a statment like: For all but a finite number of n there exists a lang Ln such that
There IS a CFG of size O(n)
ANY DFA is of size triple-exp-in-n.
I also looked at finite sets. For COMPLIMENT of { ww : |w|=2n }
There IS a CFG of size O(n)
ANY DFA (in fact any DPDA) is of size double-exp-in-n.
This is stated in my paper here though the real interesting math needed to get it are in here where they get lower bounds on the size of CFG's, generalizing techniques from here.
SO my question still stands- can we get a triple exp separation? I have shown that this CANNOT be achieved with finite sets:
THEOREM: If L is a finite lang then there exists n such that (1) Any CFG in Chomsky Normal Form for L has to be of size \ge log n, AND (2) there IS a DFA of size 2n.
Proof: Let n be such that the longest string in L if of length n. In order for a Chomsky Normal Form grammar to generate this string it needs \ge log n nonterminals. Since the lang has at most 2n
strings in it, there is a DFA of size 2n for it.
End of proof.
So I have shown that one approach won't work. I am hoping that YOU know an approach to get
a trip-exp-sep and leave a comment about it.
There IS a CFG of size O(n)
ANY DFA is of size double-exp-in-n
I was wondering if we can increase this gap. That is, a statment like: For all but a finite number of n there exists a lang Ln such that
There IS a CFG of size O(n)
ANY DFA is of size triple-exp-in-n.
I also looked at finite sets. For COMPLIMENT of { ww : |w|=2n }
There IS a CFG of size O(n)
ANY DFA (in fact any DPDA) is of size double-exp-in-n.
This is stated in my paper here though the real interesting math needed to get it are in here where they get lower bounds on the size of CFG's, generalizing techniques from here.
SO my question still stands- can we get a triple exp separation? I have shown that this CANNOT be achieved with finite sets:
THEOREM: If L is a finite lang then there exists n such that (1) Any CFG in Chomsky Normal Form for L has to be of size \ge log n, AND (2) there IS a DFA of size 2n.
Proof: Let n be such that the longest string in L if of length n. In order for a Chomsky Normal Form grammar to generate this string it needs \ge log n nonterminals. Since the lang has at most 2n
strings in it, there is a DFA of size 2n for it.
End of proof.
So I have shown that one approach won't work. I am hoping that YOU know an approach to get
a trip-exp-sep and leave a comment about it.
Monday, January 11, 2016
A question in Formal Lang Theory about Size of Desc of Languages.
(This post is inspired by me thinking more about this blog entry and this paper.)
Upper bounds on n are really O(n).
Lower bounds of f(n) are really Omega(f(n))
DPDA = Deterministic Push Down Automata
NDFA = Nondet. Finite Aut.
DFA = Det. Finite Aut.
It is known that FOR ALL n there is a lang Ln such that there is an NDFA of size n, but ANY DFA for Ln is of size at least 2n : Ln = (a,b)*a(a,b)n. for DFA-exactly 2n, NDFA- n+2. One can also get 2n and n.
It is known that FOR ALL n there is a lang Ln such that there is a DPDA of size n, but ANY NDFA for Ln is of size at least roughly 2n size: Ln={a2n }. (ADDED LATER TO CLARIFY- NOTE THAT
Ln is a one-element set, hence regular.)
Can we acheive both at the same time? Is the following true: for all n there exists Ln such that
1) There is a size n DPDA for Ln.
2) Any NDFA for Ln requires size 2n.
3) There IS an NDFA for Ln of size 2n.
4) Any DFA for Ln requires size 22n.
If we replace DPDA with PDA then { (a,b)*a(a,b)2n } works.
Wednesday, January 06, 2016
Rūsiņš Freivalds (1942-2016)
Rūsiņš Freivalds did much of the early work in probabilistic algorithms and automata, and in inductive inference, a computability-theoretic approach to learning. In the 70's he found fast probabilistic algorithms for checking multiplication of integers and matrices. More recently he has looked at quantum finite automata. He supervised several PhD students including Andris Ambainis, one of the leading researchers in quantum algorithms.
Mostly I remember Freivalds as the leader of the Latvian theoretical computer science community and just an extremely nice and friendly colleague. I am glad to have known and worked with him.
Sunday, January 03, 2016
Predictions for 2016
This is the first post of 2016! (that is not a factorial). Hence I will make some predictions and at the end of the year I'll see how I did
1) The USA Prez election will have two main candidates: Hillary Clinton and Ted Cruz. Clinton will win by floating rumors that Cruz was born in a foreign country.
2) There will be a proof that there exists a constant c such that the methods that got a lower bound of (3+1/86)n on an explicit function cannot be extended to cn.
3) P vs NP will not be solved. But see next point.
4) I will be asked to look at at least two resolutions of P vs NP. This year I was asked to look at two
proofs that P=NP. Neither was correct.
5) GI will still not be in known to be in P.
6) There will be a proof that Babai's techniques cannot get GI into P.
7) The fact that 2016=25327 will be useful in a math competition.
8) Posting a video of a talk (as Babai did) will become an acceptable way to claim a result, rather than having a paper on arXiv. Getting a paper in a Journal will become less and less relevant for big results.
(This is a topic for a later blog post.)
9) The Unique Game conjecture... When it was first stated it seemed like maybe it could be proven or disproven. The longer it stays open the harder it seems. Clyde Kruskal tells me that a good method for guessing how long a problem will stay open is how long its been open. So I'll predict it won't be resolved this year. However, by that reasoning, I will always predict it will not be resolved in the following year.
10) There will be a big data breach. The security protocols used were never proven secure, though that won't be what caused the breach. Nor was it caused because of a really fast factoring or DL algorithm.
11) Comp Sci enrollment will continue to rise.
12) Leave your predictions in the comments!
1) The USA Prez election will have two main candidates: Hillary Clinton and Ted Cruz. Clinton will win by floating rumors that Cruz was born in a foreign country.
2) There will be a proof that there exists a constant c such that the methods that got a lower bound of (3+1/86)n on an explicit function cannot be extended to cn.
3) P vs NP will not be solved. But see next point.
4) I will be asked to look at at least two resolutions of P vs NP. This year I was asked to look at two
proofs that P=NP. Neither was correct.
5) GI will still not be in known to be in P.
6) There will be a proof that Babai's techniques cannot get GI into P.
7) The fact that 2016=25327 will be useful in a math competition.
8) Posting a video of a talk (as Babai did) will become an acceptable way to claim a result, rather than having a paper on arXiv. Getting a paper in a Journal will become less and less relevant for big results.
(This is a topic for a later blog post.)
9) The Unique Game conjecture... When it was first stated it seemed like maybe it could be proven or disproven. The longer it stays open the harder it seems. Clyde Kruskal tells me that a good method for guessing how long a problem will stay open is how long its been open. So I'll predict it won't be resolved this year. However, by that reasoning, I will always predict it will not be resolved in the following year.
10) There will be a big data breach. The security protocols used were never proven secure, though that won't be what caused the breach. Nor was it caused because of a really fast factoring or DL algorithm.
11) Comp Sci enrollment will continue to rise.
12) Leave your predictions in the comments!
Monday, December 28, 2015
Complexity Year in Review 2015
We had an incredible year for theorems in 2015, the strongest year in computational complexity in the last decade. Topping the list as the theorem of the year is László Babai's quasipolynomial-time algorithm for graph isomorphism. A little risky because nobody, including Babai himself, has fully verified the proof but given Babai's reputation and the warm reception of his lectures, I have little doubt it will all work out. Again I strongly recommend watching the video of Babai's first lecture. Not only is Babai a great researcher but he's an excellent speaker as well. For the those up to the challenge, the paper is now available as well.
Beyond Babai's result we had a great year including random oracles separating the polynomial-time hierarchy, the (3+1/86)n size lower bound for general circuits, Terry Tao's solution to the Erdős discrepancy problem, explicit constructions of Ramsey Graphs and new bounds on threshold circuits.
We celebrated 50 years of computational complexity and Moore's law, the centenary of Hamming and the bicentenaries of Ada Lovelace and George Boole, the latter giving us the Boolean algebra. In 2016 we will celebrate the centenary of the person who used those bits to describe information.
We thank our guest posters Dylan McKay and Thomas Zeume. Dylan's metric group problem is still open if anyone wants to try and crack it.
In 2015 we continue to see CS departments struggle to meet the demand of students and industry. US universities faced many difficult challenges including dealing with race relations, freedom of speech and safety issues. Too much tragedy in the US and abroad, and a far too divisive political campaign. Here's hoping for reason to prevail in 2016.
We remember Gene Amdahl, Alberto Apostolico, Barry Cooper, George Cox, Jiří Matoušek, John Nash, Leonard Nimoy, Hartley Rogers Jr., Donald Rose, Karsten Schwan and Joe Traub.
Wishing everyone a Happy New Year and a successful 2016.
Beyond Babai's result we had a great year including random oracles separating the polynomial-time hierarchy, the (3+1/86)n size lower bound for general circuits, Terry Tao's solution to the Erdős discrepancy problem, explicit constructions of Ramsey Graphs and new bounds on threshold circuits.
We celebrated 50 years of computational complexity and Moore's law, the centenary of Hamming and the bicentenaries of Ada Lovelace and George Boole, the latter giving us the Boolean algebra. In 2016 we will celebrate the centenary of the person who used those bits to describe information.
We thank our guest posters Dylan McKay and Thomas Zeume. Dylan's metric group problem is still open if anyone wants to try and crack it.
In 2015 we continue to see CS departments struggle to meet the demand of students and industry. US universities faced many difficult challenges including dealing with race relations, freedom of speech and safety issues. Too much tragedy in the US and abroad, and a far too divisive political campaign. Here's hoping for reason to prevail in 2016.
We remember Gene Amdahl, Alberto Apostolico, Barry Cooper, George Cox, Jiří Matoušek, John Nash, Leonard Nimoy, Hartley Rogers Jr., Donald Rose, Karsten Schwan and Joe Traub.
Wishing everyone a Happy New Year and a successful 2016.
Tuesday, December 22, 2015
Guns and Crypto
“We believe it would be wrong to weaken security for hundreds of millions of law-abiding customers so that it will also be weaker for the very few who pose a threat,” said a spokesperson from Smith & Wesson on the recent calls for increased gun control.
The quote was actually from Apple on a proposed British law that would require the company to devise methods to break into iPhone communications.
In the wake of Paris and San Bernardino we've heard calls for controls on both guns and cryptography with eerily similar arguments coming from defenders. If we ban guns/crypto then only bad people will have guns/crypto. Any attempts to limit guns/crypto is a slippery slope that takes away our constitutional rights and our freedoms.
I don't have a gun but I do use encryption, built into the iPhone and the various apps and browsers I use to communicate. Fat lot of good it does me as hackers stole my personal information because I shopped at Home Depot and Target. Because my health insurance runs through Anthem. Because I voted in the State of Georgia.
I am a strong believer in individual rights, a person should be able to use cryptography to protect their communications and a gun, if they wish, to protect their family. But I do see the value in gaining access in communications to stop a terrorist as well as making it harder for them to get the weapons to carry out their acts. Why can't the fingerprint technology that unlocks my iPhone also unlock a gun? The gun/crypto advocates don't trust to government to implement any restrictions reasonably and thus fight any changes.
No laws can completely eliminate or even restrict guns or crypto but they can make it harder to use. The challenges aren't technological, we can create guns or crypto protocols that perform as we want them to perform. The challenges are social, finding the right balance between rights and security and governments we can trust to enforce that balance.
The quote was actually from Apple on a proposed British law that would require the company to devise methods to break into iPhone communications.
In the wake of Paris and San Bernardino we've heard calls for controls on both guns and cryptography with eerily similar arguments coming from defenders. If we ban guns/crypto then only bad people will have guns/crypto. Any attempts to limit guns/crypto is a slippery slope that takes away our constitutional rights and our freedoms.
I don't have a gun but I do use encryption, built into the iPhone and the various apps and browsers I use to communicate. Fat lot of good it does me as hackers stole my personal information because I shopped at Home Depot and Target. Because my health insurance runs through Anthem. Because I voted in the State of Georgia.
I am a strong believer in individual rights, a person should be able to use cryptography to protect their communications and a gun, if they wish, to protect their family. But I do see the value in gaining access in communications to stop a terrorist as well as making it harder for them to get the weapons to carry out their acts. Why can't the fingerprint technology that unlocks my iPhone also unlock a gun? The gun/crypto advocates don't trust to government to implement any restrictions reasonably and thus fight any changes.
No laws can completely eliminate or even restrict guns or crypto but they can make it harder to use. The challenges aren't technological, we can create guns or crypto protocols that perform as we want them to perform. The challenges are social, finding the right balance between rights and security and governments we can trust to enforce that balance.
Monday, December 21, 2015
Blog Followers
If you follow this or any other Blogger-based blog via a non-Google account, you'll need to follow with a Google account instead. Details.
Wednesday, December 16, 2015
Simons and Berkeley
The Simons Institute for the Theory of Computing in Berkeley held two programs this fall, Fine-Grained Complexity and Algorithm Design and Economics and Computation both ending this week. It would have been a great fall for me to spend there as I have strong connections in both groups, but as a department chair and father to a high-school senior it would have been impossible to be away from Atlanta for an extended period of time.
But I did manage some short trips, my first to Simons. In September I went to attend the workshop on Connections Between Algorithm Design and Complexity Theory but had to leave early and to make up for it I visited for a few days last week as well. The Simons Institute has taken over Calvin Lab, a circular building on the UC Berkeley campus. Lecture rooms on the first floor, wide-open working spaces on the second floor where most people congregate and visitor offices on the third floor. It's not just the space but an incredible group of complexity theorists and EC types there this fall.
The year marks thirty years since I spent that not-so-happy year of grad school in Berkeley. The city of Berkeley has mellowed a bit. When I lived there before, many recent graduates didn't leave and instead remained in their cheap rent-controlled apartments. Now you see almost a bimodal distribution centered around college-aged students and late middle-aged residents. Berkeley housing is getting hard to find again, this time because of overflow from limited affordable housing in San Francisco. Still Berkeley seems like a city that has never grown up. I guess people like it that way.
But I did manage some short trips, my first to Simons. In September I went to attend the workshop on Connections Between Algorithm Design and Complexity Theory but had to leave early and to make up for it I visited for a few days last week as well. The Simons Institute has taken over Calvin Lab, a circular building on the UC Berkeley campus. Lecture rooms on the first floor, wide-open working spaces on the second floor where most people congregate and visitor offices on the third floor. It's not just the space but an incredible group of complexity theorists and EC types there this fall.
The year marks thirty years since I spent that not-so-happy year of grad school in Berkeley. The city of Berkeley has mellowed a bit. When I lived there before, many recent graduates didn't leave and instead remained in their cheap rent-controlled apartments. Now you see almost a bimodal distribution centered around college-aged students and late middle-aged residents. Berkeley housing is getting hard to find again, this time because of overflow from limited affordable housing in San Francisco. Still Berkeley seems like a city that has never grown up. I guess people like it that way.
Thursday, December 10, 2015
Ada Lovelace (1815-1852)
Ada, daughter of the poet Lord Byron, showed an early talent in mathematics. At age 17 she became a friend of Charles Babbage, inventor of the difference engine and proposed a more complex analytic engine. Babbage gave a talk at the University of Turin and Ada had the task of translating an article written by Italian engineer Luigi Menabrea based on the talk. Ada did more than just translate, she added her own thoughts which included a program that could be run on the analytic engine computing a specific sequence of numbers. The analytic engine never got built and Ada never saw her program run but nevertheless she gets credit as the first computer programmer and most notably in 1980 the US Department of Defense named their preferred computer language "Ada" in her honor.
Here's to Lady Ada Lovelace, born way too far ahead of her time.
Monday, December 07, 2015
crank math and crank people
In the same week I got email claming:
1) Babai had shown that GI is in quasipoly time.
2) Opeyemi Enoch, a Nigerian Mathematician, solved the Riemann hypothesis.
I believe Babai's claim since (a) he's worked on the problem a lot and is a known expert, (b) the result is believable, and (c) the math we know now seem up to the task.
I do not believe Enoch's claim. It would be unfair for me to not believe it because I never heard of him. However, my sources in math say that RH is not in a position to be solved. Also, articles I've read on the web including this article seem to say things that cast doubts on it such as this quote:
motivation to solve the problem came from his students, who brought it to him with the hope of making 1 million ``off the internet''
Bobby Jindall (who?) dropped out of the Prez race (why am I bringing this up? Is he working on RH?) Bobby Jindall was a Rhodes Scholar. (why am I bringing this up? Wait and see.)
In 2003 I was in a Taxi (ask your grandparents what Taxi's were before we had Uber) and the driver began telling me lots of conspiracies--- there is no Gold in Fort Knx, the novel Goldfinger was Ian Flemmings attempt to tell us this, Reagan was shot because he knew this, and the American Presidency is controlled by a secret cabal. I pointed out that if that's the case how did George Bush Sr (clearly a member of the Cabal) lose to Bill Clinton. His answer: Bill Clinton was a Rhodes Scholar and the Cabal is controlled by Rhode Scholars.
Its hard to argue with a conspiracy theorist about the past. But you can challenge him to predict the future. So I told him ``if you can predict who will be the Democratic Nominee for Prez in 2004 then I will look at your website and take it seriously. If not, then not'' He smiled and just said ``Wesley Clark was a Rhodes scholar'' If I had asked him who would be the republican nominee in 2016 then he might have said ``Bobby Jindal is a Rhodes Scholar''
The way to expose conspiracy theorists, astrologers, and The National Inquirer is to take their predictions seriously and test them. I wonder why anyone believes these things given their track record.
Judging math things is different- A math paper can be read and should stand on its own.
Thursday, December 03, 2015
Moonshots
In 1961 Kennedy said "this nation should commit itself to achieving the goal, before this decade is out, of landing a man on the Moon and returning him safely to the Earth". In 1969 Neil Armstrong and Buzz Aldrin walked on the moon and returned to earth safely. In the 46 years hence man has gone no further.
Bill Gates founded Microsoft to place "a computer on every desk and in every home". Once he succeeded, Microsoft lost its way. Their current mission takes a different tact "Empower every person and every organization on the planet to achieve more."
In my life I've seen many a moonshot succeed, from a computer chess program that can beat the best humans, the classification of finite simple groups and the proof of Fermat's last theorem. They all seem to end up with a "Now what?". If you have a finite mission, you can only achieve finite things. We often think of moonshots as an amazing challenge but they serve as limits as well.
In computing we have a law that has become a mission, that the power of computing doubles roughly every two years. These improvements have enabled the incredible advances in learning, automation and connectivity that we see today. We keep making better computers because we don't have a limit just a goal of getting better than where we are today.
When I first started writing grant proposals, a senior faculty member chided me for saying "the ultimate goal of computational complexity is prove P ≠ NP". He said that would kill any grant proposals after we did settle the question. While we are in no danger of solving this problem in the near future, eventually we will and I'd hate to see the field whither afterwards.
My favorite mission comes from Star Trek, particularly the TNG version with "Its continuing mission: to explore strange new worlds, to seek out new life and new civilizations, to boldly go where no one has gone before."
Bill Gates founded Microsoft to place "a computer on every desk and in every home". Once he succeeded, Microsoft lost its way. Their current mission takes a different tact "Empower every person and every organization on the planet to achieve more."
In my life I've seen many a moonshot succeed, from a computer chess program that can beat the best humans, the classification of finite simple groups and the proof of Fermat's last theorem. They all seem to end up with a "Now what?". If you have a finite mission, you can only achieve finite things. We often think of moonshots as an amazing challenge but they serve as limits as well.
In computing we have a law that has become a mission, that the power of computing doubles roughly every two years. These improvements have enabled the incredible advances in learning, automation and connectivity that we see today. We keep making better computers because we don't have a limit just a goal of getting better than where we are today.
When I first started writing grant proposals, a senior faculty member chided me for saying "the ultimate goal of computational complexity is prove P ≠ NP". He said that would kill any grant proposals after we did settle the question. While we are in no danger of solving this problem in the near future, eventually we will and I'd hate to see the field whither afterwards.
My favorite mission comes from Star Trek, particularly the TNG version with "Its continuing mission: to explore strange new worlds, to seek out new life and new civilizations, to boldly go where no one has gone before."
Sunday, November 29, 2015
One more sign that a Journal is bogus. Or is it?
I went to an AMS conference with my High School Student Naveen (who presented this paper) and an ugrad from my REU program Nichole (who presented this paper). Note that this is a math conference- low prestige, mostly unrefereed, parallel sessions, but still a good place to pick up some new problems to work on and learn some things, and meet people.
Both Naveen and Nichole later got email from a journal urging them to submit their work! They were also invited to be on the Editorial Board! I can understand inviting a HS student who is a SENIOR to be on an editorial board, but Naveen is a SOPHMORE!
Naveen and Nichole both emailed me asking what this was about and I looked around and found, not to my surprise, that the journal is an author-pays journal of no standards. On the other hand, it was open access, on the other other hand, they had an article claiming that R(4,6)=36 (might be true, but if this was known, I would know it, see this blog post for more on that proof technique).
The pay-to-publish model is not necc. bad, and is standard in some fields, but is unusual for math. Of perhaps more importance is that the journal had no standards. And they prey on the young and naive.
Or do they?
Consider the following scenario: Naveen publishes in this journal and this publication is on his college application, hence he gets into a good college with scholarship. He knows exactly what he is buying. Or Nicole does this to help her Grad school Application. Or I do this for my resume and get a salary increase. Or an untenured prof does this to get tenure ( Deans can count but they can't read). And it gets worse--- the school gives the prof tenure, knowing the papers are bogus, but now they can say they have a prof who publishes X papers a year! At some point I don't know who is scamming who.
This blog post (not mine) gives several criteria for when a journal is bogus. I'll add one: When they ask a 15 years old to be on their editorial board.
Both Naveen and Nichole later got email from a journal urging them to submit their work! They were also invited to be on the Editorial Board! I can understand inviting a HS student who is a SENIOR to be on an editorial board, but Naveen is a SOPHMORE!
Naveen and Nichole both emailed me asking what this was about and I looked around and found, not to my surprise, that the journal is an author-pays journal of no standards. On the other hand, it was open access, on the other other hand, they had an article claiming that R(4,6)=36 (might be true, but if this was known, I would know it, see this blog post for more on that proof technique).
The pay-to-publish model is not necc. bad, and is standard in some fields, but is unusual for math. Of perhaps more importance is that the journal had no standards. And they prey on the young and naive.
Or do they?
Consider the following scenario: Naveen publishes in this journal and this publication is on his college application, hence he gets into a good college with scholarship. He knows exactly what he is buying. Or Nicole does this to help her Grad school Application. Or I do this for my resume and get a salary increase. Or an untenured prof does this to get tenure ( Deans can count but they can't read). And it gets worse--- the school gives the prof tenure, knowing the papers are bogus, but now they can say they have a prof who publishes X papers a year! At some point I don't know who is scamming who.
This blog post (not mine) gives several criteria for when a journal is bogus. I'll add one: When they ask a 15 years old to be on their editorial board.
Monday, November 23, 2015
Star Trek Computing
In the wake of Leonard Nimoy's death last February, I decided to rewatch the entire original Star Trek series, all 79 episodes. I had watched them each many times over in high school in the 70's, though the local station removed a scene or two from each episode to add commercial time and I often missed the opening segment because I didn't get home from school in time. Back in those stone ages we had no DVR or other method to record shows. I hadn't seen many episodes of the original series since high school.
Now I can watch the entire episodes whenever I want in full and in order through the magic of Netflix. I finished this quest a few days ago. Some spoilers below.
I could talk about the heavy sexism, the ability to predict future technologies (the flat screen TV in episode 74), the social issues in the 23rd century as viewed from the 60's, or just the lessons in leadership you can get from Kirk. Given the topic of this blog, let's talk about computing in Star Trek which they often just get so wrong, such as when Spock asks the computer to compute the last digit of π to force Jack-the-Ripper to remove his consciousness from the ship's computers.
Too many episodes end with Kirk convincing a computer or robot to destroy itself. I'd like to see him try that with Siri. In one such episode "The Ultimate Computer", a new computer is installed in the Enterprise that replaces most of the crew. A conversation between Kirk and McCoy sounds familiar to many we have today (source).
MCCOY: Did you see the love light in Spock's eyes? The right computer finally came along. What's the matter, Jim?
KIRK: I think that thing is wrong, and I don't know why.
MCCOY: I think it's wrong, too, replacing men with mindless machines.
KIRK: I don't mean that. I'm getting a Red Alert right here. (the back of his head) That thing is dangerous. I feel. (hesitates) Only a fool would stand in the way of progress, if this is progress. You have my psychological profiles. Am I afraid of losing my job to that computer?
MCCOY: Jim, we've all seen the advances of mechanisation. After all, Daystrom did design the computers that run this ship.
KIRK: Under human control.
MCCOY: We're all sorry for the other guy when he loses his job to a machine. When it comes to your job, that's different. And it always will be different.
KIRK: Am I afraid of losing command to a computer? Daystrom's right. I can do a lot of other things. Am I afraid of losing the prestige and the power that goes with being a starship captain? Is that why I'm fighting it? Am I that petty?
MCCOY: Jim, if you have the awareness to ask yourself that question, you don't need me to answer it for you. Why don't you ask James T. Kirk? He's a pretty honest guy.
Later in the episode the computer starts behaving badly and Kirk has to convince it to shut itself down. But what if the computer just did its job? Is that our real future: Ships that travel to stars controlled only by machine. Or are we already there?
Now I can watch the entire episodes whenever I want in full and in order through the magic of Netflix. I finished this quest a few days ago. Some spoilers below.
I could talk about the heavy sexism, the ability to predict future technologies (the flat screen TV in episode 74), the social issues in the 23rd century as viewed from the 60's, or just the lessons in leadership you can get from Kirk. Given the topic of this blog, let's talk about computing in Star Trek which they often just get so wrong, such as when Spock asks the computer to compute the last digit of π to force Jack-the-Ripper to remove his consciousness from the ship's computers.
Too many episodes end with Kirk convincing a computer or robot to destroy itself. I'd like to see him try that with Siri. In one such episode "The Ultimate Computer", a new computer is installed in the Enterprise that replaces most of the crew. A conversation between Kirk and McCoy sounds familiar to many we have today (source).
MCCOY: Did you see the love light in Spock's eyes? The right computer finally came along. What's the matter, Jim?
KIRK: I think that thing is wrong, and I don't know why.
MCCOY: I think it's wrong, too, replacing men with mindless machines.
KIRK: I don't mean that. I'm getting a Red Alert right here. (the back of his head) That thing is dangerous. I feel. (hesitates) Only a fool would stand in the way of progress, if this is progress. You have my psychological profiles. Am I afraid of losing my job to that computer?
MCCOY: Jim, we've all seen the advances of mechanisation. After all, Daystrom did design the computers that run this ship.
KIRK: Under human control.
MCCOY: We're all sorry for the other guy when he loses his job to a machine. When it comes to your job, that's different. And it always will be different.
KIRK: Am I afraid of losing command to a computer? Daystrom's right. I can do a lot of other things. Am I afraid of losing the prestige and the power that goes with being a starship captain? Is that why I'm fighting it? Am I that petty?
MCCOY: Jim, if you have the awareness to ask yourself that question, you don't need me to answer it for you. Why don't you ask James T. Kirk? He's a pretty honest guy.
Later in the episode the computer starts behaving badly and Kirk has to convince it to shut itself down. But what if the computer just did its job? Is that our real future: Ships that travel to stars controlled only by machine. Or are we already there?
Thursday, November 19, 2015
A Silly String Theorem
First a note on a serious theorem: Babai has posted a video (mp4, 1h 40 m, 653MB) of his first talk on his Graph Isomorphism algorithm.
I was giving a talk on the Kleene star operator (don't ask) and came across this cute little problem. Say a language L commutes if for all u,v in L, uv=vu.
Problem: Show that L commutes if and only if L is a subset of w* for some fixed string w.
Here w* is the set of strings consisting of zero or more concatenations of w with itself. The if case is easy, but I found the other direction pretty tricky and came up with an ugly proof. I found a cleaner proof in Seymour Ginsburg's 1966 textbook The Mathematical Theory of Context Free Languages which I present here.
⇐ If u=wi and v=wj then uv=vu=wi+j.
⇒ Trivial if L contains at most one string. Assume L contains at least two strings.
Proof by induction on the length of the shortest non-empty string v in L.
Base case: |v|=1
Suppose there is an x in L with x not in v*. Then x = vibz for b in Σ-{v}. Then the i+1st character of xv is b and the i+1st character of vx is v, contradicting xv=vx. So we can let w=v.
Inductive case: |v|=k>1.
Lemma 1: Let y=vru be in L (u might be empty). Then uv=vu.
Proof: Since yv=vy we have vruv=vvru=vrvu. So vu=uv.
Let x in L be a string not in v* (if no such x exists we can let w=v). Pick j maximum such that x=vjz with z not the empty string. By Lemma 1 we have zv=vz. Note |z| < |v| otherwise v is a prefix of z, contradicting the maximality of j.
Lemma 2: For all y in L we have yz=zy.
Proof: As before let y = vru. By Lemma 1, uv=vu
Since yx=xy we have vruvjz = vjzvru
vruvjz = vruzvj by swapping v's and z's.
vjzvru = zvruvj by swapping v's and z's, and v's and u's.
So we have vruzvj = zvruvj
or yzvj = zyuj and thus yz=zy.
By Lemma 2, the set L∪{z} commutes. Since 0 < |z| < |v| by induction L∪{z} is a subset of w* for some w so L is a subset of w*.
I was giving a talk on the Kleene star operator (don't ask) and came across this cute little problem. Say a language L commutes if for all u,v in L, uv=vu.
Problem: Show that L commutes if and only if L is a subset of w* for some fixed string w.
Here w* is the set of strings consisting of zero or more concatenations of w with itself. The if case is easy, but I found the other direction pretty tricky and came up with an ugly proof. I found a cleaner proof in Seymour Ginsburg's 1966 textbook The Mathematical Theory of Context Free Languages which I present here.
⇐ If u=wi and v=wj then uv=vu=wi+j.
⇒ Trivial if L contains at most one string. Assume L contains at least two strings.
Proof by induction on the length of the shortest non-empty string v in L.
Base case: |v|=1
Suppose there is an x in L with x not in v*. Then x = vibz for b in Σ-{v}. Then the i+1st character of xv is b and the i+1st character of vx is v, contradicting xv=vx. So we can let w=v.
Inductive case: |v|=k>1.
Lemma 1: Let y=vru be in L (u might be empty). Then uv=vu.
Proof: Since yv=vy we have vruv=vvru=vrvu. So vu=uv.
Let x in L be a string not in v* (if no such x exists we can let w=v). Pick j maximum such that x=vjz with z not the empty string. By Lemma 1 we have zv=vz. Note |z| < |v| otherwise v is a prefix of z, contradicting the maximality of j.
Lemma 2: For all y in L we have yz=zy.
Proof: As before let y = vru. By Lemma 1, uv=vu
Since yx=xy we have vruvjz = vjzvru
vruvjz = vruzvj by swapping v's and z's.
vjzvru = zvruvj by swapping v's and z's, and v's and u's.
So we have vruzvj = zvruvj
or yzvj = zyuj and thus yz=zy.
By Lemma 2, the set L∪{z} commutes. Since 0 < |z| < |v| by induction L∪{z} is a subset of w* for some w so L is a subset of w*.
Monday, November 16, 2015
Is the word Quantum being used properly by civilians? Understood by them? You know the answer.
I've seen the word `civilians' expanded in use from non-military to non-X for some X. Not sure I've ever seen `civilians' mean `people who don't do math stuff' until the title of todays post. Well, there is a first time for everything.
I've blogged in the past about the use of the word Quantum (here) . The phrase Quantum Leap means a BIG leap, where as Quantum stuff is small. Though, to be fair, the discovery (invention?) of Quantum Mechanics was a big leap. So maybe that IS proper use. The James Bond movie Quantum of Solace uses Quantum to mean small, the ONLY time I've seen Quantum used to mean small, so Kudos to the title of an absolutely awful movie. Commenters on my prior blog on the subject pointed out that the original meaning of
quantum was quantity or amount without regard to size of discreteness. I think using it that way now would be very rare.
I came across a theatre called Quantum Theatre. What is Quantum Theatre? The following are actual quotes from their website.
Quantum artists mine all kinds of non-traditional spaces for the sensory possibilities they offer when combined with creative design.
We find it meaningful to place the audience and performer together, the moving parts inside the works.
We want to move people with our experiements.
The shows run the gamut from those you thought you knew but now experience like never before, to shows that didn’t exist until their elements mixed in our laboratory.
I came across this article in a place it didn't belong-- in the middle of an article about Google and NASA trying to build a quantum computer (see here.) This news might be exciting but the article was full of mistakes and bad-signs so I'm not to excited about it. Plus the reference to Quantum Theatre is just odd.
The BEST use of the word quantum that I've heard recently was in the episode The Ricks must be crazy of the excellent TV show Ricky and Morty:
The car is broken
Morty (a 14 year old): W-Whats wrong Rick? Is it the Quantum carburetor or something?
Rick (his grandfather, a brilliant scientist): Quantum carburetor? You can't just add a Sci-fi word to a car word and hope it means something.
W-what's wrong, Rick? Is it the quantum carburetor or something?
Read more at: http://transcripts.foreverdreaming.org/viewtopic.php?f=364&t=20185
Read more at: http://transcripts.foreverdreaming.org/viewtopic.php?f=364&t=20185
W-what's wrong, Rick? Is it the quantum carburetor or something?
Read more at: http://transcripts.foreverdreaming.org/viewtopic.php?f=364&t=20185
Read more at: http://transcripts.foreverdreaming.org/viewtopic.php?f=364&t=20185
Thursday, November 12, 2015
A Primer on Graph Isomorphism
I spent 14 years on the faculty at the University of Chicago. I know László Babai well, we collaborated on some of my best known work. I also know Ryerson 251, a room where I've seen hundreds of talks and given more than a few myself. So I could imagine the excitement in that room on Tuesday as Babai gave the most anticipated talk in the history of theoretical computer science, the first of several talks Babai is giving on his new algorithm for graph isomorphism [Video]. Gabriel Gaster extensively live tweeted the event. Jeremy Kun has some details. Update (12/14): Paper now posted.
For this post instead of doing a bad job trying to overview Babai's proof, I'll explain the graph isomorphism problem and why it is important in the complexity world.
Suppose we have two high schools, say HS North and HS South, each with 1000 students. Consider a diagram (or graph) containing a point for each student at HS North with lines between students who are facebook friends, and a similar diagram for HS South. Is there a 1-1 map from the students at HS North to the students at HS South so that these diagrams look identical? That's the graph isomorphism problem.
To determine whether these two graphs were isomorphic you could look at all the possible mappings between students, but that's 1000! or more than 4x102567 possible maps. There has long been known how to search a smaller number of possibilities, especially if we put some restrictions on the diagrams, but always exponential in n (the number of students) in the worst case. Ideally we'd like a polynomial-time algorithm and Babai gets very close, an algorithm that runs in time nlogkn time for some fixed k.
Graph Isomorphism is one of those few problems, like factoring, known to be in NP but not known to be in P or NP-complete. Graph non-isomorphism is the poster child for the class AM, the problems solved by a randomized verifier asking a single question to a powerful prover. Graph non-isomorphism in AM implies that Graph Isomorphism is not likely NP-complete and that under reasonable derandomization assumptions that Graph non-isomorphism is in NP. Kobler, Schöning and Toran wrote a whole book on the computational complexity issues of graph isomorphism.
Even small progress in graph isomorphism creates waves. At a theory conference in the late 80's a speaker caused a major stir when he casually mentioned he had a proof (he didn't) that Graph non-isomorphism was in NP. Babai's announced caused a huge tsunami and those of us who know him realize he wouldn't make such an announcement without being sure he has a proof. The talk put together a large number of ideas from combinatorics and graph theory. My impression is that those who saw the talk didn't leave convinced of the proof, but did feel Babai had found the pieces to make it work.
With Babai's breakthrough algorithm, the smart money says now that graph isomorphism sits in P. It took decades to get from quasipolynomial time to polynomial time for primality testing and the same time frame may be needed to get polynomial time for graph isomorphism. But it will likely happen and the complexity of graph isomorphism then gets a whole lot simpler.
A couple of thoughts: All the breakthrough results that I can remember were released as papers, ready to devour. This is the first result of this caliber I remember being announced as a talk.
Also we think of theory as a young person's game, most of the big breakthroughs coming from researchers early in their careers. Babai is 65, having just won the Knuth Prize for his lifetime work on interactive proofs, group algorithms and communication complexity. Babai uses his extensive knowledge of combinatorics and group theory to get his algorithm. No young researcher could have had the knowledge base or maturity to be able to put the pieces together the way that Babai did.
More on Babai and graph isomorphism from Scott, Luca, Bill, Tim, Dick and Ken, Reddit, Science News, New Scientist and Science.
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