Consider the following problem:
Given k, a natural number, determine if there exists x,y,z INTEGERS such that x3+y3+z3=k.
It is not obvious that this problem is decidable (I think it is but have not been able to find an exact statement to that affect; however, if it was not solvable, I would know that, hence it is solvable. If you know a ref give it in the comments.)
If k≡ 4,5 mod 9 then mod arguments easily show there is no solution. Huisman showed that if k≤ 1000, k≡1,2,3,6,7,8 mod 9 and max(|x|,|y|,|z|) ≤ 1015 and k is NOT one of
33, 42, 114, 165, 390, 579, 627, 633, 732, 795, 906, 921, 975
then there was a solution. For those on the list it was unknown.
Recently Booker (not Cory Booker, the candidate for prez, but Andrew Booker who I assume is a math-computer science person and is not running for prez) showed that
x3 + y3 + z3 =33
DOES have a solution in INTEGERS. It is
x= 8,866,128,975,287,528
y=-8,778,405,442,862,239
z=-2,736,111,468,807,040
does that make us more likely or less likely to think that
x3 + y3 + z3 =42
has a solution? How about =114, etc, the others on the list?
Rather than say what I think is true (I have no idea) here is what I HOPE is true: that the resolution of these problems leads to some mathematics of interest.
Sunday, April 28, 2019
Thursday, April 25, 2019
Geo-Centric Complexity
An interesting discussion during Dagstuhl last month about the US-centric view of theory. Bad enough that all talks and papers in an international venue are in English but we also have
- Manhattan Distance. How are foreigners supposed to know about the structure of streets in New York? What's wrong with grid distance?
- Las Vegas Algorithms. I found this one a little unfair, after all Monte Carlo algorithms came first. Still today might not Macau algorithms make sense?
- Arthur-Merlin Games. A British reference by a Hungarian living in the US (László Babai who also coined Las Vegas algorithms). Still the Chinese might not know the fables. Glad the Europeans don't remember the Pat and Vanna terminology I used in my first STOC talk.
- Alice and Bob. The famous pair of cryptographers but how generically American can you get. Why not Amare and Bhati?
I have two minds here. We shouldn't alienate or confuse those who didn't grow up in an Anglo-American culture. On the other hand, I hate to have to try and make all terminology culturally neutral, you'd just end up with technical and ugly names, like P and NP.
Monday, April 22, 2019
Quiz Show Scandals/Admissions Scandal/Stormy Daniels/Beer names:being a lawyer would drive me nuts!!!!!!
0) Charles van Doren (see here) passed away recently. For those who don't know he he was (prob most of you) he was one of the contestants involved in RIGGED quiz shows in the 1950's. While there was a Grand Jury Hearing about Quiz Shows being rigged, nobody went to jail since TV was new and it was not clear if rigging quiz shows was illegal. Laws were then passed to make them it illegal.
So why are today's so-called reality shows legal? I ask non-rhetorically.
(The person he beat in a rigged game show- Herb Stempel (see here) is still alive.)
1) The college admissions scandal. I won't restate the details and how awful it is since you can get that elsewhere and I doubt I can add much to it. One thing I've heard in the discussions about it is a question that is often posted rhetorically but I want to pose for real:
There are people whose parents give X dollars to a school and they get admitted even though they are not qualified. Why is that legal?
I ask that question without an ax to grind and without anger. Why is out-right bribery of this sort legal?
Possibilities:
a) Its transparent. So being honest about bribery makes it okay?
b) My question said `even though they are not qualified' - what if they explicitly or implicitly said `having parents give money to our school is one of our qualifications'
c) The money they give is used to fund scholarships for students who can't afford to go. This is an argument for why its not immoral, not why its not illegal.
But here is my question: Really, what is the legal issue here? It still seems like bribery.
2) Big Oil gives money to congressman Smith, who then votes against a carbon tax. This seems like outright bribery
Caveat:
a) If Congressman Smith is normally a anti-regulation then he could say correctly that he was given the money because they agree with his general philosophy, so it's not bribery.
b) If Congressman smith is normally pro-environment and has no problem with voting for taxes then perhaps it is bribery.
3) John Edwards a while back and Donald Trump now are claiming (not quite) that the money used to pay off their mistress to be quiet is NOT a campaign contribution, but was to keep the affair from his wife. (I don't think Donald Trump has admitted the affair so its harder to know what his defense is). But lets take a less controversial example of `what is a campaign contribution'
I throw a party for my wife's 50th birthday and I invite Beto O'Rourke and many voters and some Dem party big-wigs to the party. The party costs me $50,000. While I claim it's for my wife's bday it really is for Beto to make connections to voters and others. So is that a campaign contribution?
4) The creators of HUGE ASS BEER are suing GIANT ASS BEER for trademark infringement. I am not making this up- see here
---------------------------------------------------------
All of these cases involve ill defined questions (e.g., `what is a bribe'). And the people arguing either side are not unbiased. The cases also illustrate why I prefer mathematics: nice clean questions that (for the most part) have answers. We may have our biases as to which way they go, but if it went the other way we would not sue in a court of law.
So why are today's so-called reality shows legal? I ask non-rhetorically.
(The person he beat in a rigged game show- Herb Stempel (see here) is still alive.)
1) The college admissions scandal. I won't restate the details and how awful it is since you can get that elsewhere and I doubt I can add much to it. One thing I've heard in the discussions about it is a question that is often posted rhetorically but I want to pose for real:
There are people whose parents give X dollars to a school and they get admitted even though they are not qualified. Why is that legal?
I ask that question without an ax to grind and without anger. Why is out-right bribery of this sort legal?
Possibilities:
a) Its transparent. So being honest about bribery makes it okay?
b) My question said `even though they are not qualified' - what if they explicitly or implicitly said `having parents give money to our school is one of our qualifications'
c) The money they give is used to fund scholarships for students who can't afford to go. This is an argument for why its not immoral, not why its not illegal.
But here is my question: Really, what is the legal issue here? It still seems like bribery.
2) Big Oil gives money to congressman Smith, who then votes against a carbon tax. This seems like outright bribery
Caveat:
a) If Congressman Smith is normally a anti-regulation then he could say correctly that he was given the money because they agree with his general philosophy, so it's not bribery.
b) If Congressman smith is normally pro-environment and has no problem with voting for taxes then perhaps it is bribery.
3) John Edwards a while back and Donald Trump now are claiming (not quite) that the money used to pay off their mistress to be quiet is NOT a campaign contribution, but was to keep the affair from his wife. (I don't think Donald Trump has admitted the affair so its harder to know what his defense is). But lets take a less controversial example of `what is a campaign contribution'
I throw a party for my wife's 50th birthday and I invite Beto O'Rourke and many voters and some Dem party big-wigs to the party. The party costs me $50,000. While I claim it's for my wife's bday it really is for Beto to make connections to voters and others. So is that a campaign contribution?
4) The creators of HUGE ASS BEER are suing GIANT ASS BEER for trademark infringement. I am not making this up- see here
---------------------------------------------------------
All of these cases involve ill defined questions (e.g., `what is a bribe'). And the people arguing either side are not unbiased. The cases also illustrate why I prefer mathematics: nice clean questions that (for the most part) have answers. We may have our biases as to which way they go, but if it went the other way we would not sue in a court of law.
Thursday, April 18, 2019
Physics of Everday Life
Based on Scott's review, I read through Stephen Pinker's Enlightenment Now. I can't top Scott's exposition of the book, but it is pretty incredible how far humanity has gone when you step back to look at the big picture.
One line intrigued me, one that Pinker credits to a book called The Big Picture by Sean Carroll
Is it possible we could do more in everyday life if we knew more physics? I'd certainly use a teleporter in everyday life.
And is the statement even true today? We all use public key cryptography, even to read this blog. It's not completely clear if we understand the physics enough to know how or if large-scale quantum computers capable of breaking those systems can be built.
Everday life is relative.
One line intrigued me, one that Pinker credits to a book called The Big Picture by Sean Carroll
The laws of physics underlying everyday life (that is excluding extreme values of energy and gravitation like black holes, dark matter and the Big Bang) are completely known.Hasn't this statement almost always been true, in the sense that the leading minds would make this claim at many times in history. The ancient Greeks probably believed they understood physics that underlies everyday life. So did physicists after Newton. Life back then not today. My everyday life involves using a GPS device that requires understanding relativistic effects and computer chips that needed other scientific advances.
Is it possible we could do more in everyday life if we knew more physics? I'd certainly use a teleporter in everyday life.
And is the statement even true today? We all use public key cryptography, even to read this blog. It's not completely clear if we understand the physics enough to know how or if large-scale quantum computers capable of breaking those systems can be built.
Everday life is relative.
Sunday, April 14, 2019
Good article, terrible headline
About a month ago (after my P NP poll appeared) I got email from Jacob Aron asking me some questions about it. One thing he was excited about was that the number of people who thought P vs NP would be solved in the next decade had increased from 11% to 22%. I told him that this also surprised me and there had been no major advances to warrant that increase.
Then the article came out. Here is the pointer to the headline and the first few lines, the rest is behind a paywall
here
You may notice that the headline is
We could solve the biggest problem in math in the next decade
I emailed Jacob to send me the article, which he did. The article was fine, even quoting me as saying that the increase of people who thought it would be solved soon was unwarranted.
1) So, article fine, headline terrible.
2) A more honest headline would be
The Complexity Theory Community slightly more optimistic about when P vs NP will be resolved for no apparent reason.
3) More bad science:here
Headline is
Turing's Oracle: The computer that goes beyond logic
I think the article was about how a Turing Machine with an oracle for HALT can solve HALT. (If I am wrong about this let me know in the comments and I'll correct it.)
4) More bad science:here
Headline
Finally, a problem that only Quantum Computers will ever be able to solve.
This was about the oracle such that BQP is not in PH. Really!
5) I invite you to add your own.
6) OKAY, so why is the press SO BAD at reporting on our field? And is it just our field? I have one explanation, though I am sure there are many.
Our field is about showing the LIMITS of computation. Its hard to make that sexy so they ... lie? exaggerate. They themselves don't really understand our field? Note:
To explain to someone who does not really know CS why its important to have an oracle where BQP is not in PH is hard
To explain this to someone IN CS but not in Theory is still hard!
To explain this to someone IN CS and even in CS Theory, but not complexity (e.g., algorithms) might be hard, though it may depend on the person.
7) The old saying is `I don't care if you get the story wrong so long as you spell my name right' And indeed, they did spell my name right. So there is that! But more seriously and less about me or even the article that refers to my poll--- is it bad that science reporting is often wrong?
Then the article came out. Here is the pointer to the headline and the first few lines, the rest is behind a paywall
here
You may notice that the headline is
We could solve the biggest problem in math in the next decade
I emailed Jacob to send me the article, which he did. The article was fine, even quoting me as saying that the increase of people who thought it would be solved soon was unwarranted.
1) So, article fine, headline terrible.
2) A more honest headline would be
The Complexity Theory Community slightly more optimistic about when P vs NP will be resolved for no apparent reason.
3) More bad science:here
Headline is
Turing's Oracle: The computer that goes beyond logic
I think the article was about how a Turing Machine with an oracle for HALT can solve HALT. (If I am wrong about this let me know in the comments and I'll correct it.)
4) More bad science:here
Headline
Finally, a problem that only Quantum Computers will ever be able to solve.
This was about the oracle such that BQP is not in PH. Really!
5) I invite you to add your own.
6) OKAY, so why is the press SO BAD at reporting on our field? And is it just our field? I have one explanation, though I am sure there are many.
Our field is about showing the LIMITS of computation. Its hard to make that sexy so they ... lie? exaggerate. They themselves don't really understand our field? Note:
To explain to someone who does not really know CS why its important to have an oracle where BQP is not in PH is hard
To explain this to someone IN CS but not in Theory is still hard!
To explain this to someone IN CS and even in CS Theory, but not complexity (e.g., algorithms) might be hard, though it may depend on the person.
7) The old saying is `I don't care if you get the story wrong so long as you spell my name right' And indeed, they did spell my name right. So there is that! But more seriously and less about me or even the article that refers to my poll--- is it bad that science reporting is often wrong?
Thursday, April 11, 2019
Elwyn Berlekamp Died April 9, 2019
Elwyn R. Berlekamp entered this world on Sept 6, 1940, and left it on April 9, 2019. Wikipedia calls him An American Mathematician which seems to narrow to me. He worked on a variety of topics which were Math, Comp Sci, finance, and likely others --- if you know any that I missed leave comments.
Here are some of the things he worked on:
1) Factoring Polynomials over large finite fields (See here from 1970)
I quote the abstract
2) Combinatorial Games. He co-wrote the classic Winning Ways For your Mathematical Plays with John Conway and Richard Guy. These books are like NIM on steroids. They are FUN and have some serious math in them. (My blog system thinks Combinatorial is not a word. It is wrong)
3) Combinatorial Games. He was very interested in Go via Combinatorial Games. I have an article (alas, not online) by him entitled Introductory Overview of Mathematical Go Endgames. There has been further work on this that is on the web, see here. I wonder what ERB would think of the ALPHAGO program.
4) Berlekamp-Massey Algorithm finds the shortest linear feedback shift register for a given binary output sequence.
5) Berlekamp-Welch Algorithm efficiently corrects errors in Reed-Solomon codes. (Berlekamp also wrote a book on Algebraic Coding Theory.)
6) I didn't know about his work in finance until I began writing this, but the following quote from Wikipedia is impressive
Here are some of the things he worked on:
1) Factoring Polynomials over large finite fields (See here from 1970)
I quote the abstract
This paper reviews some of the known algorithms for factoring polynomials over finite fields and presents a new deterministic procedure for reducing the problem of factoring an arbitrary polynomial over the Galois field GP(pm) to the problem of finding roots in GF(p) of certain other polynomials over GP(p). The amount of computation and the storage space required by these algorithms are algebraic in both the degree of the polynomial to be factored and the log of the order of the field.
Certain observations on the application of these methods to factorization of polynomials over the rational integers are also included.I think he meant what we would call polynomial time when he says algebraic.
2) Combinatorial Games. He co-wrote the classic Winning Ways For your Mathematical Plays with John Conway and Richard Guy. These books are like NIM on steroids. They are FUN and have some serious math in them. (My blog system thinks Combinatorial is not a word. It is wrong)
3) Combinatorial Games. He was very interested in Go via Combinatorial Games. I have an article (alas, not online) by him entitled Introductory Overview of Mathematical Go Endgames. There has been further work on this that is on the web, see here. I wonder what ERB would think of the ALPHAGO program.
4) Berlekamp-Massey Algorithm finds the shortest linear feedback shift register for a given binary output sequence.
5) Berlekamp-Welch Algorithm efficiently corrects errors in Reed-Solomon codes. (Berlekamp also wrote a book on Algebraic Coding Theory.)
6) I didn't know about his work in finance until I began writing this, but the following quote from Wikipedia is impressive
In 1989, Berlekamp purchased the largest interest in a trading company named Axcom Trading Advisors. After the firm's futures trading algorithms were rewritten, Axcom's Medallion Fund had a return (in 1990) of 55%, net of all management fees and transaction costs. The fund has subsequently continued to realize annualized returns exceeding 30% under management by James Harris Simons and his Renaissance Technologies Corporation.
Sunday, April 07, 2019
Problems with a point- NOT a plug, just some thoughts on books and book writing
Problems with a Point: Exploring Math and Computer Science by Gasarch and Kruskal, available on amazon here, came out a while back and I plugged it in my blog post here. Regan-Lipton reviewed it here.
This post is NOT a plug, its random points about the book and book writing
1) On Feb 28 I checked my Amazon ranking twice. At noon it was at 400,000 and at 4:00PM it was at roughly 80,000. Either (a) someone bought a copy, or (b) the number of sales of such books is so small that amazon does rankings arbitrarily. A counter argument to that: Bounded Queries in Recursion Theory, which I wrote in 1998 and, despite an awesome review on amazon that I wrote, sold literally 0 copies last year, is ranked 5,000,000. So even within the low numbers there is SOME logic to the rankings. (Update- Today, April 7, PWAP is at 800,000 and BQIRT is at 12,000,000. I suspect that nothing has changed with BQIRT and the 5,000,000 or 12,000,000 is arbitrary. As for PWAP--- since its still a new book the ranking might mean something. Oh well- fame is fleeting!)
1.5) At one time my book was 29th in the category Teen and Young Adult Computer Programming. Really? While teens and young adults might read it, I don't think of it as geared towards them (only one use of OMG and two uses of awesome in the entire book). And as for Computer Programming... uh... no.
2) I emailed many people about it and many people are buying it. Some tell me that the cover looks great so YES, they are judging a book by its cover. My mom asked how it was selling. I told her that (a) about 20 people told me they bought it, and (b) I doubt that's how JK Rowling responds when she's asked how her Harry Potter books are selling.
3) Many people could actually read this book and would care about its content. When I wrote Bounded Queries in Recursion Theory (with Georgia Martin) I didn't email a lot of people about it since it was rather specialized.
4) Some people said `Gee bill, we didn't know you were working on a book'. That's true- I didn't talk about it much. Clyde knew (my co-author, so he had to know), Lance knew. Darling knew. Mom Knew. That might be about it. Why so few? see next point.
5) I had heard there are two kinds of people:
Those that talk about writing a book but don't do it
Those that write a book instead of talking about it.
I chose to be the second type.
6) When I got an advanced copy I couldn't stop reading it. Ego? Disbelieve? Some of each?
7) ADVICE for book writers: just get it done. You can polish it later but first get it done so you have something to polish.
8) More advice: You can never have too many proofreaders. Even though its been gone over a lot I still read it and find minor things to fix OR think `gee why didn't I include BLAH in the book'
9) Title: `Problems with a Point' I came up with very early and I liked it a lot. World Scientific (our publisher) pointed out, correctly, that we need a subtitle since PWAP didn't really tell the reader whats in it. I wanted to make it alliterative but it was hard. Here were some possibilities
PWAP: Mathematical Musing Made Magnificent (can replace Musings with Morsels)
PWAP: Mathematical Meditations and Computer Science Cogitations
PWAP: Mathematical Musings and the Math that Makes those Musings Matter
All in all I am happy with the non-alliterative, but informative
PWAP: Exploring Math and Computer Science.
10) Odd thing I learned- if I quote someone who made a comment on my blog or emailed me I needed permission. Actually I doubt I NEEDED it but the laws here are murky so I got it just to make sure. That explains this blog from a while back: here
11) ADVICE for book writers: Write the preface last since then you have a better idea of why you wrote the book.
This post is NOT a plug, its random points about the book and book writing
1) On Feb 28 I checked my Amazon ranking twice. At noon it was at 400,000 and at 4:00PM it was at roughly 80,000. Either (a) someone bought a copy, or (b) the number of sales of such books is so small that amazon does rankings arbitrarily. A counter argument to that: Bounded Queries in Recursion Theory, which I wrote in 1998 and, despite an awesome review on amazon that I wrote, sold literally 0 copies last year, is ranked 5,000,000. So even within the low numbers there is SOME logic to the rankings. (Update- Today, April 7, PWAP is at 800,000 and BQIRT is at 12,000,000. I suspect that nothing has changed with BQIRT and the 5,000,000 or 12,000,000 is arbitrary. As for PWAP--- since its still a new book the ranking might mean something. Oh well- fame is fleeting!)
1.5) At one time my book was 29th in the category Teen and Young Adult Computer Programming. Really? While teens and young adults might read it, I don't think of it as geared towards them (only one use of OMG and two uses of awesome in the entire book). And as for Computer Programming... uh... no.
2) I emailed many people about it and many people are buying it. Some tell me that the cover looks great so YES, they are judging a book by its cover. My mom asked how it was selling. I told her that (a) about 20 people told me they bought it, and (b) I doubt that's how JK Rowling responds when she's asked how her Harry Potter books are selling.
3) Many people could actually read this book and would care about its content. When I wrote Bounded Queries in Recursion Theory (with Georgia Martin) I didn't email a lot of people about it since it was rather specialized.
4) Some people said `Gee bill, we didn't know you were working on a book'. That's true- I didn't talk about it much. Clyde knew (my co-author, so he had to know), Lance knew. Darling knew. Mom Knew. That might be about it. Why so few? see next point.
5) I had heard there are two kinds of people:
Those that talk about writing a book but don't do it
Those that write a book instead of talking about it.
I chose to be the second type.
6) When I got an advanced copy I couldn't stop reading it. Ego? Disbelieve? Some of each?
7) ADVICE for book writers: just get it done. You can polish it later but first get it done so you have something to polish.
8) More advice: You can never have too many proofreaders. Even though its been gone over a lot I still read it and find minor things to fix OR think `gee why didn't I include BLAH in the book'
9) Title: `Problems with a Point' I came up with very early and I liked it a lot. World Scientific (our publisher) pointed out, correctly, that we need a subtitle since PWAP didn't really tell the reader whats in it. I wanted to make it alliterative but it was hard. Here were some possibilities
PWAP: Mathematical Musing Made Magnificent (can replace Musings with Morsels)
PWAP: Mathematical Meditations and Computer Science Cogitations
PWAP: Mathematical Musings and the Math that Makes those Musings Matter
All in all I am happy with the non-alliterative, but informative
PWAP: Exploring Math and Computer Science.
10) Odd thing I learned- if I quote someone who made a comment on my blog or emailed me I needed permission. Actually I doubt I NEEDED it but the laws here are murky so I got it just to make sure. That explains this blog from a while back: here
11) ADVICE for book writers: Write the preface last since then you have a better idea of why you wrote the book.
Thursday, April 04, 2019
Cuckoo Cycles
Guest Blog by John Tromp (Thanks for the opportunity, Lance)
In the past few months, cycle finding has become one of the most widely run graph theory problems. An estimated quarter-to-half million GPUs are constantly looking for 42-cycles in random bipartite graphs on a billion or more nodes. Customized chips have been designed to perform this specific application an order of magnitude more efficiently, thanks to integrating as much as 1GB of SRAM on a single chip, where Intel/AMD CPUs top out at 64 MB.
The purpose of all this? To compete for block rewards in various cryptocurrencies using the Cuckoo Cycle Proof of Work.
It is named after the Cuckoo Hashtable, in which each data item can be stored at two possible locations, one in each of two arrays. A hash function maps the pair of item and choice of array to its location in that array. When a newly stored item finds both its locations already occupied, one is kicked out and replaced by the new item. This forces the kicked-out item to move to its alternate location, possibly kicking out yet another item. Problems arise if the corresponding Cuckoo graph, a bipartite graph with array locations as nodes and item location pairs as edges, has cycles. While n items, whose edges form a cycle, could barely be stored in the table, any n+1st item mapping within the same set of locations (a chord in the cycle) would not fit, as the pigeonhole principle tells us.
In the Proof-of-Work problem, we typically set n ≥ 29, and use edge indices (items) 0 .. N-1, where N = 2n. The endpoints of edge i are (siphash(i|0) % N, siphash(i|1) % N), with siphash being a popular keyed hash function. A solution is a cycle of length L in this Cuckoo graph, where typically L = 42.
While looking for cycles takes a lot of time and memory, solutions can be instantly verified. This sets Cuckoo Cycle apart from many other memory hard proofs of work that require computing a memory hard hash function, such as scrypt, both for proving and verifying. Some luck is also required to find a 42-cycle; Slides 18 and 19 of this Grincon.US presentation sketch a proof of the
Cuckoo Cycle Theorem: The expected number of L-cycles tends to 1/L
While there is a known linear time-memory trade-off, it suffers from a large constant factor penalty. A large bounty is offered on the project webpage for better trade-offs.
Cuckoo Cycle solvers spend nearly all cycles on edge trimming; identifying and removing edges that end in a leaf node (of degree 1), as such edges cannot be part of a cycle. Trimming rounds alternate between the two node partitions. This can be done using N bits for the edge bitmap, and N log2(3) bits for the (truncated) node degrees. The random accesses to the latter, while not a problem for custom chips using enough SRAM, cause large memory latencies on commodity hardware. These can be mostly avoided by storing remaining edges explicitly and (bucket) sorting them to determine node degrees. Despite using an order of magnitude more memory than the edge bitmap, this is still over 4x faster.
The fraction fi of remaining edges after i trimming rounds (in the limit as N goes to infinity) appears to obey the
So far I have only been able to prove the conjecture for i ≤ 3. For instance, for i = 1, the probability that an edge endpoint is not the endpoint of any other edge is (1-1/N)N-1 ~ 1/e. Here's hoping someone finds an elegant proof...
In the past few months, cycle finding has become one of the most widely run graph theory problems. An estimated quarter-to-half million GPUs are constantly looking for 42-cycles in random bipartite graphs on a billion or more nodes. Customized chips have been designed to perform this specific application an order of magnitude more efficiently, thanks to integrating as much as 1GB of SRAM on a single chip, where Intel/AMD CPUs top out at 64 MB.
The purpose of all this? To compete for block rewards in various cryptocurrencies using the Cuckoo Cycle Proof of Work.
It is named after the Cuckoo Hashtable, in which each data item can be stored at two possible locations, one in each of two arrays. A hash function maps the pair of item and choice of array to its location in that array. When a newly stored item finds both its locations already occupied, one is kicked out and replaced by the new item. This forces the kicked-out item to move to its alternate location, possibly kicking out yet another item. Problems arise if the corresponding Cuckoo graph, a bipartite graph with array locations as nodes and item location pairs as edges, has cycles. While n items, whose edges form a cycle, could barely be stored in the table, any n+1st item mapping within the same set of locations (a chord in the cycle) would not fit, as the pigeonhole principle tells us.
In the Proof-of-Work problem, we typically set n ≥ 29, and use edge indices (items) 0 .. N-1, where N = 2n. The endpoints of edge i are (siphash(i|0) % N, siphash(i|1) % N), with siphash being a popular keyed hash function. A solution is a cycle of length L in this Cuckoo graph, where typically L = 42.
While looking for cycles takes a lot of time and memory, solutions can be instantly verified. This sets Cuckoo Cycle apart from many other memory hard proofs of work that require computing a memory hard hash function, such as scrypt, both for proving and verifying. Some luck is also required to find a 42-cycle; Slides 18 and 19 of this Grincon.US presentation sketch a proof of the
Cuckoo Cycle Theorem: The expected number of L-cycles tends to 1/L
While there is a known linear time-memory trade-off, it suffers from a large constant factor penalty. A large bounty is offered on the project webpage for better trade-offs.
Cuckoo Cycle solvers spend nearly all cycles on edge trimming; identifying and removing edges that end in a leaf node (of degree 1), as such edges cannot be part of a cycle. Trimming rounds alternate between the two node partitions. This can be done using N bits for the edge bitmap, and N log2(3) bits for the (truncated) node degrees. The random accesses to the latter, while not a problem for custom chips using enough SRAM, cause large memory latencies on commodity hardware. These can be mostly avoided by storing remaining edges explicitly and (bucket) sorting them to determine node degrees. Despite using an order of magnitude more memory than the edge bitmap, this is still over 4x faster.
The fraction fi of remaining edges after i trimming rounds (in the limit as N goes to infinity) appears to obey the
Cuckoo Cycle Conjecture:
fi = ai-1 * ai, where a-1 = a0 = 1, and ai+1 = 1 - e-ai
fi could equivalently be defined as the fraction of edges whose first endpoint is the middle of a (not necessarily simple) path of length 2i.So far I have only been able to prove the conjecture for i ≤ 3. For instance, for i = 1, the probability that an edge endpoint is not the endpoint of any other edge is (1-1/N)N-1 ~ 1/e. Here's hoping someone finds an elegant proof...
Monday, April 01, 2019
An Interdisciplinary approach to P vs NP
I have a grant with some brain scientists on the following exciting approach to P vs NP.
We want to prove:
There is no algorithm for SAT in P
This seems hard. So lets scale down our goals to:
Lance cannot find an algorithm for SAT in P
You can replace Lance with someone else, but Lance has volunteered to have brain scans done. The idea is to scan the P vs NP part of his brain and see what we can discern.
I know what you are thinking. Lance truly believes that SAT is NOT in P so perhaps even if SAT is in P, he would have a mental block. Hence I am hoping to also get some serious theorists that think SAT is in P to participate. This might also shed light on the following conjecture: is it easier to prove a theorem if you believe it is true.
Lance has proposed another line of research. Barriers. Try to establish
Lance cannot prove that SAT is NOT in P
using brain scans and he volunteered.
For this one the mental block problem is gone; however, as you can clearly see from his brain scan, Lance will never prove that SAT is not in P. Seven years as department chair has caused irreversible damage to his ability to reason logically.
We hope to recruit other theorists to the project. The sticking point might be privacy: if we prove that professor X cannot solve P vs NP will the affect their being hired? For this reason we will restrict the study to professors with Tenure. But this is a weakness of the study since we had wanted to study if younger people have a better chance to resolve P vs NP. So we need to find very young tenured professors.
After we get the technology working on Theorists and P vs NP we will try to expand it for home use. For example, if a spouse claims I could never learn to cook or I could never learn to drive (that's me) their partner will be able to verify the statement. It is not clear if this will make the divorce rate go up or down. Or if a child says I just can't do algebra the parents can test the child and say Oh yes you can!
We want to prove:
There is no algorithm for SAT in P
This seems hard. So lets scale down our goals to:
Lance cannot find an algorithm for SAT in P
You can replace Lance with someone else, but Lance has volunteered to have brain scans done. The idea is to scan the P vs NP part of his brain and see what we can discern.
I know what you are thinking. Lance truly believes that SAT is NOT in P so perhaps even if SAT is in P, he would have a mental block. Hence I am hoping to also get some serious theorists that think SAT is in P to participate. This might also shed light on the following conjecture: is it easier to prove a theorem if you believe it is true.
Lance has proposed another line of research. Barriers. Try to establish
Lance cannot prove that SAT is NOT in P
using brain scans and he volunteered.
 |
| Brain Scan of Lance's Proving Ability |
For this one the mental block problem is gone; however, as you can clearly see from his brain scan, Lance will never prove that SAT is not in P. Seven years as department chair has caused irreversible damage to his ability to reason logically.
We hope to recruit other theorists to the project. The sticking point might be privacy: if we prove that professor X cannot solve P vs NP will the affect their being hired? For this reason we will restrict the study to professors with Tenure. But this is a weakness of the study since we had wanted to study if younger people have a better chance to resolve P vs NP. So we need to find very young tenured professors.
After we get the technology working on Theorists and P vs NP we will try to expand it for home use. For example, if a spouse claims I could never learn to cook or I could never learn to drive (that's me) their partner will be able to verify the statement. It is not clear if this will make the divorce rate go up or down. Or if a child says I just can't do algebra the parents can test the child and say Oh yes you can!
Thursday, March 28, 2019
Scooped
At Dagstuhl I got a few ideas for future posts but then...
We had some discussions about STOC allowing program committee members to submit papers that I planned to post about. But then Suresh wrote a fine post on the same issue for SODA. My thoughts: PC members should not submit--no matter how you try to avoid the conflict of interest, there will always be a cloud on the process. Suresh suggest blind reviews that other non-theory conferences use, but I prefer the tiered PC system. It's fine to have PC members submit papers if the top of the tier, a senior PC or the PC chairs, makes the final calls.
I was also going to post about Yann LeCun's Facebook rant about stodgy CS departments but then Yann goes ahead and wins a Turing award with Geoffrey Hinton and Yoshua Bengio for their work on machine learning. I knew Yann from when we worked together at NEC Research in the early 2000's and let's just congratulate him and the others and let them bask in glory for truly transforming how we think of computing today. I'll get back to his post soon enough.
We had some discussions about STOC allowing program committee members to submit papers that I planned to post about. But then Suresh wrote a fine post on the same issue for SODA. My thoughts: PC members should not submit--no matter how you try to avoid the conflict of interest, there will always be a cloud on the process. Suresh suggest blind reviews that other non-theory conferences use, but I prefer the tiered PC system. It's fine to have PC members submit papers if the top of the tier, a senior PC or the PC chairs, makes the final calls.
I was also going to post about Yann LeCun's Facebook rant about stodgy CS departments but then Yann goes ahead and wins a Turing award with Geoffrey Hinton and Yoshua Bengio for their work on machine learning. I knew Yann from when we worked together at NEC Research in the early 2000's and let's just congratulate him and the others and let them bask in glory for truly transforming how we think of computing today. I'll get back to his post soon enough.
Sunday, March 24, 2019
Random Thoughts on the admissions scandal
In light of the recent academic scandal I am going to list ways I've heard to help get your kid into college and thoughts on how ethical they are (hint: bribing a coach to claim your kid is on the rowing team is not ethical).
1) Your kid likes (1) helping the homeless and (2) Ramsey Theory and (3) helping the homeless learn Ramsey Theory. Or perhaps they like rowing or Latin or fencing or Pig Latin or.... Great! encourage them, get them books and tutors, and whatever they need. You have an eye towards how this will look on for college admissions; however, it is your kids choice as to what they like. Also note that these activities are in addition to doing well in school, SATs, etc, not instead of it. Perfectly ethical, though I note that this avenue is not open to poor families and in some cases even middle class families.
2) Item 1 is the extreme on a spectrum in terms of how much are the extra things the kid does there idea OR planned by you to GET INTO A GOOD COLLEGE. This item will be the other extreme, but realize there is a continuum (actually I doubt there are a continuum number of options here, but there are many in between. Maybe its omega + omega*.) You have heard that being on the chess boxing teams is good on a college application so you TELL YOUR KID that they likes both Chess AND Boxing and should be on the team. You also hear this about being on the fencing team and knowing pig latin, so your kid can taunt their opponents like this:
youah, aint-kay ence-fay orth-way eans-bay
This might not be as bad as it sounds if the kid learns to like Chess-Boxing. But it may be worse than it sounds if the kid rebels against all of this and becomes a crack whore.
Is this ethical? It may be bad for the kid, but it may push him into things he ends up liking. One drawback: you've HEARD that being on the chess boxing team is good for college admissions, but is it true?
And again, not open to some families.
3) Item 2 (or even 1) but with an addition: Hire a college adviser to help you. Someone who knows (or claims to know) what colleges look for- Trombone, Latin, Chess-boxing, whatever. Still ethical but I again worry about the kids future rehab bills.
4) Here is where it gets murky. The college adviser helps:
a) Polish the kids essay (my parents, who are both in English, helped polish my essay to get into graduate school (I don't recall if there was one for ugrad). They told me to use lots of `ing' words so it sounds like I am actively doing things. They also helped me figure out when recursion-theoretic is hyphenated. I got into Harvard but not MIT, so make of that what you will.) Polishing, proofreading, that could be okay. But it can slip into b or c below.
b) The adviser (or the parents) talk to the kids to find out what to write, but then writes it. Maybe the kid proofreads and polishes. Maybe not even. The adviser is a ghostwriter. Clearly unethical but the line between helping-to-polish and adviser-wrote-it is again a continuum.
c) Adviser writes it and it is completely fictional. I once heard a rumor that a particular sample of an essay to get into med school was used by several med school applicant. Gee,they can't all have gotten inspired by watching their grandfather in his pajamas die of cancer. This is awful of course, but I wonder- what if the student writes a fictional essay all by themselves! Some combination of how much the essay is true and how much help you got on it is unethical. But some might be okay. Is it bad to polish stories that are basically true to make them flow more easily? Prob not. But there is a 2-dim continuum based on both accuracy and how much help the student got.
As a side note- how much does the personal statement matter for admissions? I suspect that if an elite school gets LOTS of REALLY QUALIFIED applicants, the essay may be all that distinguishes them.So it can be important. I also wonder if people on admissions can tell if an essay is not written by the applicant. Or maybe if there is an interview that can help detect it.
5) Parents give X amount of money to the college and the kid gets in. This is talked about a lot though I don't know how common it is for someone NOT qualified to GET IN based on money. College admissions has many factors so its not quite so clear cut what NOT qualified means. Even so, if seems odd that this is not in any way shape or form illegal. It IS transparent, so I guess thats a plus. The argument I've heard is that the money is used for scholarships to fund students who get in but can't afford to go. I do not know if this is true. And this one is only available to the top Z %, not sure what Z is, but there are people who can do 1,2,3,4 who can't do 5. I would call this unethical though colleges don't seem to think so. Or they do but they do it anyway.
6) Before I list the current scandal I want to list another issue: having a psychologist (or whoever it is who judges these things) say your kid is Learning Disabled so they get more time on the SATs. Perhaps bribing them, or perhaps its understood what you want. Again, I do not know how common this is. An alternative if you can't afford some of the above options, or done in conjunction with a lot of the above options.
7) The current scandal. Obviously unethical. A few things I wonder about:
a) One story was that they bribed someone to say their daughter was Learning Disabled and had to take the SAT (or whatever it was) in a separate room, making it easier to change the answers to the correct ones. So twice unethical.
b) Some of the students were clearly NOT qualified.
c) A parent does unethical things to get the kid into college.
The kid later lies to the parents about their grades or their plans
The parents are SHOCKED that their kids lie and wonder where the learned such behaviour!
8) Actually Item 2 --Parent has kid do things to plan to get into college-- is interesting for another reason. Are you your resume?
Imagine that Alice helps the homeless her Sophmore year in High School NOT because she cares about the homeless but because its good for college admissions.
Alice goes on to do other things that look good for college, NOT because she likes them or cares, but just to get into a good college.
She gets into an elite college
Did they take her in the hope she would KEEP doing these things or because she is the KIND OF PERSON who does these things?
And it gets weirder- she DOES keep doing these things since she's heard its good for Business School (disclaimer- I do not know if its good for B-school)
More generally, she keeps doing things she doesn't care about to advance. So her outward self really is doing good deeds and such, but her heart is not in it. So if her college or B-school or Job hired her because she DOES these things, that is NOT a lie. If they hired her because they want this KIND OF PERSON then... its a lie but I'm not sure what to make of that.
9) Is there ANY reason to have legacy matter for admissions? This seems like the dumbest and most easily fixed aspect of the whole process. I have never heard a good argument for it. Ever.
10) College Sports--- that's an entire blog post or book all by itself, so I won't go there.
11) One can argue whether helping the homeless, or being on the rowing team, or teaching the homeless how to row, should matter for college anyway. But lets assume that its legit to want people at your college who have lead interesting lives. So charity work or sports might be a MEASURE of that. But beware Goodhart's law:
When a measure becomes a target is ceases to be a measure.
The recent scandal is Goodhart's law on steroids.
(NOTE- I had in earlier version `Goodwin's law' but a commenter corrected me and reminded me that Goodwin's law is that if an internet discussion goes on long enough someone will be compared to Hitler. I wonder if that also applies in discussions by Nazi's.)
12) Does getting into an Elite College really increase your income or happiness (these are two very different questions) over the course of your life? I do not know-- if you do, please comment.
13) (ADDED LATER) A commenter pointed out that one could also go to a school outside of the USA that values proficiency in the chosen discipline over the items above. Excellent question that raises a few more:
Do schools outside of the USA hold Americans (or for that matter any non-citizen of their country) to a higher standard for admissions? I ask non-rhetorically.
If you get a degree from outside of the USA will that help or hurt your job prospects in the USA? Of course this depends on the school you goto, but even with that I do not know the answer.
1) Your kid likes (1) helping the homeless and (2) Ramsey Theory and (3) helping the homeless learn Ramsey Theory. Or perhaps they like rowing or Latin or fencing or Pig Latin or.... Great! encourage them, get them books and tutors, and whatever they need. You have an eye towards how this will look on for college admissions; however, it is your kids choice as to what they like. Also note that these activities are in addition to doing well in school, SATs, etc, not instead of it. Perfectly ethical, though I note that this avenue is not open to poor families and in some cases even middle class families.
2) Item 1 is the extreme on a spectrum in terms of how much are the extra things the kid does there idea OR planned by you to GET INTO A GOOD COLLEGE. This item will be the other extreme, but realize there is a continuum (actually I doubt there are a continuum number of options here, but there are many in between. Maybe its omega + omega*.) You have heard that being on the chess boxing teams is good on a college application so you TELL YOUR KID that they likes both Chess AND Boxing and should be on the team. You also hear this about being on the fencing team and knowing pig latin, so your kid can taunt their opponents like this:
youah, aint-kay ence-fay orth-way eans-bay
This might not be as bad as it sounds if the kid learns to like Chess-Boxing. But it may be worse than it sounds if the kid rebels against all of this and becomes a crack whore.
Is this ethical? It may be bad for the kid, but it may push him into things he ends up liking. One drawback: you've HEARD that being on the chess boxing team is good for college admissions, but is it true?
And again, not open to some families.
3) Item 2 (or even 1) but with an addition: Hire a college adviser to help you. Someone who knows (or claims to know) what colleges look for- Trombone, Latin, Chess-boxing, whatever. Still ethical but I again worry about the kids future rehab bills.
4) Here is where it gets murky. The college adviser helps:
a) Polish the kids essay (my parents, who are both in English, helped polish my essay to get into graduate school (I don't recall if there was one for ugrad). They told me to use lots of `ing' words so it sounds like I am actively doing things. They also helped me figure out when recursion-theoretic is hyphenated. I got into Harvard but not MIT, so make of that what you will.) Polishing, proofreading, that could be okay. But it can slip into b or c below.
b) The adviser (or the parents) talk to the kids to find out what to write, but then writes it. Maybe the kid proofreads and polishes. Maybe not even. The adviser is a ghostwriter. Clearly unethical but the line between helping-to-polish and adviser-wrote-it is again a continuum.
c) Adviser writes it and it is completely fictional. I once heard a rumor that a particular sample of an essay to get into med school was used by several med school applicant. Gee,they can't all have gotten inspired by watching their grandfather in his pajamas die of cancer. This is awful of course, but I wonder- what if the student writes a fictional essay all by themselves! Some combination of how much the essay is true and how much help you got on it is unethical. But some might be okay. Is it bad to polish stories that are basically true to make them flow more easily? Prob not. But there is a 2-dim continuum based on both accuracy and how much help the student got.
As a side note- how much does the personal statement matter for admissions? I suspect that if an elite school gets LOTS of REALLY QUALIFIED applicants, the essay may be all that distinguishes them.So it can be important. I also wonder if people on admissions can tell if an essay is not written by the applicant. Or maybe if there is an interview that can help detect it.
5) Parents give X amount of money to the college and the kid gets in. This is talked about a lot though I don't know how common it is for someone NOT qualified to GET IN based on money. College admissions has many factors so its not quite so clear cut what NOT qualified means. Even so, if seems odd that this is not in any way shape or form illegal. It IS transparent, so I guess thats a plus. The argument I've heard is that the money is used for scholarships to fund students who get in but can't afford to go. I do not know if this is true. And this one is only available to the top Z %, not sure what Z is, but there are people who can do 1,2,3,4 who can't do 5. I would call this unethical though colleges don't seem to think so. Or they do but they do it anyway.
6) Before I list the current scandal I want to list another issue: having a psychologist (or whoever it is who judges these things) say your kid is Learning Disabled so they get more time on the SATs. Perhaps bribing them, or perhaps its understood what you want. Again, I do not know how common this is. An alternative if you can't afford some of the above options, or done in conjunction with a lot of the above options.
7) The current scandal. Obviously unethical. A few things I wonder about:
a) One story was that they bribed someone to say their daughter was Learning Disabled and had to take the SAT (or whatever it was) in a separate room, making it easier to change the answers to the correct ones. So twice unethical.
b) Some of the students were clearly NOT qualified.
c) A parent does unethical things to get the kid into college.
The kid later lies to the parents about their grades or their plans
The parents are SHOCKED that their kids lie and wonder where the learned such behaviour!
8) Actually Item 2 --Parent has kid do things to plan to get into college-- is interesting for another reason. Are you your resume?
Imagine that Alice helps the homeless her Sophmore year in High School NOT because she cares about the homeless but because its good for college admissions.
Alice goes on to do other things that look good for college, NOT because she likes them or cares, but just to get into a good college.
She gets into an elite college
Did they take her in the hope she would KEEP doing these things or because she is the KIND OF PERSON who does these things?
And it gets weirder- she DOES keep doing these things since she's heard its good for Business School (disclaimer- I do not know if its good for B-school)
More generally, she keeps doing things she doesn't care about to advance. So her outward self really is doing good deeds and such, but her heart is not in it. So if her college or B-school or Job hired her because she DOES these things, that is NOT a lie. If they hired her because they want this KIND OF PERSON then... its a lie but I'm not sure what to make of that.
9) Is there ANY reason to have legacy matter for admissions? This seems like the dumbest and most easily fixed aspect of the whole process. I have never heard a good argument for it. Ever.
10) College Sports--- that's an entire blog post or book all by itself, so I won't go there.
11) One can argue whether helping the homeless, or being on the rowing team, or teaching the homeless how to row, should matter for college anyway. But lets assume that its legit to want people at your college who have lead interesting lives. So charity work or sports might be a MEASURE of that. But beware Goodhart's law:
When a measure becomes a target is ceases to be a measure.
The recent scandal is Goodhart's law on steroids.
(NOTE- I had in earlier version `Goodwin's law' but a commenter corrected me and reminded me that Goodwin's law is that if an internet discussion goes on long enough someone will be compared to Hitler. I wonder if that also applies in discussions by Nazi's.)
12) Does getting into an Elite College really increase your income or happiness (these are two very different questions) over the course of your life? I do not know-- if you do, please comment.
13) (ADDED LATER) A commenter pointed out that one could also go to a school outside of the USA that values proficiency in the chosen discipline over the items above. Excellent question that raises a few more:
Do schools outside of the USA hold Americans (or for that matter any non-citizen of their country) to a higher standard for admissions? I ask non-rhetorically.
If you get a degree from outside of the USA will that help or hurt your job prospects in the USA? Of course this depends on the school you goto, but even with that I do not know the answer.
Thursday, March 21, 2019
Back at Dagstuhl
 |
| Participants and Their Research Interests |
This week I'm in Germany for the Dagstuhl workshop on Computational Complexity of Discrete Problems well timed for Georgia Tech spring break. No Bill, so no typecast, no family, just a bunch of fellow theorists. New this year, beer.
Dagstuhl always had bottled beer (and wine), after all this is Germany. However, Ronen Shaltiel is living his lifelong dream of bringing a keg to Dagstuhl. Turns out Dagstuhl had a refrigerator/tap for a beer keg, one only needs to order and pay for the beer. Ronen's daughter designed a special logo, though the keg contains Bitburger, a fine German pilsner. Prost to Ronen and thanks for the beer.
Of course the fun is hanging with colleagues old and new. Talking about open problems old and new. Used to solve more of them back in the day, now it seems harder.
I did learn a new old theorem, planarity testing, whether you can embed a given graph in the plane so no two edges cross, is computable in log space. In 2000 Eric Allender and Meena Mahajan showed that you can test for planarity in symmetric log space, basically the complexity class whose complete problem is undirected connectivity. In 2005, Omer Reingold famously showed that undirected connectivity is computable in log-space. Thus planarity testing is in log-space, a result you might have missed if you didn't know both papers.
This came out in Eric's talk on his work with Archit Chauhan, Samir Datta and Anish Mukherjee showing that checked whether there is a directed path from a given node s to a given node t in planar graphs can be computed by concurrent-read exclusive-write on parallel random-access machines in logarithmic time, and thus likely weaker than directed s-t paths on general graphs.
Sunday, March 17, 2019
Third Poll on P vs NP and related Questions is out now! And the winner is Harambe!
I took a poll of the theory community (and others) about P vs NP and related issues in 2002, 2012, and 2019 (sorry its not an arithmetic sequence --- read the third poll article to see why). The 2002 and 2012 polls are on the web (see here and here ), and now the third poll is out and its here or here . All three appear in Lane's Complexity Column in SIGACT News.
I'll give some highlights and thought about the third poll, but for more read the article. Or read all three and see how opinions have changed over time.
0) How many respondents: 100 the first time, and 152 the second time, 124 the third time.
1) P≠NP is at about 80%. When restricted to people who have thought about the problem a lot (my judgement) then it goes to 99%. The strongest P≠NP is by Lance who says
People who think P=NP are like people who believe Elvis is alive.
I disagree. I think Elvis is alive and is trying to prove P=NP.
2) When will it be solved? 66% thought it would be solved before 2100, a bit more than in prior years. But also more thought it would never be solved. Some commented that a computer might solve it. I doubt that, but I do think this poll will be done by a computer in 10 years.
3) Because I used Survey monkey (1) each question got more answers, and (2) most questions got a forced YES or NO so less funny comments or room for creative answers. People could still leave comments, and some did.
4) Related to point (3): The last time I did the poll P=BPP was thought by everyone who answered the question. This year far more people answered it and quite a few thought P≠BPP. This may be because Survey monkey had a psychological affect of making people vote even if they didn't really know (people who have thought a lot about P vs NP all thought P=BPP). Has there been evidence that P≠BPP that I am unaware of? Or since there has not been that much progress on it maybe they are unequal. 10 years ago I would have thought we would have L=RL by now.
5) The last time I did the poll 10 people thought factoring IS in P and the NSA or the KGB knows that. This time around nobody thought that. Why? Those 10 people have vanished mysteriously.
6) Five people thought P vs NP will be resolved by Harambe. That is more than any person got.
7) Is this poll worth doing? I would say yes (gee, I have to having done his poll three times) because it is good to have an objective record of subjective opinion.
8) I'll give some of my answers: P≠NP, Elvis is dead, and both will be proven in the year 2525. For more about the future see this.
Thursday, March 14, 2019
Captain Einstein
If the president of the United States uses "complexity" in a tweet, I can't leave it alone.
I also agree with the Donald that a complex environment can often lead to confusion, especially in an emergency. HCI matters.
But that's where it stops. Better technology in the plane and on the ground, combined with better procedures have all but eliminated deadly major commercial airplane crashes in the US and quite rare in the world at large. I'll call that more than a "little gain". I remember the "old and simpler" days where we had deadly airline crashes every few months. No thanks.
I've had great discussions with some of the aerospace engineering professors at Georgia Tech. The amount of effort to get the "front-of-the-plane" software right via formal methods and rigorous testings has become harder to engineer than the plane itself. The "back-of-the-plane" software that controls the video screens and WiFi gets less attention. I can live with safety over movies.
When technology makes things much safer, whether it be planes or self-driving cars, the rare accidents become far more noticeable. while we should use these opportunities to learn and make things safer, the rare accidents help us realize just how much more safer technology can make us.
Airplanes are becoming far too complex to fly. Pilots are no longer needed, but rather computer scientists from MIT. I see it all the time in many products. Always seeking to go one unnecessary step further, when often old and simpler is far better. Split second decisions are....— Donald J. Trump (@realDonaldTrump) March 12, 2019
I got my MIT degree in Applied Mathematics so I don't count, but I know plenty of MIT computer scientists and the one undeniable truth is that I don't want any of them flying my plane.....needed, and the complexity creates danger. All of this for great cost yet very little gain. I don’t know about you, but I don’t want Albert Einstein to be my pilot. I want great flying professionals that are allowed to easily and quickly take control of a plane!— Donald J. Trump (@realDonaldTrump) March 12, 2019
I also agree with the Donald that a complex environment can often lead to confusion, especially in an emergency. HCI matters.
But that's where it stops. Better technology in the plane and on the ground, combined with better procedures have all but eliminated deadly major commercial airplane crashes in the US and quite rare in the world at large. I'll call that more than a "little gain". I remember the "old and simpler" days where we had deadly airline crashes every few months. No thanks.
I've had great discussions with some of the aerospace engineering professors at Georgia Tech. The amount of effort to get the "front-of-the-plane" software right via formal methods and rigorous testings has become harder to engineer than the plane itself. The "back-of-the-plane" software that controls the video screens and WiFi gets less attention. I can live with safety over movies.
When technology makes things much safer, whether it be planes or self-driving cars, the rare accidents become far more noticeable. while we should use these opportunities to learn and make things safer, the rare accidents help us realize just how much more safer technology can make us.
Sunday, March 10, 2019
Richard Karp: His influence and how to honor him
When I first saw the definition of NP-Complete I first thought if there are NP-complete problems I suspect they are contrived. When I saw the proof that SAT is NP-complete I thought okay, so there is one natural problem NP-complete by a trick of encoding, but I suspect there aren't any more. I then read Karp's article that had 22 NP-complete problems. I thought okay, there are 22, but I suspect there are no more. No I didn't think that. But the point is that Cook and Karp together birthed modern complexity theory by introduction NP-completeness and showing that there are MANY natural problems that are NP-complete.
How many people has Richard Karp inspired? I don't know but I suspect it's a cardinal between countable and the reals so it may depend on your model of set theory. Later in this post I will present Samir Khuller's story of how Richard Karp inspired him.
How to honor him? The Simons inst. is currently welcoming contributions to the Richard M Karp Fund, which honors the scientific contributions of Founding Director Dick Karp. The fund will provide vital support for the mission and activities of the Simons institute. They have a deadline of Pi-Day. You can go here to contribute and/or here for a letter from Shafi Goldwasser, Prabhakar Raghavan, and Umesh Vazirani about the fund.
OKAY, Samir's story:
How many people has Richard Karp inspired? I don't know but I suspect it's a cardinal between countable and the reals so it may depend on your model of set theory. Later in this post I will present Samir Khuller's story of how Richard Karp inspired him.
How to honor him? The Simons inst. is currently welcoming contributions to the Richard M Karp Fund, which honors the scientific contributions of Founding Director Dick Karp. The fund will provide vital support for the mission and activities of the Simons institute. They have a deadline of Pi-Day. You can go here to contribute and/or here for a letter from Shafi Goldwasser, Prabhakar Raghavan, and Umesh Vazirani about the fund.
OKAY, Samir's story:
Mentoring and Influence via a Train Journey...
Almost to the day, 33 years ago I was running from my dorm room to catch an unreliable bus to the train station as I headed home for a mid-semester break. On the way,I stopped by the mailroom, and not knowing where to put the CACM that had just arrived, stuffed it into the side of my bag and in the process, dropping my notebook in the process. On the train, when I looked for my notebook to go over some material I realized it was missing! All I had was a CACM and a very long train ride. Skimming through the journal, I found Karp’s Turing Award article “Combinatorics, Complexity, and Randomness“. I started reading this article, and was riveted! It was one of those life changing moments for me, when my immediate future became clear - I wanted to study Theoretical Computer Science and specifically the complexity of combinatorial problems. Without ever having met Dick Karp, he changed my life forever.
I met Dick long after he influenced me via his Turing Award Lecture and it was a real privilege to have had the opportunity to interview him via a fireside chat at Simons. See here.
I am delighted about the fund Bill mentions above as the money that goes to it will help inspire others as Karp inspired me.
Friday, March 08, 2019
QED
Via Scott, John Horgan wrote a blog post following on his 1993 Scientific American article The Death of Proof. The article talked about computer-generated proofs, experimental mathematics and even mentioned some of our work on transparent proofs (basically time-efficient probabilistically checkable proofs). I got (sarcastically I think) accused of killing mathematics after that article but of course we had traditional proofs about probabilistically check proofs. Andrew Wiles got a sidebar titled "A Splendid Anachronism?" for merely solving the most famous open problem in all of mathematics. Somehow traditional mathematical proofs survived the article.
Meanwhile in CACM, Moshe Vardi writes Lost in Math?
TCS is surely blessed with mathematical beauty. As a graduate student a long time ago, it was mathematical beauty that attracted me to TCS, and continued to lead my research for many years. I find computational complexity theory (or complexity theory, for short), with its theorems (for example, the time-hierarchy and space-hierarchy theorems) and its open questions (for example, P vs NP), to be hauntingly beautiful. Beauty, yes; but what about truth?Here here on the beauty of complexity. So what about truth? I cover much of this in my P v NP survey. In short the P v NP captures the most critical aspects of what is efficiently computable but it is not the endpoint. Nevertheless P v NP captures both truth and beauty and that's why the problem has so captivated and guided me throughout my research career.
Monday, March 04, 2019
How are there 20 copies of my new book through other booksellers and why are they so highly priced
(This is about my book PROBLEMS WITH A POINT: exploring Math and computer science
by Gasarch and Kruskal, here. This post is NOT a plug.)
I often see weird pricing on amazon but don't have enough inside information to explain it. This time I have seen some weird pricing on amazon but DO have enough information to be truly baffled.
My book is on amazon. You can buy it
Softcover new: $38.00
Hardcover new: $68.00
Okay, that all makes sense.
OR you can buy it from some other source.
There are 10 copies of the softcover, they claim new, ranging from $38.00 to $57.00.
There are 2 copies of the softcover, they claim used (how used could it be?), $49 and $53
There are 9 copies of the hardcover, they claim new, ranging from $68.00 to $91.00
There are 2 copies of the hardcover, they claim used, $84.00 and $88.00
This raises two questions
1) I have a copy, Clyde got a copy, and two copy's were send to reviewers. Neither Clyde nor I have sold our copy. So how did 20 or so copies get out there?
2) Why do they cost MORE than list price.
I've asked around and got some answers, but I INVITE you to give more possible answers. If your answer is not just speculation (e.g., `I used to be in the black market book business and here's the real scoop') then that would be great; however, I will take specuation
POSSIBLE EXPLANATIONS FOR HOW SOMEONE GOT COPIES
1) Clyde and I haven't gotten our free copies yet. Hijacked? Here's hoping the hijackers read it and enjoyed it before posting it. However, this theory is unlikely since I inquired with World Scientific (our publisher) and found out we'll be getting our free copies in 2 weeks. So this theory will either be confirmed or denied in 2 weeks. I would bet against OR those are some really bad hijackers.
2) Complete ripoff. You pay the money and get NOTHING. I think this was more likely in the early days of amazon (or e-bay or....) but now things are pretty well monitored. I have sold books on amazon and am very impressed with how foolproof it is. Still --- could be.
3) Someone hacked my computer. Really? Where did they get the orange covers? Why would they bother? This is not Harry Potter. Plus I asked our staff and NO this could not happen.
4) Some of the shipment dates are fairly far in the future, so they plan to get an order, then
order the book, and have it send to the buyer, at a profit. Seems like too much sugar for a satoshi, BUT could be profitable if its a bot and its automated to do this with lots of books.
5) My publisher also send the book to other sellers. That does not explain the higher prices.
WHY THE HIGHER PRICES?
1) They are counting on people assuming that other sellers are cheaper without checking. Would people do that? People still fall for the Nigerian billionaire scam, so maybe yes.
2) The seller themselves, which is possibly an army of bots, didn't bother checking but priced it
similar to other books. This is plausible since $38.00 is fairly cheap, so looking at similar books may get you a higher price.
-----
Any speculation is welcome so long as it does not involve space aliens or astrology.
by Gasarch and Kruskal, here. This post is NOT a plug.)
I often see weird pricing on amazon but don't have enough inside information to explain it. This time I have seen some weird pricing on amazon but DO have enough information to be truly baffled.
My book is on amazon. You can buy it
Softcover new: $38.00
Hardcover new: $68.00
Okay, that all makes sense.
OR you can buy it from some other source.
There are 10 copies of the softcover, they claim new, ranging from $38.00 to $57.00.
There are 2 copies of the softcover, they claim used (how used could it be?), $49 and $53
There are 9 copies of the hardcover, they claim new, ranging from $68.00 to $91.00
There are 2 copies of the hardcover, they claim used, $84.00 and $88.00
This raises two questions
1) I have a copy, Clyde got a copy, and two copy's were send to reviewers. Neither Clyde nor I have sold our copy. So how did 20 or so copies get out there?
2) Why do they cost MORE than list price.
I've asked around and got some answers, but I INVITE you to give more possible answers. If your answer is not just speculation (e.g., `I used to be in the black market book business and here's the real scoop') then that would be great; however, I will take specuation
POSSIBLE EXPLANATIONS FOR HOW SOMEONE GOT COPIES
1) Clyde and I haven't gotten our free copies yet. Hijacked? Here's hoping the hijackers read it and enjoyed it before posting it. However, this theory is unlikely since I inquired with World Scientific (our publisher) and found out we'll be getting our free copies in 2 weeks. So this theory will either be confirmed or denied in 2 weeks. I would bet against OR those are some really bad hijackers.
2) Complete ripoff. You pay the money and get NOTHING. I think this was more likely in the early days of amazon (or e-bay or....) but now things are pretty well monitored. I have sold books on amazon and am very impressed with how foolproof it is. Still --- could be.
3) Someone hacked my computer. Really? Where did they get the orange covers? Why would they bother? This is not Harry Potter. Plus I asked our staff and NO this could not happen.
4) Some of the shipment dates are fairly far in the future, so they plan to get an order, then
order the book, and have it send to the buyer, at a profit. Seems like too much sugar for a satoshi, BUT could be profitable if its a bot and its automated to do this with lots of books.
5) My publisher also send the book to other sellers. That does not explain the higher prices.
WHY THE HIGHER PRICES?
1) They are counting on people assuming that other sellers are cheaper without checking. Would people do that? People still fall for the Nigerian billionaire scam, so maybe yes.
2) The seller themselves, which is possibly an army of bots, didn't bother checking but priced it
similar to other books. This is plausible since $38.00 is fairly cheap, so looking at similar books may get you a higher price.
-----
Any speculation is welcome so long as it does not involve space aliens or astrology.
Thursday, February 28, 2019
Flying Pigs Unsafe at Any Speed
Take a moment and imagine a flying pig. Do you see a pig with tiny wings lazily guiding along. But pigs are not particularly slow animals as anyone has seen a pig race at a county fair can attest to that. So why not in the air shouldn't a pig go faster than an unladen African swallow?
I have a point discussing the fastness of a flying pig. Consider my favorite flying pig, P = NP. I would put more probability on an actual flying pig (thanks CRISPR) than polynomial-time algorithms for traveling salesman.
Sometimes I get in the following conversation:
AI person: The P v NP problem is irrelevant as we have machine learning and SAT solvers and we can for all practical purposes solve NP-complete problems.
Me: If P = NP we can solve AI!
AI: If P = NP it's likely to have a very slow poly-time algorithm with high exponents and constants and be for all practical purposes useless.
Let's break that down. The claim is E(running time of SAT|running time of SAT is poly) = cnk for some large c and k. First of all you are conditioning on a probability zero event. Beyond that nearly all the known problems we know that sit in polynomial time have practical efficient solutions. So under the assumption that SAT is one of these problems, why shouldn't the same rule apply. You've already made the assumption we can reduce its running time from exponential to polynomial, so why would it stop at some high polynomial?
If we are going to talk about the hypothetical flying pig P = NP world, let my algorithms run fast.
I have a point discussing the fastness of a flying pig. Consider my favorite flying pig, P = NP. I would put more probability on an actual flying pig (thanks CRISPR) than polynomial-time algorithms for traveling salesman.
Sometimes I get in the following conversation:
AI person: The P v NP problem is irrelevant as we have machine learning and SAT solvers and we can for all practical purposes solve NP-complete problems.
Me: If P = NP we can solve AI!
AI: If P = NP it's likely to have a very slow poly-time algorithm with high exponents and constants and be for all practical purposes useless.
Let's break that down. The claim is E(running time of SAT|running time of SAT is poly) = cnk for some large c and k. First of all you are conditioning on a probability zero event. Beyond that nearly all the known problems we know that sit in polynomial time have practical efficient solutions. So under the assumption that SAT is one of these problems, why shouldn't the same rule apply. You've already made the assumption we can reduce its running time from exponential to polynomial, so why would it stop at some high polynomial?
If we are going to talk about the hypothetical flying pig P = NP world, let my algorithms run fast.
Tuesday, February 26, 2019
Problems with a Point: Exploring Math and Computer Science
As you can see from Lance's tweet
Problems with a Point: Exploring Math and Computer Science
by Gasarch and Kruskal
(ADDED LATER- the World scientific website has more info than amazon and has a table of contents, so here it is: here)
is now available! The tweet says its $68.00 but that's hardcover- paperback is $38.00 (are covers that expensive) and if you like your books already broken in, there are some used copies for $89.00. Makes sense to me (no it doesn't!). Could be the topic of a blog post (probably already was).
Okay, so whats in the book? One of my favorite types of blog posts is when I make a point ABOUT math and then do some math to underscore that point. I went through all of my blogs (all? No, I doubt I did that) and picked out blogs of that type. With Clyde's help we EXPANDED and POLISHED and GOT THE MATH RIGHT (in some cases I didn't have any math so we had to supply it).
When I first got a copy (about a month ago) I just couldn't stop reading it. I really like it! This is a non-trivial remark -- often authors get tired of their book, or after a while and wonder things like ``why did I write 300 page on the muffin problem? What was I thinking?'' So the fact that I am very pleased with it is not obvious. Does it mean you will?
If you ever thought `I wish bill would clean up his posts spelling and grammar AND expand on the math AND make it a more cohesive whole' then buy the book!
I will in future posts describe more about writing the book, but this is probably my last post where I plug the book.
bill g.
Thursday, February 21, 2019
Extra! Extra! Read all about it!
Last weekend I saw the documentary Joseph Pulitzer: Voice of the People. Pulitzer, as you probably know from the prize named after him, was a major newspaper publisher in the late 19th and early 20th century. He ran two papers, the St. Louis Post-Dispatch and The New York World. The World at one point took on massive proportions, including sheet music of the latest tunes and dress patterns of new fashion that one could make at home. The World was the Internet of the turn of the 20th century.
The movie mentioned the many editions of the paper during the day, including the extra edition. An extra edition came out because of some major breaking news story. Back then newspapers would drum up minor stories to sell extra editions but they tended to disappear over time.
Which brings me to Monday, August 19, 1991. Hard-line members of the communist party of the USSR attempted a coup to take over the government from Mikhail Gorbachev. To us in the US, this seemed like the cold war which appeared to be coming to an end might rekindle. At the time I lived in Chicago and on that Monday the Chicago Tribune ran an extra afternoon edition talking about the coup. The return to the cold war didn't happen. Within a couple of days the coup failed and if anything hastened the dissolution of the Soviet Republic.
That was probably the last of the extra editions. By the time of the next major historical event, ten years and twenty-three days later, we had the Internet and cell phones and one no longer needed a newspaper to tell you the world has changed.
The movie mentioned the many editions of the paper during the day, including the extra edition. An extra edition came out because of some major breaking news story. Back then newspapers would drum up minor stories to sell extra editions but they tended to disappear over time.
Which brings me to Monday, August 19, 1991. Hard-line members of the communist party of the USSR attempted a coup to take over the government from Mikhail Gorbachev. To us in the US, this seemed like the cold war which appeared to be coming to an end might rekindle. At the time I lived in Chicago and on that Monday the Chicago Tribune ran an extra afternoon edition talking about the coup. The return to the cold war didn't happen. Within a couple of days the coup failed and if anything hastened the dissolution of the Soviet Republic.
That was probably the last of the extra editions. By the time of the next major historical event, ten years and twenty-three days later, we had the Internet and cell phones and one no longer needed a newspaper to tell you the world has changed.
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