Sunday, May 20, 2018

COMPUTER PROOF vs computer proof- Quadratic VDW theorem

Quad VDW Theorem: For all c there exists W=W(c) such that for all c-colorings of {1,...,W} there exists a,d such that a and a+d2 are the same color.

What is known about W(c)?

The first proof of Quad VDW was nonconstructive.

The second proof was constructive but used VDW's theorem and gave terrible bounds, even for W(2).

EASY: Show W(2)=5

ON A HS MATH COMP: Show that for all 3-colorings of {1,...,2003} there exists two numbers a square apart that are the same color.

One can get a bound less than 100.

With a lot more detailed work one can get W(3)=29.

None of above was done with a computer.

SO- what about W(4)?  There are three proofs that W(4) exists:

a) Prove the QUAD VDW theorem. This gives terrible bounds.

b) I had some students use SAT solvers and they obtained W(4)=58. I am sure they are right and I am glad to know it (especially since it confirms the general mantra that the bounds from proofs in Ramsey theory tend to be much bigger than the reality) but its somehow unsatisfying. I want a HS proof. I wanted to put it on the MD Math Comp for 2017 but wiser people prevailed.

c) I gave the problem to a Grad Student Zach Price and he came up with a HS proof-- sort of.  Look at the following graph: here.  It shows that in any 4-coloring of N which does not have two numbers a square apart the same color:

COL(x) = COL(x+290085289)

from this one can obtain

W(4) ≤ 2900852892  (which is MUCH better than the bound from the proof of Quad VDW) and hence a proof of QVDW that does not need a program, except that to FIND this gadget used a program.  I doubt a HS student could do this on an exam.

Is Zach's proof the HS proof I am looking for?

PRO- once you see it its easy to verify and does not need a program.

CON- coming up with it needed a program.

More to the point: which proof do you like better:

1) The SAT Solver proof.  Not a proof you can see or feel, but gives exact bounds.

or

2) Zach's gadget proof. Much worse bound but you can see and feel the proof, and verify it after you know the gadget.

I prefer Zach's proof.


Thursday, May 17, 2018

The Complexity of the Firm

In 1937, a year after Turing had his seminal paper, Ronald Coase published a paper The Nature of the Firm to give a framework to why we have companies and how large they become. In a perfect market economy we shouldn't need a firm at all, everyone is just an independent contractor and market pricing will drive efficient use of labor. Coase notes there are costs to creating contracts and one can gain efficiencies by avoiding these contracts by having both parties inside the same organization.

Much of those costs come from complexity, it is impossible to create a contract to cover all possible outcomes and thus contracts are incomplete leading to loopholes and lawsuits.

In the other direction, having the central organization of a firm has its own costs, from inefficiencies from not using markets to balance supply and demand, to the complexity of the organization processes themselves. In Coase's model, a firm grows to a size that balances the organization and contract costs at the margin.

So what happens to the sizes of firms when we reduce complexity, as happens with modern computing, from better communication, optimization and data analytics. We see relatively new companies like Google and Amazon getting larger. We also see a number of startups and small companies with very few workers, sometimes just one.

Computing drops both the cost of central organization and the cost of contracts so the size of a firm depends on circumstance. One can have a tech startup with a small number of workers since they can write apps that run on other people's platforms (like web browsers and on phones) and have the processing done on the cloud where they can scale or not scale as needed. Meanwhile a large company can more easily coordinate and connect using modern technology making it cost efficient to expand.

As we enter a new age of machine learning and automation, how will that affect the very nature of companies in the future? How about universities, a special kind of firm in itself? How can we harness the tools and ideas from computational complexity to help understand these and other societal changes.

Monday, May 14, 2018

What does it mean for a student to guess an answer?

On my final in Aut Theory I wanted to ask a TRUE/FALSE/UNKNOWN TO SCIENCE
question but did not want them to guess. Hence I had +4 for a right answer, -3 for a wrong answer.
Here is the question:

------------------------------------------------------------------------------------------
For each of the following say if its TRUE, FALSE, or UNKNOWN TO SCIENCE. No Proof Required BUT you get +4 for every right answer and -3 for every wrong answer and 0 for an answer left blank. So

                DO NOT GUESS!!!!!!!!!!!!!!!!!!!!!!!!!!!

a) If A is regular and F is a finite set then A UNION F is regular.

b) If A is in P and F is a finite set then A UNION F is in P.

c) If A is in NP and F is a finite set then A UNION F is in NP.

d) If A is decidable and F is a finite set then A UNION F is decidable.

e) If A is undecidable and F is a finite set then A UNION F is undecidable
-------------------------------------------------------------------------------------------------

Note that they are all TRUE. A student who (as many did) answered T-T-T-T-F had the following conversation with me:

BILL: You guessed! I told you DO NOT GUESS!!!!!!!!!!!!

STUDENT: No. I REASONED that you would not make them all T, so by this reasoning the last one had to be F. I now see that my reasoning is wrong--- you would make them all T, but it was not guessing, it was reasoning.

                I claim the student was guessing, he claims he was not. What do you think?

Having said that, the following IS a rational strategy:

If I don't know the answer but it has nothing to with P vs NP then it has to be T or F. In this case guess since exp val is positive.  If the answer has to do with P vs NP then do not guess.

This also raises a question- if they honestly thought (say) e was F I want to just give them 0,  whereas if they are guessing or using reasoning about `Bill wouldn't ...' I want to give them -3. But alas, we cannot read their minds or souls.




Friday, May 11, 2018

Richard Feynman (1918-1988)


When I took cryptography from Manuel Blum, he handed out copies of the chapter "Safecracker Meets Safecracker" from Richard Feynman's book Surely You're Joking Mr. Feynman. Feynman, the Nobel Prize winning physicist who was born a hundred years ago today, wrote this book not about physics but just a series of stories from different times in his life. This chapter described how Feynman learned how to open locked safes in Los Alamos during the Manhattan Project.

We all have interesting stories to tell but Feynman finds a way to keep things compelling in a way most scientists could not--even if he sometimes comes off being a bit of a jerk, hence the title. This book inspired me to tell my own stories which occasionally show up in this blog.

His most important stories form The Feynman Lectures on Physics (free to read online), an amazing explanation of deep physical concepts.

Richard Feynman's biggest contribution to theoretical computer science comes from a 1981 keynote address.
What kind of computer are we going to use to simulate physics? The full description of quantum mechanics for a large system cannot be simulated with a normal computer. And therefore, the problem is, how can we simulate the quantum mechanics? Can you do it with a new kind of computer—a quantum computer?
which begat a proof-of-concept paper by David Deutsch and Richard Jozsa which begat Daniel Simon's exponential separation which begat Peter Shor's factoring algorithm which begat billions of research dollars and considerable expectations, real and imagined, for Feynman's vision.

Wednesday, May 09, 2018

Second of N posts on G4G13. Maybe

(Don't forget to vote for SIGACT posistions:here  9th workshop on Flexible network design, May 22-25 at College Park, here.)

My first poston G4G13 is arguably here. To see why its debatable, see that post.

FOXTROT HALF-EMPTY, HALF-FULL PROBLEM, INCLUDING 13 by Thomas Francic Banchoff

In a Foxtrot cartoon (see here) Foxtrot has a glass which looks like it is half-full (or half-empty)
and asks people if its half-full or half empty. But the jokes on them!
The class is slightly angled so its actually 5/12 full (or 7/12 empty)
Given the area of the top and bottom What is the angle?
Generalize to other dimensions.

--------------------------------------
HEX PRIMES by Spandan Bandyopadhyay

Here is an alternative definition of primes that
lends itself to a generalization.

A number x is PRIME if when there is a rectangle with
integer sides and area x, one of the sides is 1.

Lets generalize this!

A number x is TRIPRIME if when there is a triangle with
integer sides of area x, one of the sides is 1.

Rather than use these prefixes we will go with

A number x is n-PRIME if when there is a convex n-gon with
integer sides and area x, one of the sides is 1.

HEXPRIMES are of course 6-primes.

The problem with 5-minute talks (maybe it should have been 6 minutes)
is that the concept is intersting but I didn't get to hear much
about them. And I could not find a paper on line. Note that this
conference has many non-academics for whome PAPERS are not the
basic currency so things are more informal. This is GOOD in that
its more of a free-for-all, but bad for follow up.
ALL of G4G12's are on You-Tube, so when that happens for G4G13,
I can follow up on this.

One thing I did manage to write down- 7 is the first HEX-composite.

Thursday, May 03, 2018

Broader Impacts Redefined

The ACM Future of Computing Academy suggests that "peer reviewers should require that papers and proposals rigorously consider all reasonable broader impacts, both positive and negative." Here is the broader impacts section of a future imagined grant proposal.

My latest cryptocurrency paper will allow people to sell all sorts of paraphernalia, illegal, immoral and fattening, while avoiding paying taxes. Dr. Evil and his minions can move money around the world anonymously to more easily implement their dastardly plans. With this new work bitcoin will increase in value, causing far more mining setups that will deplete the energy resources of the world.

Since I already own bitcoin, I will get filthy rich, increasing the gap between the one-percenters and the middle-class. I'll buy a fancy car and reprogram the auto-pilot to run over people who get in my way. And all those pesky bicyclists clogging the roads. That would be a positive impact.

For the rest of humanity my new quantum gravitation algorithm will put all those people out of work. But their misery will be short lived because if anyone tries to run the algorithm, it may cause a small black hole that devours the earth.

My proposed generic oracle separation would seem to have no impact whatsoever. But what happens if an alien race threatens to destroy the planet unless we find such a separation. I will have saved the earth, a very positive broader impact. Of course a different alien race could visit the earth and deem it mostly harmless until they discover my oracle, realized we have advanced too far and then blow us up before we become a threat.

In other broader impacts, I will train graduate student to be just like me. Whether this is a positive or negative impact I leave to the reviewers.

Monday, April 30, 2018

Gathering for Gardner 13 - the first or third of many posts

I attended G4G13 (Gathering for Gardner- meeting 13).  Martin Gardner was the Scientific American Mathematical Recreations columnist from 1956 until 1981.  He had a great influence on many math-people of my generation.

Most of the talks were 5 minutes so they could tell you a problem or thought of interest but not much more. This is GOOD in that I UNDERSTOOD most of the talks!  There were also some talks on Magic and on science literacy (or perhaps science illiteracy) which were also interests of Martin Gardner.

I posted on one problem and its solution so this is either my first or third post on G4G13.

Withouth further ado, here are my descriptions of some of the talks.  OR you could just go here


My descriptions are long- I may have lots of posts on G4G13!

EXTENDING THE 10958 PROBLEM by Bill Ames.

(paper on arxiv here)

In 2014 Taneja proposed the following math problem:

Take the digits 1,2,3,4,5,6,7,8,9.

You can put any of +, -, *, /, and powers you want between the digits, or erase the comma to get (for example) 34. Use parenthesis freely.  Which natural numbers can you produce?

Here are some examples:

1 = 1 + (2 - 3) - (4 - 5) - (6 - 7) + (8 - 9)

30 = 1 + 2^3     -  4*5    +  6*7    + (8-9)

Which numbers can you get?

Taneja got all the numbers between 0 and 11,111 EXCEPT 10958.

(Is 10958 possible? The talk didn't say.)

Using other operations- square roots and factorials- you can get 10958. The talk was about adding more operations and getting all numbers between 0 and 111,111

THIRTEEN BOUNCES by Gary Antonick.

Since this was G4G13 there were several talks about the number 13. The hotel did not have a 13th floor, and no room was of the form X13 (I was in room 515 and sometimes overshot my room since I assumed it would go 511-513-515, not 511-515). Martin Gardner would have OBJECTED to the lack of a 13th floor and would have condemned triskaidekaphobia (fear of  the number 13 and the longest word I ever spelled correctly) as irrational. And he would be right. But I am preaching to the converted (also known as preaching to the choir).

The talk was about firing a cannoball at a perfect reflector.  If  the cannon is positioned just right the ball will go back into the cannon. Can you make it do this after taking 13 bounces?  What if you an only use a compass and straightedge for your calculations?

A GRAMMATICAL APPROACH TO THE CURLING NUMBER CONJECTURE by Duane Bailey.

(For more on the Curling Conjecture Google "Curling Number Conjecture"-- I was going to give a list of papers but there are just too many.  There is no wikipedia entry on it, though I did learn about the sport of Curling!)

Take any finite sequence of integers (we will just use natural numbers) for example

2 3 2 3

Write it as X Y Y Y ... Y with as many Y's as possible.

for example

(2 3)2

The number 2 is the Curling Number of the sequence. Append it to the sequence

2 3 2 3 2

Now take the Curling number of this sequence. It is 2 since the sequence is

2 (3 2)2

Append it to the sequence:

2 3 2 3 2 2

Now the Curlng number of this sequence is 1 and we stop.

The conjecture is that if you start with any sequence you will eventually get to 1.

The talk relates the conjecture to aperiodic grammars.


Thursday, April 26, 2018

The 8 digit number I asked for

(On June 29th, co-located with STOC, there will be a workshop to celebrate Vijay Vazarani's 60th birthday. See here. As computer scientists shouldn't we use 64 as the milestone?)



At Gathering for Gardner 13 Peter Winkler gave a great talk entitled

Problems that Solve Themselves.

(The title kind-of gives away how to solve it. And its just the kind of thing Peter Winkler would talk about. Hence I omitted the title and author when I first posted about it)

I blogged about one of them, asking for the answer. I repeat the problem and answer it:

Find an 8-digit number

d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0

such that

d_0 is the number of 0's in the number

d_1 is the number of 1's in the number

d_2 is the number of 2's in the number

.....

d_6 is the number of 6's in the number

but

d_7 is NOT necc the number of 7's

d_7 is the number of DISTINCT DIGITS in d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0

WRONG APPROACH: Work out algebra for the digits, try to force it Actually, I had thought this was the wrong approach but some of the comments on my last blog DID do this.

RIGHT APPROACH:  Take an aribtrary number like

1 1 1 1 1 1 1 1

NOW- look at it and count the number of 0's, 1's, ... 6'ls and the number of distinct numbers
For the new number let

d_0 be the number of 0's in the old number

....

d_6 be the number of 6's in the old number

d_7 be the number of distinct digits

To get

1 0 0 0 0 0 8 0

iterate the process to get

3 0 0 0 0 0 0 6

3 1 0 0 0 0 0 6

4 1 0 0 1 0 1 5

4 0 1 1 0 0 2 3

5 0 0 1 1 1 2 2

4 0 1 0 0 2 3 2

5 0 0 1 1 2 1 2

4 0 1 0 0 2 3 2

5 0 0 1 1 2 1 3

5 0 1 0 1 1 3 2

5 0 1 0 1 1 3 2

AH- this number works!

Peter Winkle claims that if you start with any 8 digits number this process will converge to a solution within at most 15 steps. I do not think he proved this. But the key here is that by NOT being clever you can easily find the answer. The problem, as the title of the talk says, solves itself!

How many such numbers are there? Proving the process works? Getting statistics on how long it will take on average? I leave all of this to the reader. (I do not know the answers.) And one can ask all of this for other bases and other condidtions on the digits (though calling them digits is odd since that smacks of base 10- what to use instead of ``digits'' when in another base?)








Monday, April 23, 2018

Find an 8-digits number such that ....

(June 29th, co-located with STOC, will be a workshop to celebrate  Vijay Vazarani's 60th birthday. As computer scientists shouldn't we use 64 as the milestone? See here for details about the workshop, not about 60 vs 64.)

I will likely post a lot about Gathering For Gardner 13, but for now I will just give one problem I saw there. I will not say the speaker or the title of the talk as that might be a clue. I'll give the answer in my next post which will be this week


Find an 8-digit number

d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0

such that

d_0 is the number of 0's in the number

d_1 is the number of 1's in the number

d_2 is the number of 2's in the number

.....

d_6 is the number of 6's in the number

but

d_7 is NOT necc the number of 7's

d_7 is the number of DISTINCT DIGITS in d_7 d_6 d_5 d_4 d_3 d_2 d_1 d_0

You could prob find such a number by computer search- but don't.

Feel free to comment. I will not block comments (except those we usually block- usually spam)  but by the same token, if you don't want hints, don't read the comments.


Thursday, April 19, 2018

Memory is Hot

A good number of the faculty candidates interviewing at Georgia Tech have a common theme: Memory. Memory connected to databases, to programming languages, to architecture, to operating systems, to networks and in security. Why all the interest in memory?

I started asking the candidates. The short answer: We no longer get faster performance from the CPU but new memory technologies can make a large difference.

Intel and Micron developed a new memory technology they call 3D XPoint ("3D Cross-Point") as diagrammed above, with the memory, in yellow, addressable by choosing the adjacent horizontal and vertical bars. 3D XPoint gives fully bit addressable high-density high-speed non-volatile memory without transistors or electrons. Non-volatile means the memory does not disappear when the power goes off, like a "flash" drive. Intel has announced a 3D XPoint main memory card under the Optane brand available this fall. One could use this memory as a full replacement or in conjunction with traditional DRAM in a heterogeneous system.

What's the big deal? The non-volatility means we can reduce power needed for memory, power being perhaps the biggest bottleneck in computing today. We can have large-scale databases in memory, fast performance with quick crash recovery since the memory isn't lost. 3D XPoint can enable edge or fog computing that brings the power of the cloud closer to the user for applications like virtual reality or self-driving cars where the time to reach a data center can cause unacceptable lag. Like most transformative technologies it will bring opportunities and challenges we can't even imagine now.

As theorists we need to take a leading role. How can we model 3D XPoint-like memories so we can properly develop algorithms and analyze complexity to understand what these memories can or cannot enable? Theoretical Computer Science can play a large role in adapting to new technologies if it is willing to get into the game.

Monday, April 16, 2018

Is DTIME(n) closed under concat? star? of course not but...

(STOC 2018 will offer child care for the first time. I was emailed the following and asked to
pass it on:

We are pleased to announce that we will provide pooled, subsidized
child care at STOC 2018. The cost will be $40 per day per child for
regular conference attendees, and $20 per day per child for students.
For more detailed information, including how to register for STOC 2018
childcare, see http://acm-stoc.org/stoc2018/childcare.html

Ilias Diakonikolas and David Kempe (local arrangements chairs)

I tell my class that poly time is nice mathematically since its closed under lots of operations including
concat and *.  That is:

L1 , L2  ∈ P  implies L1 L2 ∈ P.

unlike DTIME(n) which, as you can see, is NOT closed under concat! After all, the proof that P is closed under concat uses that if p(n) is a poly then np(n) is a poly which does not hold for linear time. If p(n) is O(n) then np(n) is NOT necc O(n).

But--- thats not a proof that DTIME(n) is not closed under concat! Thats just the observation that the argument for P being closed under concat does not extend to DTIME(n). Perhaps some other argument does.

I do not believe that. I believe there exists L1 , L2  ∈ DTIME(n) such that  L1 L2 is not in DTIME(n).

I have not been able to prove this. In fact, the question I pose is not well defined since I need to specify a machine model.

I pose the following question which may well be known - if so then please leave a comment:

Find a reasonable machine model (RAM? k-tape TM?) such that DTIME(n) on that model is NOT closed under concat.  (Prob use DTIME(O(n))).

Similar for *

These are likely hard questions since if L is in DTIME(n) then L* is in NTIME(n), (similar for concat),
so I would be separating DTIME(n) from NTIME(n), which HAS been done, but not with nice natural problems of the type that I seek.

Friday, April 13, 2018

Lance and Bill Gather for Gardner

Every two years in Atlanta, recreational mathematicians gather to honor Martin Gardner, whose Scientific American column Mathematical Games through the 60's and the 70's. Those columns inspired budding mathematicians of a certain age including Bill and I.

Bill came down to this years Gathering for Gardner 13. Talks are only six minutes long. Bill talked on the Muffin Problem right after an 8-year old and right before Stephan Wolfram.

We also did a short vidcast from the exhibition room.

Monday, April 09, 2018

Whan a deep theorem of your Uncles becomes standard should you be sad?

(An exposition of Nash-Williams's proof of  the Kruskal Tree Theorem is here)

Andrew Vazsonyi (the mathematician, see here, not the folklorist, see here for that folklorist's wife) conjectured that the set of trees, under the minor ordering, is a well quasi order. I do not know when he made the conjecture, but he was born in 1916 and the conjecture was solved in 1960, so you can take a guess based on that. We only know he conjectured it since when Joe Kruskal proved it, he credits Vazsonyi and calls it `Vazonyi's conjecture' I do not know if anyone else ever called it that.
After a conjecture is solved it's usually called by the solvers name, not the poser's name, so it's hard to find out anything about Vazsonyi's conjecture when it was a conjecture.

Joe Kruskal's proof was quite hard and quite deep (are they necc the same? Prob not). See here for that paper. Nash-Williams three years later, in 1963, provided a much easier, though still deep proof.  (I could not find a free online copy to point to- if you know of one please email me or comment.) Nash-Williams prove is in my writeup.

Joe Kruskal is Clyde Kruskal's uncle (Joe is also known in our circles for MST).  I told Clyde that I made his Uncle's Theorem a problem on my TAKE HOME midterm in graduate Ramsey Theory. He PONDERED- is it sad that this once great theorem is now merely a problem in a course?

I asked  some random students from both my Ramsey Theory class and my Aut theory class for their take on this. Here are the responses.

Dolapo: (Aut Student) Clyde should stop worrying about his Uncle's legacy and start building his own!

Ben:   (Ramsey Student) Bill proved in class that SUBSEQUENCE is a wqo. GIVEN that, the problem wasn't that hard. Had he not given it to us the problem would be impossible.

Clyde: Bill, next time you teach the course give them the problem cold- without any prior theorems about wqo.

Bill: I'd rather not

Nishant:(Ramsey Student) Clyde should be happy, and know it, and clap his hands! People are still talking about his Uncle's Theorem!

Bill: If you're happy and you know it clap your hands! Alice is not clapping here hands. Bob is. What can you deduce about Alice and Bob? Never mind, that will be a different post.

Bill:  If I gave the problem on an in-class exam in a grad course or a take-home exam in an ugrad course THEN you should be sad. But take-home exam in grad-course. That's just right.

Ben: (Ramsey Student) When are you going to do a post about this?

Bill: Since my post will include a pointer to a proof of the Kruskal Tree Theorem, I won't post until after you all hand in your take-home midterms.

Nishant and Ben together: Darn!

Joshua (TA for Aut Theory): I heard the grad students complaining about the problem being too hard is a  sign of a great mathematician. If your  grad students complain then  kudos to Joe Kruskal!

Bill: They Didn't comlain

Clyde: Darn!

Ajeet (Ramsey Student) It's much harder to CREATE new math then to LEARN new math. I feel our working out this problem, given what you already gave us, was more a LEARNING thing rather than a CREATE thing. Like P vs NP: easier to verify then generate.

Katherine (Ramsey Student and Clydes Algorithms TA): A bigger problem for Joe Kruskal's legacy is that people in the algorithms class refer to The Kruskal MST algorithm as Clyde's Algorithm.

Saddiq (Aut Theory Student) If Clyde can test on Kruskal MST on his final (which he did) then you can test on Krusakl Tree theorem on your midterm (which you did).

Anyway, one lesson here is that fame is fleeting. Very few people are remembered 100 years after their death. So Clyde - I am helping keep your Uncle's memory alive!

Clyde: Oh Joy

SO- what do you think? Should Clyde be happy that his Uncle's theorem is on my take him midterm? Should he know it? Should he clap his hands?



Thursday, April 05, 2018

Challenge about NFA for {a^y : y\ne 1000} answered.


Recall that in a prior post I asked

Is there an NFA for  { ay : y ≠ 1000 } with substantially less than 1000 states.

I will now show that any NFA for this set requires 999 states, so essntially 1000. The proof uses Ramsey Theory. I will tell you the little bit of Ramsey Theory that you need.

NO- the above is false.

There is an NFA with 60 states.  I have a complete exposition here and I have a paper (with coauthors) on cofinite unary languages and NFA's here). Generally if you want to only avoid
an the NFA an be done in sqrt(n) + O(\log n) states, but requires sqrt(n) states.

I sketch the ideas here for the 1000.

Convention- `reject 988' would mean rejecting a988.

It is easy show the following:
For all n \ge 992 there exists x,y\in N such that n = 32x +33y

For all x,y  32x+ 33y is NOT = 991.

If you have  an NFA with a 33-loop and a shortcut so you can also to 32 and back to the start
state, this NFA

accepts all y ≥ 992

rejects 991

we have no comment on anything else.

So if you prepend 9 states to that NFA you will have an NFA that

accepts all y ≥ 1001

rejects 1000

How to get all the numbers < 1000?

Use mod 3, mod 5, mod 7, mod 11 loops that only accept if the number is NOT equiv to 1000
mod 3, 5, 7, 11, Since 3*5*7*11 > 1000 we have

if y is rejected then y ≤ 1000, but y \equiv 1000 mod 3*5*7*11, so must have y=1000.

so 1000 is the only reject.

Number of states: 1 + 33 + 3 + 5 + 7 + 11 = 60 states.

I think you can do this in 58, but what is the BEST you can do?
My paper has lower bounds of sqrt(n) so in this case 32.

This is a GREAT theorem to teach to students in a course that covers regular langs and also P vs NP since the students THINK that an NFA cannot do better - AH BUT IT CAN! - so it gives a concrete example of

lower bounds are hard since someone may come along with a very clever trick you didn't think of

This semester when I had the class VOTE on whether or not there was a small NFA

48 thought that ANY NFA would require around 1000 states

One of the best students proved this! or perhaps ``proved'' this

Only 2 students knew that it could be done with MUCH LESS than 1000 states- and those two are exceptions- one is co-author on the paper (and he tells me that he originally thought it required 1000 when I first showed him the problem) and one is someone who often goes to my old course websites looking for more information and problems to work on (I like that ambition!) and came across material on this. 

I tell them `you thought this was the last lecture on reg langs. It was not. it was the first lecture on P vs NP'






Tuesday, April 03, 2018

Challenge: Is there a small NFA for { a^i : i\ne 1000} ?


(Added later- a reader left a comment pointing to a paper with the answer and saying that the problem is not original. My apologies-  upon rereading I can see why one would think I was claiming it was my problem. It is not. I had heard the result was folklore but now I have a source! So I thank the commenter and re-iterate that I am NOT claiming it is my problem.)


Consider the language

{L =  ai   : i ≠ 1000 }

There is a DFA for L of size 1002 and one can prove that there is no smaller DFA.

What about an NFA?  Either:

a) Show that any NFA for L requires roughly 1000 states

or

b) Show that there is a small NFA for L, say less than 500 states

or

c) State that you think the question is unknown to science.

I will reveal the answer in my next post, though its possible that (c) is the answer and the comments will convert it to either (a) or (b).

Feel free to leave comments with your answers. if you want to work on it without other information then do not read the comments.






Thursday, March 29, 2018

Almost Famous Quantum Polynomial Time

I have been playing with a new complexity class AFQP, defined in a yet-to-be-published manuscript by Alagna and Fleming. A language L is in AFQP if there is a polynomial-time quantum Turing machine Q such that for all inputs x,
  • If x is in L, then Q(x) accepts with high probability.
  • If x is not in L, then Q(x) rejects with high probability.
  • Q(x) only has O(log |x|) quantumly entangled bits as well as a polynomial amount of "deterministic" memory. 
AFQP is meant to capture the state of the art of current quantum technology.

AFQP has several nice properties, capturing the complexity of many open problems.
  1. If BQP is in AFQP then factoring is in BPP.
  2. If one can solve satisfiability in AFQP then the polynomial-time hierarchy collapses.
  3. AFQP is contained in the fifth level of the polynomial-time hierarchy.
  4. If AFQP is in log-space then matching has a deterministic NC algorithm.
  5. Graph isomorphism is in a quasi-polynomial time version of AFQP.
  6. AFQP = PPAD iff Nash Equilibrium has solutions we can find in polynomial time.
The proofs use a clever combination of Fourier analysis and semidefinite programming. 

Where does the name AFQP come from? The authors claim that they didn't name the class after themselves, and instead say it stands for "Almost Famous Quantum Polynomial Time" as it won't get the fame of BQP. More likely it is because it's April the first and I'm feeling a bit Foolish making up a new complexity class that is just P in disguise. 

A Reduced Turing Award

A Turing machine has an extremely simple instruction set: Move left, move right, read and write. If you want to do real programming, you need something a bit more powerful. So computer architects created chips with more and more operations. As an undergrad at Cornell in the early 80's I worked on coding an email system written in IBM 370 assembly language. As I wrote in a 2011 post,
IBM Assembly language was quite bloated with instructions. There was a command called "Edit and Mark" that went through a range data making modifications based on some other data. This was a single assembly language instruction. We used to joke that there was a single instruction to do your taxes.
After Cornell I spent one year at Berkeley for graduate school where I heard about this new idea: Let's move in the other direction and have a "reduced instruction set computer". Make the chip as simple as possible to simplify the design, save power and increase performance. Fast forward to last week and ACM tapped RISC pioneers John Hennessey and David Patterson for the Turing Award, the highest honor in computer science. They won an award named after the man who created the simplest instruction set of them all.

RISC didn't take over--Intel never really embraced the concept and kept adding new features from real-number operations to AES encryption/decryption built into the processor. GPUs added vector operations and Tensor Processing Units can do machine learning instructions. Added complexity has its complications such as the famous 1994 division bug.

RISC processors do play a role in mobile devices which need to have low power, one can find ARM (Advanced Risc Machines) processors in many mobile phones and tablets. Based on quantity alone there are far many more RISC processors than CISC (C = complex).

These days the lines between the processor and the operating systems blur and we are programming chips. David Patterson continues to work on RISC, with a recent open RISC-V architecture that also supports chip programming. John Hennessey was otherwise occupied--he was president of Stanford from 2000 to 2016.

Monday, March 26, 2018

Why do we give citations? How should we give citations?


Why do we cite past work? There are many reasons and they lead to advice on how we should cite past work

  1. Give credit where credit it due. Some people over cite and that diminishes any one citation. I once saw a paper that had in the first paragraph: ''Similar work in this field has been done by [list of 20 citations].'' One of the papers in that list was extremely relevent, the rest much less so. This is not really helpful. There should be less citations and more about each one.
  2. If you are using a result in a prior paper the user should be able to read that paper. For that reason I try to give the websites of where the paper is. (this might be  less crucial now then it used to be since if a paper is on lnie someplace for free its usually easy to find). Some students ask me if its okay to cite papers on arXiv. Of course it is, especially if it's to guide the reader to a place to read the result. Note also that papers on arXiv are not behind paywalls.
  3. At some point a result is so well known that it does not need a citation. It's not clear when this is. I think people write `by the Cook-Levin theorem' without citing the original source. Nor do people ever cite Ramsey's original paper either. See next item for why this might be a mistake.
  4. A reader might want to know WHEN a result was discovered. For this reason, perhaps people should give references for Cook-Levin or for Ramsey. The original source is often NOT a good place to read a result so I often do  ``by a theorem of Curly [1] (see also Moe's simplification [2] or Larry's survey [3])'' so I give the reader WHO did it first, when they did it, but also an alternative place to read it. However, when does it end? `By the Pythagorean theorem [1] (also see [2])'
  5. A reader should know the context of the result.  Is the problem new? Is the problem related to other problems? Has there been much work in this field? Inquiring minds want to know!

Wednesday, March 14, 2018

Stephen Hawking (1942-2018)

Stephen Hawking passed away earlier this morning in Cambridge, England. As a brilliant theoretical physicist and best-selling author all while dealing with a debilitating disease, Hawking rose to be the best-known scientist of his time. 

I'll leave it to the physics blogs to talk about his research. Hawking inspired me through his 1988 book, A Brief History of Time. Hawking told Time magazine before the magazine's publication "Someone told me that each equation I included in the book would halve the sales. In the end, however, I did put in Einstein’s famous equation E = mc2. I hope that this will not scare off half my potential readers.”

So you read the book and he manages to describe his highly mathematical-based view of the universe without resorting to mathematics, by far the best written popular science book I have read.

A Brief History of Time came out when I was in grad school so it didn't play a role in me becoming an academic but it did make me realize that science has a story to tell. From the preface of my own book.
I took inspiration from Stephen Hawking's A Brief History of Time, which explains physics not through formulas and technicalities but with examples and stories. I attempt to do the same here to explore the spirit and importance of the P versus NP problem.
I am under no illusion that I came even close to Hawking's level of exposition.

A poll taken last year showed most Americans could not name a single living scientist but among the 19% that could, the scientist they named most often was Stephen Hawking. We lost not only a brilliant physicist but one of the great symbols for science of our generation.

Monday, March 12, 2018

Wanted: An Easy Proof of a Weak Theorem in NT so that I can get another VDW--> Primes infinite proof to be really easy.

(All math in this article is here)

A while back I posted about a proof that Van Der Waerden's theorem implies the number of primes
is infinite (see the post here). That post brought up some logical issues.

More recently there is another proof that the primes are infinite that raises (for me at least) some number theory results and proofs. The proof uses the following theorem:

There are no 4-length Arithmetic Progressions consisting of squares.

This is attributed to Fermat. All of the proofs I've seen of it look difficult.

Here is a sketch of the proof of infinite number of primes from VDW and the theorem above.
Assume that {p1,...,pm} are all of the primes.  Let vi(n) be the power of pi in the factorization of n.

We define the following 2m coloring of N

COL(n) = ( v1(n) mod 2, ... vm(n) mod 2)

There exists a monochromatic 4-AP.  We leave it to the reader to show that you can use this to get a 4-AP of squares.

Using Fermats 4-AP theorem is hard!  In the write up I show how to use a stronger VDW theorem and a weaker (at least easier to prove in my opinion) result in Number Theory to get a different proof.

VDWPlus: for all k,c there exists W such that for all c-colorings of [W] there exists a,d such that

a, a+d, a+2d, ..., a+(k-1)d AND d (that's the PLUS part) are all the same color.

Now to prove primes are infinite. Assume not. p1,...,pm are all of the primes.

Let vi(n) be as above

COL(n) = (v1(n) mod 4, ..., vm(n) mod 4)

just apply this to the k=1 case so we have

a, a+d, d all the same color.  Say the color is (b1,...,bm) where bi in {0,1,2,3}

mult all three by p1^{4-b1} ... pm^{4-bm}.

now we have A, A+D, D all four powers. call them x^4, z^4, y^4 and you
contradict the n=4 case of Fermat'ls last theorem (this is the easiest case and was proven by Fermat)

TO DO: find an even easier theorem in Number Theory to use with a perhaps more advanced form of VDW theorem to get a proof that primes are infinite.

Coda: Cute that I replace one theorem by Fermat with another theorem by Fermat.

Friday, March 09, 2018

Data-Driven Programming

Five years ago I posted about Flash Fill, a then new Microsoft Excel feature that would reformat data based on examples. I just checked the latest version of Excel and Flash Fill is still there just working by making suggestions when appropriate. It uses a programming language approach finding the simplest string manipulation program consistent with the examples.

I loved the idea of example-based programming but it did seem pretty limited at the time. With the machine learning revolution we've gone full circle. We solve a number of problems with data and examples alone: voice recognition, face recognition, spam detection, playing Chess and Go from scratch in ways we wouldn't even know how to program.

That's not completely true: Designing a good machine learning program requires programming to get data in a good format and creating the right network structure to learn is still considerably an art. But the real magic does happen when the deep learning process happens creating a neural net that works well in ways we can't fully understand even when we look at the final network produced.

Now "it's all about the data". This hit me when I got an email invitation for the Uber visa card. Unlike other branded cards it gives you no particular discounts or rewards for Uber. Rather it gives you significant cash back for dining and travel, in other words for giving Uber the data about you that can make Uber itself more valuable to you.

Nothing in this post should be new to you readers. But every now and then just take a breadth and realize who much our notions of computing and data have changed over even the past five years.

Programmers aren't going away. Machine learning works well for things humans do well instinctively. But for actual number crunching, simulations and manipulation, even what Flash Fill does, machine learning has much more to do.

Wednesday, March 07, 2018

When the Wrong People raise the issue

On my discrete math final in Spring 2017 I had a question:

Prove that sqrt(2/3) is irrational.

A student emailed me the folloing (I paraphrase and am prob not as elegant or as long as he was)

Dr. Gasarch

I received only 2 points out of 20 on Problem 3 of the final- the one that asked us to prove that sqrt(2/3) is irrational. This was graded rather harshly, and the reason is endemic to the entire course. It was unclear what we could and could not assume in this problem. That has been unclear all semester.

I responded (I leave out the name for privacy)

Mr. Student

Come by my office and we will look over  your final. At the same time, bring by your HWs and Midterms to look at. The deadline for regrading those is up; however, either you are correct and I will learn how to run a better course by looking over your HWs and midterms, or you incorrect and you will be enlightened by us looking over them.

We agreed to meet. The Student came all ready to  argue not just points on the final but also about the course in general. I looked forward to this since either I would learn about how to run the course better OR I would enlighten a student!

STUDENT:  I used the obvious fact that the ratio of two irrationals is irrational. How was I supposed to know I had to prove something so obvious!!!!!!!! And only 2 points!!!!!!! Really!!!!!!

BILL: What is the ratio of \sqrt{8} to \sqrt{2}.

STUDENT:  sqrt(8)/sqrt(2) = sqrt(8/2) = sqrt(4) = 2. Why is that relevant?

BILL: You just showed that the ratio of two irrationals could be rational.

STUDENT: Oh.

BILL: Lets look over your HW and midterm and see there were times when it was not clear what you could assume OR if not then I can clear up some misconception you had.

STUDENT: Uh, Uh. Nevermind.

BILL: You are here with your HWs and midterms, lets take a look.

He really didn't want to. I think he really just wanted more points on the final But since he phrased it as a philosphical debate about how the course was run, I took him at his word.  Everything he showed me was either clearly wrong or clearly unclear. None of it fell into the category of `not clear what you can assume'.

This disappointed me. I suspect someone could make a case that my course sometimes does not make it clear what you can assume. Other students with a similar story as above claim my course is to pedantic. But the examples they show me of this are usually just wrong, not marked wrong for being to pedantic.

There is a more general problem here. The students who complain about a  course may well have a valid point to make!  But its usually students who are not very good making the complaint, and hence they are not the ones who could make a good argument.  One thing I have done when a student has a complaint about how I run the course is then ask my TAs about it. This has been helpful sometimes but they are on the other end -- the course is easy for them so its hard for them to see the problems.

Having said all of this I will own up to one flaw I've had which the students complained about incoherently  (see here) and my TA pointed out the fair objection.  I had taught ONE type of Pigeon hold argument and tested them on a very closely related but different type -- as a way of testing if they understood pigeon hole and were not just memorizing. It was a fair question BUT my TA said
(correctly) I was more interested in getting a good test question then, you know, TEACHING them the Pigeon hole principle -- so I should in the future (and I have) teach them LOTS of versions, make sure they understand them, and then if on the exam I give them a variant its more fair. But more importantly he pointed out that I (and this is correct, or was) have a QUESTION I really want to put on the midterm and then teach the course so that it makes sense to do so. The question is fair, but this is NOT the point (which is why the students objections were incoherent- the question was fair). I am setting (some of them) up to fail.  I have CHANGED my Teaching style and exam style since them.

But my point is that the students could not possibly have raised that objection-- partly because the students complaining are not very good, but also because they do not see what goes on behind the scenes.

UPSHOT- if your students have an incoherent complaint there may be something to it and you should ask your TAs.


Thursday, March 01, 2018

Me and My Bolt


On Tuesday I got a knock on my car's window. Rolling it down someone asked if I liked my car as he was thinking of buying one himself. Was I driving the latest Tesla? No, the Chevy Bolt, General Motor's newest fully electric car.

On March 31, 2016 me and 180,000 of my closest friends put $1000 down sight unseen on the Tesla 3. As an east coast resident with no prior Tesla I am well down the list even though I made a reservation on the first day. I got disillusioned by early production problems and delays and bait-and-switch emails.
Now that some more details regarding Model 3 came out, I wanted to gauge your interest in coming in for a test drive in a Model S. An incredibly well equipped Model S can be had for under $80k and with the $7500 federal tax credit and $1000 referral code from a current owner, you can get into a Model S for close to $70k, meanwhile a Model 3 can cost up to $60k. Model S is considerably quicker and has an incredible amount of space to seat 5 comfortably. It is our flagship vehicle and has 5 years of manufacturing behind it to perfect the build quality of the car. Not to mention a quick turnaround time on delivery. I would love to host you in our showroom so I can showcase some of the incredible features in the Model S.
Meanwhile a GM executive on the Georgia Tech College of Computing's advisory board suggested I take a look at the Bolt. I was skeptical but he arranged for a local dealer to bring down a car to test drive. I got sold and bought my own Bolt in December.

It's an all electric car with no transmission so quick smooth acceleration. The Bolt has a one pedal driving mode as the "gas" pedal also works as a break which slows down the car by pulling power to the battery. The car doesn't even pretend to be mechanical, a screen boots up when I turn the car on, you can shift into park by pressing a button. The rear view "mirror" is really a camera. When driving slowly there is a simulated view from above. A little freaky but it helps with parking. With Android Auto I basically have Google Assistant and Maps in the car which I prefer to an in-car navigation the Bolt doesn't have.

I get over 200 miles on a full charge and can plug in overnight to get enough power to get to work and back. The car has considerable safety features that warn about cars in blind sports or getting too close to the car in front of you or if you drift from lanes. The Bolt lacks any autonomous features even adaptive cruise control or parking assist. I suspect you could get significant autonomous behavior with a software upgrade but I doubt GM would ever do so.

The car is missing some basic features like power seats and a garage door button. No problem, I rigged up Google Assistant to open the garage door when I say the magic words. Far cooler than pressing a button.

It's not as sleek looking as a Tesla and nobody will shoot it into space but I'm driving the future and it is amazing.

Sunday, February 25, 2018

When students are wrong but have a valid point, encourage them

I often have the class VOTE on a statement (the choices are usually TRUE/FALSE/UNKNOWN TO SCIENCE/Stewart-Colbert-2020)

I ask the students who voted incorrectly (some say anything except Stewart-Colbert-2020 is an incorrect vote, though I'm partial to Bee-Oliver, though John Oliver was not born in this country so he's not eligible, though our current president might say that neither was Obama so thats not a bar to the presidency) why they vote the way they do to get a good discussion going. I give some examples. I use real first names but decided to not give last names for reasons of security. The people involved all agreed. If you have examples of students being WRONG but HAVING A POINT, leave an intelligent comment about it!

1) In Discrete Math they  vote on: is Q, the rationals, countable?One of the NO votes was named Alice (really!).

Alice: The rationals, like there are SO many of them. They are just so crammed in their. I mean really!

Bill: I think you want to say that between any two rationals is a rational.

Alice: Thats what I said!!

Bill: Great! And realize that  a set is countable if there is a bijection from the  set to N. So what you are really saying is-----

Alice: There is a bijection between N and Q but not an order preserving Bijection!

2) In Formal Lang theory they vote if alpha is a reg expression and L(alpha) is the lang generated by alpha then is there a reg exp beta such that L(beta) is the complement of L(alpha). One of the NO votes was named Andrea (nickname Andy).

Andy: Given alpha I just don't see an easy way, or for that matter any way, to create beta.

Bill: I will soon prove that regular expressions are equivalent  to DFA's. So what one would do is take the L(alpha), form the DFA, complement the DFA, then convert back to a regular expression.

Andy: Oh, yes that works, but it seems complicated.

Bill: It is! You though it would be complicated so it was false. Even though its true, your intuition that its complicated is correct! In fact, there are reg expressions of length n such that the complement lang has a rather large reg expression.

3) Formal Lang theory again. Not a vote this time. I had on the board the regular expression

(a UNION b)(a UNION  b )^*

A student Dolapo thought it could be simplified:

Dolapo: Can't you write that as the regular expression (a union b)^+ ?

Bill: Yes and no. Yes the lang is (a UNION b)^+, but + is not allowed in regular expressions.

Dolapo: Why not?

Bill: First off, they really could have been defined that way and perhaps should have. So why weren't they? I go with historical accident, but some people disagree- see my blog post here

Thursday, February 22, 2018

NP is Hard

You don't get much press by stating conventional beliefs--there is no "round earth society". Nevertheless there are serious researchers out there trying to say that it is reasonably possible if not likely that P = NP. Don't believe it.

Emanuele Viola just outright states I Believe P = NP. He starts with a similar argument that we've seen from Richard Lipton: We've had surprising results in the past so why should the convention wisdom that P ≠ NP be true? I have two major problems with this line of reasoning:
  • These arguments cherry pick certain results and ignore the vast majority of theorems that went the way we expected. It's akin to saying that because there have been lottery winners in the past, the ticket I hold in my hand is likely to be a winner.
  • All the major surprises in complexity have occurred early in first two decades of the P v NP era. We've seen no surprises at that level since the early 90's. We have seen surprise proofs of results like Primes in P, Undirected Connectivity in Log Space and NEXP not in ACC0 but the theorems themselves surprised no one. 
Viola also makes the argument that we've seen fewer significant lower bound results than upper bounds. This doesn't surprise me--an upper bound requires a single algorithm, a lower bound requires showing an infinite number of algorithms can't work. I fully agree that we're unlikely to see a proof that P ≠ NP in the near future. But this isn't a reason to think they are the same. And we've seen no non-trivial algorithms for general NP problems like circuit-SAT.

Which takes me to Ryan Williams Some Estimated Likelihoods For Computational Complexity where he gives 20% probability (which I consider nearly 20% too high) that P = NP based on similar logic. Ryan surprises me even more by giving a whopping 80% chance that NEXP = coNEXP. That would imply every easily describable tautology has a short proof of being a tautology (for the appropriate definitions of describable and short) which I find hard to fathom.

If I would guess collapses that might happen, I would suggest L = NL (directed connectivity in log space) or one that Ryan gives even odds, TC0 = NC1. TC0 captures constant-depth neural networks and as we've seen from the ML community, those neural nets look awfully powerful. 

Sunday, February 18, 2018

The web bad/People using it bad.

I was going to write a post about how hard it was to find what grades mean at different schools (e.g., at UMCP W (for withdraw) means the student dropped the course, but at UT Austin its Q (for Quit?)) but then I found that my Googling

Name of school grade key

I could find it.  Okay, so grade keys are not that hard to find.

WEB BAD:

However, course descriptions are. Both questions (what grades mean and course desc) came up when I do admissions for my REU program.  If a student has a course in Discrete Mathematics that could mean a course where you learn how to prove simple things OR it could mean a course where you proof  the graph minor theorem (actually I doubt that) or something in between. Similar for Principles of Programming languages and Software Engineering in that they could mean a variety of things. I think math and physics are more standardized, but even there you have the problem that Advanced calc could be either multivar or a course with proofs or something else. There is a great related XKCD here.

PEOPLE USING THE WEB BADLY:

Fellow blogger Scott Aaronson recently posted about Vivek Wadhwa's Washington post column on quantum computing (the post is here) The article by Vivek told the tired and utterly false story that QC can be used to solve TSP by looking at all possibilities. Should Vivik have known better by looking at the web? At first I thought YES he should know better. But then I thought -- I should try to see what he might have done. Maybe the web has this one wrong. In fact, some of my students Iwho, I should add, are not writing articles on QC for the Washington Post)  tell me that they don't need to learn the proof of the Cook-Levin theorem since Quantum Computers can solve SAT anyway.  So I decided to pretend I didn't know anything about QC and went to the web to see what I could find.  First stop: Wikipedia. I found the following quote in the QC article:

BQP is suspected  to be disjoint from NP-complete and  a strict superset of P though this is not known.

So, the first place I look I find that BQP is suspected to NOT solve SAT or TSP or... Realize that Wikipedia is not some obscure source of  hidden knowledge. It is literally the first place one would look.  Hence, no excuse, even if other experts told him otherwise (which seems to be the case, though `experts'  should be in quotes) the Wikipedia quote should at least give pause.

So even when Wikipedia gives the right information, not everyone looks there.


Thursday, February 15, 2018

Corporate Education

First of all read the #metoo testimonial going around the TCS blogosphere. Our field is not immune.

Last Sunday Frank Bruni wrote an op-ed column Corporations will Inherit the Earth, an article on how corporations have taken on what governments used to do. Quite a bit is focused on education.
The nimbleness of corporations gives them an edge over hoary, complacent institutions, including those in higher education...In an effort to make sure that employees have up-to-the-minute technical skills--or are simply adept at critical thinking and creative problem solving -- more companies have developed academies of their own. That's likely to accelerate. "I think enterprises like Amazon and Google are going to build universities that teach coding and things the nation needs" said Margaret Spellings, former education secretary and current president of the University of North Carolina system.
Already in the universities we've seen a move towards more majors and minors in STEM and computer science in particular. The changing corporate workplace has already changed academia, putting more an emphasis on technical areas and less on the liberal arts. Will companies though take the next step, running their own schools that focus on the needs of industry? If we see continued decreased government funding for universities and academic research, we may end up with a handful of rich universities and the rest of the population getting education on the job, a future that would be a loss for us all.


Saturday, February 10, 2018

A ``new'' ``application'' of Ramsey Theory/A Truly ``bad'' math song

Ian Parberry once told me (though I doubt he originated it- The first link I found says it was Mark Twain)


to a man with a hammer, everything looks like a nail

Indeed. Since I am teaching a grad course Ramsey theory and its ``Applications'' (I got 24 students, which is more than I thought I would- including around 10 ugrads who are taking it because `all the cool kids are taking it' ) I have been taking the hammer of Ramsey Theory and looking for nails to apply it to. (I'm also using a webiste I found of applications of Ramsey Theory here and an survey article on applications of Ramsey here.)

Thinking about the infinite Ramsey Theorem I came up with this ``application'' :

If p1, p2, p3, ... is an infinite sequence of points in Rn then there exists a subsequence q1, q2,q3,... such that for each coordinate 1 ≤ i ≤ n the projection onto that coordinate is either (a) strictly incresing, (b) strictly decreasing, or (c) constant.

Proof:  For i< j color (i,j) with one of 3^n colors - indicating for each coordinate i if the ith projection is increaing, decreasing, or constant. The infinite homog set gives you the sequence.

End of Proof

One can also proof this from the Bolzano-Weierstrass  theorem (an infinite bounded sequence of points in Rn has a convergent subsequence). We leave that proof to the reader; however, the proof of BW looks like the proof of the infinite Ramsey Theorem, so I'm not sure if my proof is new or not.

I wanted to look into the BW theorem so I googled "Bolzano-Weierstrass"  I think Google knows me better than I know myself since the second hit was https://www.youtube.com/watch?v=dfO18klwKHg which is a Rap Song about the BW theorem (I am a fan of novelty songs, and of math, so does it follow I am a fan of Math Novelty songs. Not sure if its follows, but I AM!)

One of the problems on the HW was BW-rap- good, bad, or really bad?

Answers were:

1) Made my ears bleed

2) Lyrics good, singing really bad

3) So bad its good

4) No, just bad.

I'm in the Lyrics good/singing is `so bad its good' camp. The class was okay with the lyrics, but mostly thought it was so bad its... bad. One person thought it was awesome!

I would like to see  real rapper perform  it on you tube. I doubt I will. Oh well.

Thursday, February 08, 2018

For the Love of Algorithms

Wired magazine labelled 2017 as The Year We Fell Out of Love with Algorithms. The article goes on to talk about how algorithms give us filter bubbles, affect elections, propagate biases and eliminate privacy. The article at the end argues that we shouldn't blame the algorithm but the people and companies behind them.

Every day algorithms decide what music we listen to, what posts we see on Facebook and Twitter, how we should drive from point A to point B, what products we should buy. Algorithms feed my nostalgia in Facebook and Google Photos showing me what once was. Algorithms recognize my voice and my face. We've even created new currencies from algorithms. Algorithms form the backbone of the top six world's largest public companies (Apple, Alphabet, Microsoft, Amazon, Facebook and Tencent). It's been a long time since I only trusted algorithms to format my papers.

Algorithms have taken over nearly every aspect of our lives and our economy. With new machine learning techniques, no longer can the creator of an algorithm fully understand how algorithms work. But we aren't in a Skynet world, algorithms are not sentient and not even inherently bad, any more than we label a group of people inherently bad because of a few examples.

This societal transition from human-oriented decision to algorithmic decision making will continue and will have great benefits such as much safer cars and medical care. We must be vigilant, of course, to how algorithms will change society, but (in a serious sense) I welcome our new machine overlords.

Monday, February 05, 2018

"How can you use your research for ugrad projects?'- the wrong question

I was helping a math PhD who worked in computable ramsey theory prepare his teaching and research statements for his job application. One of the questions various schools wanted to know was

How can you use your research as the basis for undergraduate projects?

His answer was kind of a cheat- Ramsey Theory was combinatorics which ugrads can work on without too much prior knowledge (true), computability theory could be picked up (not true), and the chance to combine the two should thrill ugrads (it doesn't even thrill most Ramsey Theorists!).

I don't fault the PhD student, I fault the school. What they really want to know is:

Can you come up with ugrad projects?

They should realize that some math requires too much bacground and is not appropriate for ugrads AND that a math PhD will have picked up OTHER knowledge for projects. They may relate some to the research (e.g., in the case above just Ramsey Theory) or they may not even be close.

In Comp Sci it is more common for an ugrad to work on research related to a profs research since Comp Sci is a younger field and needs less background knowledge. And students can often code something up of interest related to the profs research.

SO- do you have ugrads work on your research program OR on something else?




Thursday, February 01, 2018

Flying Blind

Many computer science conferences have made a number of innovations such as a rebuttal phase, multi-tiered program committees, a hybrid journal/conference model with submission deadlines spread through the year. Not theoretical computer science which hasn't significantly changed their review process in the major conferences since allowing electronic submissions in the early 90's and an ever growing program committee now at 30 for FOCS.

Suresh Venkatasubramanian learned this lesson when he ran a double blind experiment for ALENEX (Algorithmic Engineering and Experiments) and laid out an argument for double blind at broader theory conferences to limit the biases that go along with knowing the authors of a paper. The theory blogosphere responded with posts by Michael Mitzenmacher, Boaz Barak and Omer Reingold and a response by Suresh. I can't stay out of a good conference discussion so here goes.

Today major orchestra auditions happen behind a screen with artists even told to remove footwear so sounds won't give away the gender of the musician. On the other extreme, the value of a piece of art can increase dramatically in price if it is discovered to be the work of a famous artist, even though it is the same piece of art. Where do research papers lie? It's more a work of art than a violinist in a symphony.

Knowing the authors gives useful information, even beyond trusting them to have properly checked their proofs. Academics establish themselves as a brand in their research and you learn to trust that when certain people submit to a conference you know what you get, much the way you would more likely buy a new book from an author you love.

Suresh rightly points out that having authors names can and do produce biases, often against women and researchers at lesser known universities. But we should educate the program committee members and trust that they can mitigate their biases. We can completely eliminate biases by choosing papers at random but nobody would expect that to produce a good program.

Having said all that, we should experiment with double blind submissions. Because everything I said above could be wrong and we won't know unless we try.

Monday, January 29, 2018

Publishing: Why are we doing it right and other sciences aren't?

(NOTE- this is NOT a `we hate Elsevier and the others' post- though I suspect the comments will be about that.)

Alexandra Elbakyan has created a repository of science papers for scientists to share. She did this because too many papers were behind paywalls. An article about her (and the inspiration for this post) is here. Note that she has had some legal problems.

But it got me thinking: I have rarely had a problem getting an article. Between authors websites and arXiv most of what I want is out there.  Lance and others I've asked agree--- while there are problems with editors and publishes etc, access is NOT one of them.

Why don't other scientists just routinely do this?

some speculation but if someone actually knows please comment

1) Some scientists purposely don't post on line to avoid being scooped. Or the fear of that

2) Some scientists don't post for fear of legal issues with publishing. I do not know of a single computer scientist or mathematician that has gotten in any kind of trouble for posting a paper on their website or arXiv.

3) Some scientists are lazy.

4) (This contrasts with Comp Sci) other fields do things the way they do since they ALWAYS did it that way. Comp Sci is a younger field and hence is less tied to tradition.

5) Other scientists ARE posting their papers (gee, then why did Alexandra have to create the reporistory).

6) Alexandar is where arxiv was X years ago. They'll catch up. But they are having legal problems-- did arXiv ever have legal problems?

Looking over these I'm not sure I believe any of them. Any other ideas?

Again, the question is: Why don't other fields routinely post to something like arXiv or their own websites?

Friday, January 26, 2018

From Art to Science

Q: Why did the neural net cross the road?
A: Who cares along as it got to the other side.

Whit Diffie and Martin Hellman in their classic 1976 paper talk about the transition of cryptography:
Theoretical developments in information theory and computer science show promise of providing provably secure cryptosystems, changing this ancient art into a science.
Indeed we have some very strong theoretical foundations for cryptography, though they still need to rely on unproven hardness assumptions. Often we trade off formal proven techniques for efficiency, in for example AES and Bitcoin. Here we use "science" in another sense: extensive testing of the hardness of these protocols. Creating and breaking cryptographic protocols used to be an art, having the right intuitions to make things work (for now). Today, breaking security rarely involves breaking a cryptographic protocol but rather due to bad implementations and easy-to-guess or phished passwords.

For neural networks we still live in an art regime. We have great methods to minimize the error for a given network (though no formal proofs as such), one still needs to design the right network with the appropriate topology and various tricks like regularization to prevent overfitting. Several members of the theoretical CS community have taken on the task to turn the art into a science but getting us to where we sit in cryptography will take some time. Crytography took thousands of years to get from an art to science, hopefully we can get there faster for machine learning.

This debate came to a head at the recent NIPS conference. Ali Rahimi in his test-of-time talk described the current state of ML research as "alchemy". Yann LeCun gave a strong response
Sticking to a set of methods just because you can do theory about it, while ignoring a set of methods that empirically work better just because you don't (yet) understand them theoretically is akin to looking for your lost car keys under the street light knowing you lost them someplace else.
Imagine if we never did any cryptography until we had the science in place.

Proving theorems in mathematics is an art more than a science. Because of incompleteness in some sense it has to be. If mathematics ever became automated it would be a whole lot less fun.

Wednesday, January 24, 2018

How many degrees are in a Martian Year?

James Tanton gave a great talk at the JMM (Joint Math Meeting) in San Diego on

how many degrees are in a Martian Year?

but he didn't quite answer his title question. I found an excerpt of the talk on YouTube but still didn't quite answer the question, though he could have.

The talk was

How many degrees are in a Martian Year?

Here is an excerpt on You Tube: here

I feel the urge to explain and complete the talk.

Why do we earthlings have 360 degrees in a circle? Because there are 365.25 days in a year and nobody wants to work with that number so they round off. They could have chosen 360 or 365 or 370 but since 360 has the most divisors, it wins. (There is a different measure, gradians, of which there are 400 in a circle- also a nice round number, but 360 is rounder via the criteria we will soon explain.) So why pick 360? It has the most divisors.

There are 668 martian days in a martin year. So you might think they would pick 670 or 660 for there degreees. But thats Base 10 thinking!  So the answer depends on what Base the Martians use. Lets assume that, like us, it depends on the number of fingers they have on their hands. We also assume they have two hands with the same number of fingers on them (would aliens look like us or not? Here are some thoughts:  here and here). Two fingers on a hand seems like two few, and ten seems like too many so lets assume there base is either

3 fingers per hand: base 6. 668-base 10 is 3032 base 6. 3020 is answer- divided by 2,3,5

4 fingers per hand: base 8. 668-base 10 is 1234 base 8. 1240 is answer- divided by 2,3,4,8.

5 fingers per hand: base 10 668. Lets go with 660- divisible by 2,3,5

6 fingers per hand: base 12 668-base 10 is 478-base 12. 470 is answer- divided by 2,3,4,5

7 fingers per hand: base 14 668-base 10 is 35A in base 14. 140 is answer- leave it to you.

Any other good candidates or methods for this?

Thursday, January 18, 2018

A Sensitive Game

Last month I posted about the sensitivity conjecture and today I would like to talk about an interesting game developed by Gilmer, Koucký and Saks and independently by Drucker that could yield a proof.

Consider the following game on n bits among three players, Alice, Bob and Carol. The game works as follows: Carol picks a bit position and Alice sets that bit to 0 or 1. Carol then picks another unused bit position and Alice sets that bit as well. This repeats until the last bit which Carol gets to set herself. The bits are then revealed to Bob who has to give a set of bit positions that includes the bit Carol set at the end. Alice and Bob can work out a strategy ahead of time with the goal of minimizing the size of the set.

If Carol can force Bob to give a set of nε bits for some ε>0, this would prove the sensitivity conjecture! Basically a family of functions that give a counterexample to the sensitivity conjecture would give a strategy for Alice and Bob that yields a set of size no(1). You can find the full proof in the papers above.

How well can Alice and Bob do? They can use the following strategy to achieve n1/2: Break up the input positions into n1/2 groups of n1/2 bits each. Whenever Carol picks a bit position, Alice sets that bit to zero unless it is the last bit in that group in which case she sets it to one. Carol can set the last bit in some group anyway she wants. Bob will either see every group having one 1 and will give the set of the positions of all the 1's, or Bob will see a group of all zeros and give the positions in that
group.

Mario Szegedy uses error-correcting codes to improve the upper bound to O(n0.4732). A Georgia Tech undergraduate DeVon Ingram improves the upper bound to O(n0.4696) by generalizing Szegedy's techniques. Ingram's proof comes down to finding a one-to-one function mapping subsets of {1,...,15} of size 4 to perfect codes of length 15 with the property that the bits of the code are 1 for the indices of the subset. Perhaps one could find a clever mapping that has this property but Ingram wrote code using a matching algorithm. A true computer assisted proof. Longer code lengths alas won't yield direct improvements.

The connection to sensitivity doesn't go both ways. An upper bound of no(1) wouldn't necessarily yield a counterexample to the sensitivity conjecture though would tell us the limitation of this game approach. Nevertheless I find it cool that a game that doesn't talk about functions at all could help settle an important question about them.

Tuesday, January 16, 2018

Donald Knuth Turns 80 years and 6 days

Celebrating Donald Knuth's 80th birthday, or 80 years + 7 days birthday seems odd. Should we use powers of 2? Hmm- too few, just 32 and 64 really. And having a 32-year celebration for someone is unusual. How about numbers that end in 0 in base 8. 64 would be 100, 72 would  110, 80 would be 120 so AH- we would be celebrating! So lets Celebrate!

LANCE: One of us should blog about Don Knuth turning 80.

BILL: How about both of us, sep posts?

Lance had his post here. Now its my turn.

Donald Knuth was the first person (roughly- history is not as definite as mathematics) to use mathematics to analyse algorithms. This may seem like an obvious thing to do but that's easy to say in retrospect. And he never lost focus: He may have to use some hard math to analyze algorithms but his goal always was to analyze algorithms. He didn't get fascinated with some obscure aspect of math like, say, surreal numbers, and go write a book on it. Oh wait, he did (see here). But, for the most part everything he did was towards faster algorithm and faster typesetting.

In an early draft of a book review of  Companion \to the Papers of Donald Knuth I wrote at the end of it

 Donald Knuth has been called the father of Theoretical Computer Science and hence his thoughts are worth knowing

I usually email my reviews to the author so they can make sure I didn't say anything stupid or in some very rare cases have a rebuttal. In Knuth's case I postal mailed it (ask your grandparents what that is) since he does not use email. He mailed back the hardcopy with some corrections in pencil. One of them was to cross out

father of theoretical computer science

and replace it with

father of algorithmic analysis.

I made the correction. That is how he views himself and it would be odd to argue the point.

As a tribute to him I have gathered up all of the book reviews  in my column of books by him (between 10 and 12 depending on how you count) and books clearly inspired by him (1 book- A=B) into one file which I point to  here




Wednesday, January 10, 2018

Donald Knuth Turns Eighty

We've kept this blog going long enough that we start repeating our celebrations. Ten years ago Bill celebrated Don Knuth's 70th Birthday and today Donald Knuth turns 80. While he celebrates in Piteå, Sweden with its less than 4.5 hours of daylight, we wish him a happy birthday from stateside.

Looking back in this blog, in 2015 I wrote about the history of the history of computing including this wonderful talk by Knuth, Let's Not Dumb Down the History of Computer Science.



Donald Knuth is known for many things, important algorithms, the brilliant book series The Art of Computer Programming, TeX, the many awards he's won and the award named after him. In my favorite story, back in the 70's when Knuth started working on Volume 4 (still in progress), he wanted a good name for those hard problems that Cook and Karp found. His candidates "Herculean", "Formidable" and "Arduous" didn't get much traction so he ran a contest which had some interesting suggestions before reluctantly going with the name "NP-complete" that came out of a group from MIT.

In his SIGACT News article that described the contest, Donald Knuth was so sure that these hard problems would remain hard, he puts everything on the line, offering anyone who could prove P = NP a live turkey. May Don Knuth and his turkey continue to celebrate many more birthdays.

Sunday, January 07, 2018

A new largest prime found!

A new largest KNOWN prime has been discovered and its 23 million digits long.

Nate Silver's website had an article about it (written by Oliver Roeder) here

An article about why people do this is  here

Lance posted about finding large primes in 2006 here

I'll just make some random comments

1) The prime is 277,232,917-1

2) The prime is not much bigger than the previous champion.

3) More generally, the graph (in Oliver Roeder's article) shows from 1600 to about 1951there was slow progress but since then there has been LOTS of progress. See the table in this article. I had wanted to say every year a new prime was found  but, alas, not that simple a pattern. Even so, lots of new records.

4) I"ll list the reasons given for why people do this and my comments on them.

a) Tradition! (Note to self- write a novelty song to the tune of Fiddler-on-the-roof's Tradition about why people work on various mathematical things)

b) For the by-product of the quest. This one does make sense and I am sure has driven and helped test out some research. Reminds me of the spinoffs from going to the moon (see here). Contrary to what I heard as a kid, the orange-powder-drink Tang was not a spinoff. But there were many others. Of course- would these spinoffs have happened anyway? Hard to know.

c) People collect rare and Beautiful items. I don't really see this one. Finding a large prime doesn't make  its yours, it belongs to the ages! And I don't think people get their names on the prime either. The only prime that has someone's name on it is the famous Grothendieck prime which is 57. Oh well. There are sets of primes with peoples names on them: Mersenne primes, Gaussian  primes (which are subsets of Gaussian integers so maybe shouldn't count), Eisenstein primes, and
Sophie Germain primes. If you know of any other primes or set of primes named after someone, leave a comment please.

d) For the glory! Maybe, but given how briefly people hold the record, fame is fleeting.

e) To test the hardware. This one I didn't know! I'll quote the article as to why primes are good for this
Why are prime programs used this way?  They are intensely CPU and bus bound.  They are relatively short, give an easily checked answer (when run on a known prime they should output true after their billions of calculations).  They can easily be run in the background while other "more important" tasks run, and they are usually easy to stop and restart.
f) To learn more about their distribution. The prime number theorem was conjectured from data. We have so many primes now that I wonder if a few more really help formulate conjectures

g) For the money. The first person to get a ten-million digit prime gets $100,000. The first person to get a one billion digit prime gets $250,000.  Wow! Except that the article must be a bit old since the $100,000 prize was claimed in 2009 (see here). Still, theres that one billion digit prize out there!

5) Mersenne primes are of the form 2^n-1. It is known that n must be prime for 2^n-1 to be prime (this is not hard). There are much faster primality testing algorithms for Mersenne primes than arb primes. But see next item.

6) While writing this blog post I looked up non-mersenne primes. It seems like the largest one is

10223*2^31172165 + 1 and was discovered in 2016.

But of more interest- there is no Wikipedia page on non-Mersenne primes, there are some outdated pages that don't have the most recent information. As the kids say, its not a thing.

8) I'll add one more reason why people work on this, but its more of a tautology: People work on finding large primes because they can!. By contrast, finding VDW numbers is hard and likely to not make much progress.

9) I think that the most reason advances have come from computing power and not number theory. (if this is incorrect let me know with a polite comment)

10) Have their been spinoffs in either number theory OR computing power lately?

11) I wonder if there will come a point where progress gets hard again and the graph of largest known primes flattens out. I tend to think yes, but hard to say when.