- Does the polynomial-time hierarchy look like the arithmetic hierarchy? I mentioned this in the centenary post for Mostowski. Probably not because it would imply factoring in P (since NP∩co-NP would equal P) but we have no proof of separation and no oracle that makes them look the same.
- Does UP = NP imply the polynomial-time hierarchy collapses? What are the consequences if SAT had an NP algorithm with a unique accepting path? Remains open, again even in relativized worlds.
- Do rational functions that agree with Boolean functions on the hypercube have low decision tree complexity? I really expected someone to have come up with a proof or counterexample by now.
- What happens if two queries to NP can be simulated by a single query? Does S
^{2}=ZPP^{NP}? Both questions asked in a post on S_{2}^{P}. - Separate NP from Logarithmic space. I gave four approaches in a pre-blog 2001 survey on diagonalization (Section 3) though none have panned out. Should be much easier than separating P from NP.

# Computational Complexity

Computational Complexity and other fun stuff in math and computer science from Lance Fortnow and Bill Gasarch

## Thursday, May 05, 2016

### Open Questions

Through the years I've mentioned a few of my favorite open problems in computational complexity on this blog that have perplexed me through the years. Let me mention a few of them again in case they inspire some of the new generation of complexity theorists.

## Sunday, May 01, 2016

### Some more bits from the Gathering for Gardner

I posted about the Gathering for Gardner conference and about some of the talks I saw here. Today I continue with a few more talks.

*Playing Penney's game with Roulette*by Robert Vallen. Penney;'s game is the following: let k be fixed. Alice and Bob pick different elements of {H,T}^k. They flip a coin until one of their sequences shows up, and that person wins. Which sequences have the best probability of winning?

*New Polyhedral dice*by Robert Fathauer, Henry Segerman, Robert Bosch. This is a good example of how my mentality (and possibly yours) differs from others. When I hear ``60-sided dice'' I think ``p1,...,p60 where are all between 0 and 1 and add up to 1'' I also thought that only the platonic solids could be usedvto form fair dice (so only 4-sided, 6-sided, 8-sided, 12-sided, and 20-sided dice can be made). NOT so. These authors actually MAKE real dice and they do not have to be platonic solids. Here is their website.

*Numerically balance dice*by Robert Bosch (paper is here). Why do dice have the opposite sides sum to the same thing? Read the paper to find out!

*Secret messages in juggling and card shuffling*by Erik Demaine. Erik Demaine was one of about 4 theoretical computer scientists I met at the conference, though Erik is so well rounded that calling him a theoretical computer scientist doesn't seem quite right. I had never met him before which surprised me. In this talk he showed us some new fonts- one using juggling. See here for an example of juggling fonts, co-authored with his father Martin.

*Fibonacci Lemonade*by Andrea Johanna Hawksley. Put in the leomon and sugar in fib number increments. Here is their website. In my first post I said the talks were on a variety of topics and then presented mostly math talks. This talk is an example of that variety. There were other talks involving the Fib numbers. I was surprised by this since they aren't that special (see here).

*Penis Covers and Puzzles: Brain Injuries and Brain Health*by Gini Wingard-Phillips. She recounted having various brain injuries and how working on mathematical puzzles, of the type Martin Gardner popularized as HELPING HER RECOVER! As for the title- people with brain injuries sometimes have a hard time finding the words for things so they use other words. In this case she wanted her husband to buy some

*condoms*but couldn't think of the word so she said

*Penis Covers*instead.

Loop- Pool on an Ellipse by Alex Bellos. Similar in my mind to the Polyhedral dice talk (you'll see why). We all know that if you built an elliptical pool table with a hole at one of the foci then if the ball is placed at the other foci and hit hard enough it WILL go into the other hole. But Alex Bellos actually MAKES these pool table (see here if you want buy one for $20,000). He told us the history- someone else tried to make one in 1962 but nobody bought them (I wonder if anyone are going to buy his), and Alex had problems with friction as you may recall that it only works on a frictionless surface. So his game does require some skill. The similarity to dice is that I (and you?) are used to thinking about dice and ellipses abstractly, not as objects people actually build.

This post is getting long so I'll stop here and report more in a later post. Why so mny posts? Six minute talks that I an actually understand and are delighted to tell you about!

## Thursday, April 28, 2016

### Claude Shannon (1916-2001)

Claude Shannon was born hundred years ago Saturday. Shannon had an incredible career but we know him best for his 1948 paper A Mathematical Theory of Communication that introduced entropy and information theory to the world. Something I didn't know until looking him up: Shannon was the first to define information-theoretic security and show that one-time pads are the one and basically only code that secure.

Entropy has a formal definition, the minimum expected number of bits to represent the output of a distribution. But I view information as a more abstract concept of which entropy is just one substantiation. When you think of concepts like conditional information, mutual information, symmetry of information, the idea of an underlying distribution tends to fade away and you begin to think of information itself as an entity worth mentioning. And when you look at Kolmogorov Complexity, often called algorithmic information theory, the measure is over strings, not distributions, yet has many of the same concepts and relationships in the entropy setting.

Computational Complexity owes much to Shannon's information. We can use information theory to get lower bounds on communication protocols, circuits, even upper bounds on algorithms. Last spring the Simons Institute for the Theory of Computing had a semester program on Information Theory including including a workshop on Information Theory in Complexity Theory and Combinatorics. Beyond theory, relative entropy, or Kullback–Leibler divergence, plays an important role in measuring the effectiveness of machine learning algorithms.

We live in an age of information, growing dramatically every year. How do we store information, how do we transmit, how do we learn from it, how do we keep it secure and private? Let's celebrate the centenary of the man who gave us the framework to study these questions and so much more.

Entropy has a formal definition, the minimum expected number of bits to represent the output of a distribution. But I view information as a more abstract concept of which entropy is just one substantiation. When you think of concepts like conditional information, mutual information, symmetry of information, the idea of an underlying distribution tends to fade away and you begin to think of information itself as an entity worth mentioning. And when you look at Kolmogorov Complexity, often called algorithmic information theory, the measure is over strings, not distributions, yet has many of the same concepts and relationships in the entropy setting.

Computational Complexity owes much to Shannon's information. We can use information theory to get lower bounds on communication protocols, circuits, even upper bounds on algorithms. Last spring the Simons Institute for the Theory of Computing had a semester program on Information Theory including including a workshop on Information Theory in Complexity Theory and Combinatorics. Beyond theory, relative entropy, or Kullback–Leibler divergence, plays an important role in measuring the effectiveness of machine learning algorithms.

We live in an age of information, growing dramatically every year. How do we store information, how do we transmit, how do we learn from it, how do we keep it secure and private? Let's celebrate the centenary of the man who gave us the framework to study these questions and so much more.

## Sunday, April 24, 2016

### Some short bits from the Gathering for Gardner Conference

I attended G4G12 (Gathering for Gardner) a conference that meets every 2 years (though the gap between the first and second was three years) to celebrate the work of Martin Gardner. Most of the talks were on Recreational mathematics, but there were also some on Magic and some are hard to classify.

Martin Gardner had a column in Scientific American called Mathematical Games from 1956 to 1981. His column inspired man people to go into mathematics. Or perhaps people who liked math read his column. The first theorem I ever read outside of a classroom was in his column. It was, in our terminology, a graph is Eulerian iff every vertex has even degree.

For a joint review of six G4G proceedings see here. For a joint review of six books on recreational math including three of Gardner's, see here. For a review of a book that has serious math based on the math he presented in his column see here.

The talks at G4G are usually 6 minutes long so you can learn about a nice problem and then work on it yourself. Their were a large variety of talks and topics. Many of the talks do not have an accompanying paper. Many of them are not on original material. But none of this matters--- the talks were largely interesting and told me stuff I didn't know.

64=64 and Fibonacci, as Studied by Lewis Caroll, by Stuart Moshowitz. This was about a Lewis Caroll puzzle where he put together shapes in one way to get a rectangle of area 65, and another way to get a square of area 64, The following link is NOT to his talk or a paper of Moshowitz, but it is about the problem: here

How Math can Save your life by Susan Marie Frontczak. This was part talk about bricks and weights and then she stood on the desk and sang this song (thats not her signing it).

Twelve ways to trisect and angle by David Richeson. This was NOT a talk about cranks who thought they had trisected and angle with straightedge and compass. It was about people who used ruler, compass, and JUST ONE MORE THING. I asked David later if the people who trisected the angle before it was shown impossible had a research plan to remove the ONE MORE THING and get the real trisection. He said no- people pretty much knew it was impossible even before the proof.

The Sleeping Beauty Paradox Resolved by Pradeep Mutalik. This paradox would take an entire blog post to explains so here is a pointer to the wikipedia entry on it: here. AH, this one DOES have a paper associated to it, so you can read his resolution here

Larger Golomb Rulers by Tomas Rokicki. A Golomb Ruler is a ruler with marks on it so that the all of the distances between marks are distinct. The number of marks is called the order of the ruler. Construction a Golumb ruler is easy (e.g., marks at the 1,2,4,8,... positions I think works). The real question is to get one of shortest length. They had some new results but, alas, I can't find them on the web.

Chemical Pi by John Conway. There are people who memorize the first x digits of pi. John Conway does something else. He has memorized the digits of pi and the chemical elements in the following way:

HYDROGEN 3.141592653 HELIUM next 10 digits of pi LITHIUM etc

that is, he memorized the digits of pi by groups of 10 and separated by the chemical elements in the order they are on the Periodic table. He claims this makes it easier to answer questions like: What is the 87th digits of pi. He also claims it gives a natural stopping point for how many digits of pi you need to memorize (need? maybe want). (ADDED LATER WHEN I CORRECTED HELIUM TO HYDROGEN: here are some mnemonic devices: here.

This post is getting long so I may report on more of the talks in a later post.

## Thursday, April 21, 2016

### The Master Algorithm

We see so few popular science books on computer science, particularly outside of crypto and theory. Pedro Domingos' The Master Algorithm: How the Quest for the Ultimate Learning Machine Will Remake the World, despite the hyped title and prologue, does a nice job giving the landscape of machine learning algorithms and putting them in a common text from their philosophical underpinnings to the models that they build on, all in a mostly non-technical way. I love the diagram he creates:

Working out from the inner ring are the representations of the models, how we measure goodness, the main tool to optimize the model and the philosophies that drove that model. The book hits on other major ML topics including unsupervised and reinforcement learning.

In the bullseye you can see the "Master Equation" or the Master Algorithm, one learning algorithm to rule them all. The quest for such an algorithm drives the book, and Domingos describes his own, admittedly limited attempts, towards reaching that goal.

I diverge from Domingos in whether we can truly have a single Master Algorithm. What model captures all the inner-ring models above: circuits. A Master Algorithm would find a minimum-sized circuit relative to some measure of goodness. You can do that if P = NP and while we don't think circuit-minimization is NP-hard, it would break cryptography and factor numbers. One of Domingos' arguments states "If we invent an algorithm that can learn to solve satisfiability, it would have a good claim to being the Master Algorithm". Good luck with that.

Working out from the inner ring are the representations of the models, how we measure goodness, the main tool to optimize the model and the philosophies that drove that model. The book hits on other major ML topics including unsupervised and reinforcement learning.

In the bullseye you can see the "Master Equation" or the Master Algorithm, one learning algorithm to rule them all. The quest for such an algorithm drives the book, and Domingos describes his own, admittedly limited attempts, towards reaching that goal.

I diverge from Domingos in whether we can truly have a single Master Algorithm. What model captures all the inner-ring models above: circuits. A Master Algorithm would find a minimum-sized circuit relative to some measure of goodness. You can do that if P = NP and while we don't think circuit-minimization is NP-hard, it would break cryptography and factor numbers. One of Domingos' arguments states "If we invent an algorithm that can learn to solve satisfiability, it would have a good claim to being the Master Algorithm". Good luck with that.

## Monday, April 18, 2016

### Its hard to tell if a problem is hard. Is this one hard?

Here is a problem I heard about at the Gathering for Gardner. Is it hard? easy? boring? interesting? I don't know.

Let N={1,2,3,...}

PROBLEM: parameters are s (start point) and f (not sure why to call it f). both are in N

Keep in mind the sequence, in order, of operations:

DIVIDE BY f, SUBTRACT f, ADD f, MULTIPLY by f.

form the following sequence of numbers in N

a(0)= s

Assume a(0),...,a(n) are known. Let A = {a(0),...,a(n)}. N-A are the elements in N that are NOT in A.

If a(n)/f is in N-A then a(n+1)=a(n)/f

Else

If a(n)-f is in N-A then a(n+1)=a(n)-f

Else

If a(n)+f is in N-A then a(n+1)=a(n)+f

Else

If a(n)*f is in N-A then a(n+1) = a(n)*f

Else

If none of the above holds then the sequence terminates.

Lets do an example! Let a=14 and f=2

14, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 24, 22, 11, 9, 18, 16, 32, 30, 15, 13, 26, 28, 56, 54, 27, 25, 23, 21,

19, 17, 34, 36, 38, 40, 20, STOP since 10, 18, 22, 40 are all on the list.

Lets do another example! Let a=7, f=2

7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 9, 11, 13, 15, 17, ... (keeps going)

If f=2 and you get to an odd number x so that ALL of the odds less than x have already appeared but NONE of the odd numbers larger than x have appeared, then the sequence will go forever

with x, x+2, x+4, ...

QUESTIONS and META QUESTIONS

1) Can one characterize for which (s,f) the sequence stops.

2) Is it decidable to determine for which (s,f) the sequence stops.

3) Both (1) and (2) for either fixed s or fixed f.

4) Are the above questions easy?

5) Are the above questions interesting?

There are four categories:

Easy and Interesting- Hmmm, if its TOO easy (which I doubt) then I supposed can't be interesting.

Easy and boring.

Hard and interesting. This means that some progress can be made and perhaps connections to other mathematics.

Hard and Boring. Can't solve and are not enlightened for the effort.

Let N={1,2,3,...}

PROBLEM: parameters are s (start point) and f (not sure why to call it f). both are in N

Keep in mind the sequence, in order, of operations:

DIVIDE BY f, SUBTRACT f, ADD f, MULTIPLY by f.

form the following sequence of numbers in N

a(0)= s

Assume a(0),...,a(n) are known. Let A = {a(0),...,a(n)}. N-A are the elements in N that are NOT in A.

If a(n)/f is in N-A then a(n+1)=a(n)/f

Else

If a(n)-f is in N-A then a(n+1)=a(n)-f

Else

If a(n)+f is in N-A then a(n+1)=a(n)+f

Else

If a(n)*f is in N-A then a(n+1) = a(n)*f

Else

If none of the above holds then the sequence terminates.

Lets do an example! Let a=14 and f=2

14, 7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 24, 22, 11, 9, 18, 16, 32, 30, 15, 13, 26, 28, 56, 54, 27, 25, 23, 21,

19, 17, 34, 36, 38, 40, 20, STOP since 10, 18, 22, 40 are all on the list.

Lets do another example! Let a=7, f=2

7, 5, 3, 1, 2, 4, 6, 8, 10, 12, 14, 16, 18, 9, 11, 13, 15, 17, ... (keeps going)

If f=2 and you get to an odd number x so that ALL of the odds less than x have already appeared but NONE of the odd numbers larger than x have appeared, then the sequence will go forever

with x, x+2, x+4, ...

QUESTIONS and META QUESTIONS

1) Can one characterize for which (s,f) the sequence stops.

2) Is it decidable to determine for which (s,f) the sequence stops.

3) Both (1) and (2) for either fixed s or fixed f.

4) Are the above questions easy?

5) Are the above questions interesting?

There are four categories:

Easy and Interesting- Hmmm, if its TOO easy (which I doubt) then I supposed can't be interesting.

Easy and boring.

Hard and interesting. This means that some progress can be made and perhaps connections to other mathematics.

Hard and Boring. Can't solve and are not enlightened for the effort.

## Thursday, April 14, 2016

### Who Controls Machine Learning?

After AlphaGo's victory, the New York Times ran an article The Race Is On to Control Artificial Intelligence, and Tech’s Future.

Some of my readers' comments forced me to rethink my hasty tweet. First, Google, Microsoft and Amazon can create ML infrastructure, cloud hardware that optimizes computational power and storage for machine learning algorithms to get a level of data analysis that one couldn't replicate in software alone.

More importantly, Google etc. have access to huge amounts of data. Cloud companies can provide pretrained machine learning algorithms. Google provides image classification, voice transcription and translation. Microsoft offers face and emotion detection and speech and text analysis. One could imagine, in the absence of privacy issues, Google taking your customer data, matching with data that Google already has on the same customers to draw new inferences about how to market to those customers better.

With almost all our computing heading to the cloud, cloud computing providers will continue to compete, and provide continuing better tools in machine learning and beyond. Eventually will one company "control AI"? That would surprise me but we may still end up with an AI oligarchy.

A platform, in technology, is essentially a piece of software that other companies build on and that consumers cannot do without. Become the platform and huge profits will follow. Microsoft dominated personal computers because its Windows software became the center of the consumer software world. Google has come to dominate the Internet through its ubiquitous search bar. If true believers in A.I. are correct that this long-promised technology is ready for the mainstream, the company that controls A.I. could steer the tech industry for years to come.I then tweeted "Can a company control AI? More likely to become a commodity." The major machine learning algorithms are public knowledge and one can find a number of open-source implementations including Google's own TensorFlow that powered AlphaGo. What's to stop a start-up from implementing their own machine learning tools on the cloud?

Some of my readers' comments forced me to rethink my hasty tweet. First, Google, Microsoft and Amazon can create ML infrastructure, cloud hardware that optimizes computational power and storage for machine learning algorithms to get a level of data analysis that one couldn't replicate in software alone.

More importantly, Google etc. have access to huge amounts of data. Cloud companies can provide pretrained machine learning algorithms. Google provides image classification, voice transcription and translation. Microsoft offers face and emotion detection and speech and text analysis. One could imagine, in the absence of privacy issues, Google taking your customer data, matching with data that Google already has on the same customers to draw new inferences about how to market to those customers better.

With almost all our computing heading to the cloud, cloud computing providers will continue to compete, and provide continuing better tools in machine learning and beyond. Eventually will one company "control AI"? That would surprise me but we may still end up with an AI oligarchy.

## Sunday, April 10, 2016

### What Rock Band Name would you choose?

I looked up my colleague Dave Mount on Wikipedia and found that he was a drummer for the glam rock band Mud. He informed me that (a) on Wikipedia he is David Mount and (b) if he had a rock band it would be named

*Fried Purple Ellipsoids.*

*This set off an email discussion where people said what their rock band name would be. I noticed that many ideas for names had variants. For example, my favorite for Ramsey Theorists:*

*The Red Cliques*could be

*The Red Cliques*

*Red Clique*

*Bill Gasarch and the Red Cliques!*

*Clique!*

So below I list one variant of each name but keep in mind that there are others.

*The Hidden Subgroups*

*Amplitudes with Attitude*

*Schrodinger's cat (I wonder if this IS a rock band already)*

*The Red Cliques*

*Fried purple ellipsoids*

*Fried green ellipsoids*

*BIG A-G-T*

*The Biconnected Sets*

*PRAM!*

*BPP! (I wonder if any complexity class would work.)*

*SAT (I wonder if other one-word problems would work. TSP!)*

*Karp and the reductions*

*Avi and the derandomizers*

*Aravind and the Expanders*

*(Could replace Karp, Avi, and Aravind with others, but these are the first that*

*came to mind. Plus THE EXPANDERS was Aravind Srinivasan's idea.)*

*The MIT Logarhythms*(This is a real acapella group see here.)

*The Discrete Logarhythms*

*RSA!*

*The Oracles*

*The Interactive Proofs*

*The Natural Proofs*

*Fried Green Proofs*

*If we expand to include math we get lots more, so I'll just mention one real one: The Klein Four, an acapella group.*

SO- what would YOUR rock band name be?

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