Sunday, April 28, 2024

Math Thoughts Inspired by the TV show Succession

I watched Succession one-episode-a-day on treadmill for 39 days. I'm glad I did this in 2023 since Season 2 aired its last show on Oct 19, 2019, and Season 3 had its first show on Oct 17, 2021, so I would have been in suspense (or at least as much suspense as corporate board meetings can generate) for about 2 years. 

The show inspired the following mathematical thoughts.

1) There was a scene which I paraphrase as follows:

Alice: I'll give you two billion dollars for your shares in the company.

Bob: Don't insult me. It's worth at least 2.5 billion. 

My thought: I would take the two billion. 

My Math Thought: Let's say the shares really were worth 2.5 billion. Is it worth haggling? I think not, since I can't imagine I will ever spend that much money in my life. (See the blog post here  on the St. Petersburg paradox which is about  how much money is enough.) To be fair, if I wanted to buy Tesla (note that I mean buying the company, not buying the car) or buy the Comedy Cable Station (I would run it so that it only airs funny commercials) then I might need the extra half a billion to even begin trying (Tesla is worth around 740 billion).  SO the question of is it worth the haggling depends on your situation. 

So let's assume you don't need the money for some large purchase. Would you haggle? One reason to haggle is that you don't want the person who is ripping you off by only offering 2 billion to get away with it and/or think you are a chump. Whether one cares about this might depend on the relationship you have with that person. Even so, I would just take the 2 billion. Perhaps that's why I am in academia instead of business.

MATH QUESTION: Can we quantify what amount of money its not worth haggling over? 

2) Alice wants to buy Bob's company. After months of negotiations they agree to x dollars (and there are many side issues as well). The next day 

Alice thinks: OH, if he was willing to sell for x, he would be willing to sell for x-1. 

Bob thinks:OH, if she was willing to buy for x, he would be willing to buy for x+1.

(In the show this scenario happened many times, but usually with only one party wanting to re-negotiate and its not just  +1 and -1, its things like seats-on-the-board, who-will-be-CEO.) 

MATH QUESTION:  If Alice and Bob behave as above is there any mechanism to make them actually come to an agreement? Might involve assuming they can't factor fast or can't solve NPC problems fast.

Wednesday, April 24, 2024

Is Persistence an Anachronism?

Guest post by Martin Bullinger

Very recently, Vijay Vazirani's paper A Theory of Alternating Paths and Blossoms, from the Perspective of Minimum Length got accepted to Mathematics of Operations Research. For the first time, it gives a complete and correct proof that the Micali-Vazirani algorithm finds a maximum cardinality matching in time \(\mathcal O\left(m\sqrt{n}\right)\). I would like to give an account of the extraordinary story of this proof and how Vazirani's contribution inspires persistence.

My fascination for matching already started during my undergrad when I gave a talk on Edmonds' blossom algorithm. It was at this time that I first heard about the Micali-Vazirani (MV) algorithm. Naturally, I was quite excited when I got to know Vazirani personally years later. When I talked to him about the MV algorithm I was, however, shocked: Vazirani admitted that even to that day, there did not exist a complete proof of its correctness. How can a theoretical result be accepted to FOCS without a proof?

Now, 44 years after publication of the algorithm, a proof exists and has been peer-reviewed in great depth. But why did it take so long? Apparently, some results just need time. Sometimes a lot of time. Think of Fermat's Last Theorem, whose proof took 358 years! So what is the story behind the MV algorithm? It can without a doubt be seen as a lifework. Together with his fellow PhD student Silvio Micali, Vazirani discovered it in the first year of his PhD in 1979-80. Without even attempting a proof, it was published in the proceedings of FOCS 1980. The first proof attempt by Vazirani was published in 1994 in Combinatorica. Unfortunately, this proof turned out to be flawed. It took another 30 years until his current paper.

What kept Vazirani going for so long? In the acknowledgements of his paper, he thanks matching theory for its gloriously elegant structure. Vazirani was driven by his passion for the subject matter---but passion by itself can only go so far. Even more important was his belief in the correctness of the algorithm and the theory, which he had broadly outlined in his 1994 paper. Similar to Andrew Wiles' story, his perseverance led him to the idea which clinched the proof. In Vazirani's case, this was to use the new algorithmic idea of double depth-first search, which forms the core of the MV algorithm, and now, its proof as well. But Vazirani's result is also the story of an excellent research environment. Finding deep results requires colleagues or friends to discuss ideas with. Vazirani had these in the form of strong postdocs and PhD students. About ten years ago, he had been discussing ideas towards his proof with his former postdoc Ruta Mehta, and in the last three years, he discussed the final touches of his proof with his current PhD student Rohith Gangam. Needless to say, both of them gained a lot from these discussions.

So why should we care for the MV algorithm? I have several reasons. First, without doubt, it is a historic result within combinatorial optimization. Matching is one of the most fundamental objects in discrete mathematics and we keep finding new applications for it, for example, in health, labor markets, and modern day matching markets on the Internet, basically in every part of our lives. But there is more. Once again, one can look at Vazirani's paper where he describes the impact of matching to the development of the theory of algorithms: Matching theory has led to foundational concepts like the definition of the complexity classes \(\mathcal P\) (Edmonds, 1965a) and \(\# \mathcal P\) (Valiant, 1979), the primal-dual paradigm (Kuhn, 1955), and polyhedral combinatorics (Edmonds, 1965b). The impact of matching on complexity theory was an earlier topic of this blog.

Despite being around for decades, the MV algorithm is still the fastest known algorithm for computing a maximum cardinality matching. This is surprising, to put it mildly. Similar to many other fundamental problems in combinatorial optimization, I would have expected the discovery of better algorithms in the last four decades. Why has this not happened? Vazirani appears to have gotten to the essence of the problem: a profound theory that interleaves algorithmic invariants and graph-theoretic concepts. It seems to be the kind of theory which would play an active role in the field of combinatorial optimization.

However, Vazirani's result proves something else, possibly even more important: the massive gains to be made by single-minded persistence. In a world in which departments and promotion procedures focus on publishing large numbers of papers, it seems impossible to work on one result for more than a year, let alone for decades. Vazirani managed to achieve both: pursue his passion and get the unfinished job done, but not let it come in the way of the rest of his otherwise-active research career. As a young researcher, this inspires me! In the end, it is through such persistence that science will take big steps forward.

This blog post evolved from many enjoyable discussions, which I had with Vijay Vazirani during a research stay at UC Irvine in spring 2024. I am grateful to Ruta Mehta for feedback on the initial version of this post. Vazirani recently presented his paper in a mini series of two talks available online.

Sunday, April 21, 2024

Intelligent Comments on Bill's G.H. Hardy/Avi W post that we did not post.

I posted (see here) about Avi Wigderson being a counterexample to two of G.H. Hardy's opinions:

1) Hardy thought Math was a young man's game. I got some good comments on this. Some agreed and some disagreed.

2) Hardy thought applied math is dull. I got no comments on this one. I assume everyone agreed with my assessment that Hardy was wrong about this.

AND I got the following comment: 

Avi Wigderson's brilliance shatters the false assumptions of  G.H. Hardy, proving that intelligence knows no limits. His groundbreaking ideas challenge the status quo and inspire a new generation of thinkers.

I agree! I was going to post it! Then I realized that 
a) There were two links below it. A bad sign

b) While Avi is a brilliant thinker, is he known for challenging the status quo? The NW result pointed to P=BPP which was not thought to be true at the time. (Micheal Sipser thought P=BPP, see here, but he was an outlier. If my memory is wrong then leave a polite comment.)  Aside from that result  I don't think Avi's  work challenges the status quo. If I am WRONG leave a polite comment correcting my error. 

c) As a side note- I think of challenging the status quo as being over-rated. Wiles proof of FLT did not challenge the status quo. Many great results do not challenge the status quo. If Avi shows P NE NP that would be awesome even though it is the prevailing opinion. 

d) Upon reading the comment carefully,  it looks like it was AI generated. 

e) It fooled me for about 5 minutes. It didn't fool Lance for a second. 

3) But this comment inspired Lance to use ChatGPT to generate comments to my post: one agreeing, and one disagreeing

Avi Wigderson’s recent Turing Award is a compelling moment to reflect on the fluid nature of mathematical careers and the erroneous stereotypes surrounding them. G.H. Hardy’s assertion in *A Mathematician’s Apology* that mathematics is a young man's game is eloquently countered by Avi’s illustrious career and the achievements of others well beyond the supposed prime age. This challenges Hardy's notion and suggests it warrants reevaluation as a mere conjecture rather than a stated fact.

Moreover, Hardy’s dismissal of applied mathematics overlooks the dynamic interplay between various disciplines. Even in his era, the boundaries between pure and applied math, along with physics and computer science, were permeable and productive. Avi Wigderson’s work, though not strictly applied math, beautifully illustrates how applied considerations can drive significant theoretical advancements.

In this light, the recognition of Wigderson’s contributions is not just a celebration of his individual genius but also a testament to the evolving and interconnected landscape of mathematics, which continues to defy the narrow confines set by earlier academic opinions.

This is an intelligent comment; however, it mostly reflects back what I said. Indeed, it is a more eloquent exposition of what I said. Fortunately I can't be fired from blogging and replaced with an AI since I don't get paid anyway. 

While Avi Wigderson's Turing Award certainly highlights his exceptional contributions to theoretical computer science, invoking this as a refutation of G.H. Hardy's views on mathematics as a young man's game might be an oversimplification. Hardy's assertion, though controversial, was not without merit when considering the context of his times and the examples he witnessed firsthand. It's important to remember that exceptions like Wigderson do not necessarily disprove the rule but rather highlight remarkable outliers.

Furthermore, Hardy's critique of applied mathematics as being dull must be understood in the philosophical context of his personal commitment to pure mathematics. While we may not agree with his perspective, dismissing it entirely fails to appreciate the depth of passion that fueled his work and the work of many pure mathematicians. The interplay between disciplines enriches mathematics, indeed, but Hardy’s emphasis on the beauty of pure theory has inspired generations and continues to hold significant value in the mathematical community. 
This does not seem to be just reflecting back what I wrote. 
The argument about Avi is an outlier is reasonable. 
Here is my counterargument for defending Hardy's negative opinion of applied math:  just because the  notion that applied math is dull fueled passion does not make it correct. And one must also see what harm the notion caused.
 If I debated a human on this point they might change their mind, or give better counterarguments. Can ChatGP change its mind? Does that question even make sense? I suspect that ChatGPT could give better or at least different counterarguments.

Thursday, April 18, 2024

Favorite Theorems: Quantum Provers

March Edition

For our next favorite theorem, we look at the surprising power of provers who share entangled bits. If you can prove something to an arbitrarily computable verifier, then two entangled provers can convince a polynomial-time verifier.

Zhengfeng Ji, Anand Natarajan, Thomas Vidick, John Wright and Henry Yuen

A quick tour:

  • A powerful prover convincing a skeptical computable deterministic verifier is another way of capturing computably enumerable (traditionally called recursively enumerable or RE). You can convince a verifier that a program halts by giving the halting time, and the verifier can simulate the machine for that many steps.
  • A powerful prover convincing a skeptical polytime deterministic verifier is another way of capturing the class NP, like giving a 3-coloring of a graph that can be easily checked.
  • If you allow the verifier to ask random questions, you can convince the verifier with high confidence that a graph is not colorable, or more generally any PSPACE problem.
  • If you add two separated provers that a skeptical probabilistic verifier can play off each other, the provers can convince the verifier that any problem in NEXP, non-deterministic exponential time.
One of many quantum variations of interactive proofs, MIP* has two provers that cannot communicate but have entangled quantum bits. This change could go either way: 
  • The provers to coordinate their answers better and so they wouldn't convince the verifier for all the languages in NEXP
  • The verifier could ask more complex questions to the provers which they could answer using the entanglement, allowing the provers to convince the verifier for even more complex languages.
Turns out it's the later in a very strong way.

Ito and Vidick showed that you can create a protocol that prevents the provers coordinating better, recovering all problems in NEXP. Natarajan and Wright showed you can ask more questions, showing that provers with entangled bits can convince a verifier of everything in NEEXP, non-deterministic double exponential time (\(2^{2^{n^c}}\)), already a proof too large for the verifier to even point into. The MIP* = RE paper takes that all the way to the computably enumerable sets, all the languages you would get with a classical prover convincing a deterministic verifier unrestricted by time.

Monday, April 15, 2024

Avi Wigderson is a counterexample to TWO stupid thoughts of G.H. Hardy


1) Avi Wigderson won the Turing Award (See blog posts by Fortnow-here, Scott-here, Lipton-Regan here, and the ACM announcement here).  The last time I could find when Fortnow-Gasarch, Scott, Lipton-Regan all blogged on the same topic was when Goldwasser-Micali won the Turing Award- see the blog entries (here, here,here). We rarely coordinate. For that matter, even Fortnow and Gasarch rarely coordinate.

2) My joint book review of G.H. Hardy's  A Mathematician's Apology (1940) and L.N. Trefethen's An Applied Mathematician's Apology appeared in SIGACT News. 

These two events would seem unrelated. However, I criticize two points in Hardy's book; and those two points relate to Avi.  The book review is here

POINT ONE: Hardy says that Mathematics is a young man's game and that if you are over 40 then you are over the hill. He gives some fair example (Gauss, Newton) and some unfair ones (Galois, Ramanujan who died before they were 40.) Rather than STATE this fact he should have made it a CONJECTURE to be studied. I would make it two conjectures: 

Was it true for math that Hardy would know about? Since most people died younger in those days, there might be to small a sample size. Euler and Leibniz might be counterexamples.

Is it true now? AVI is clearly a counterexample. Other modern counterexamples: Michael Rabin, Leslie  Valiant, Roger Apery (proved zeta(3) irrational at the age of 62), Yitang Zhang (bounded gaps between primes at age 58, which, alas, is not a prime-- would have been really cool if it was a twin prime), Louis de Branges (proof of the Bieberbach conjecture at 51), Andre Weil, Jean-Pierre Serre. Is that enough people to disprove Hardy's conjecture? 

Despite the counterexamples I provided, we have all seen some mathematicians stop producing after a time. I offer two reasons for this

a) (Andrew Gleason told me this one) A mathematician works in a field, and the field dries up. Changing fields is hard since math has so much prereq knowledge.  CS has less of that problem. One can see if in the counterexamples above, and in other counterexamples, the fields they were in didn't dry up. 

b) The Peter Principle: Abosla is a great research so lets make her department chair!

My conjecture: The notion that math is a young mans game is false. 

POINT TWO: Applied Math is dull. Trefethan's book makes a good counter argument to this. I will say something else.

Even in Hardy's time he would have seen (if his head was not so far up his ass) that math, applied math, physics, compute science and perhaps other areas interact with each other. It is common to say that things done in pure math get applied. However, there are also cases where pure math uses a theorem from applied math. Or where Physics MOTIVATES a topic in pure or applied math. The boundaries are rather thin and none of these areas has the intellectual or moral high ground. There is the matter of personal taste, and if G.H. Hardy prefers pure math, that's fine for him. But he should not mistake his tastes for anything global. And is well known, he thought pure math like number theory would never apply to the real world. He was wrong about that of course. But also notice that Cryptography motivated work in number theory.  I am not sure if I would call AVI's work applied math,but it was certainty motivated by applied considerations.

Wednesday, April 10, 2024

Avi wins the Turing Award

The ACM announced that Avi Wigderson, a force in computational complexity and beyond, will receive the 2023 A. M. Turing Award (Quanta article). This is the first primarily complexity theorist to win the award since Andy Yao in 2000. Avi can add this to his AbelNevanlinna and Knuth prizes. Avi has served on the faculty at the Institute for Advanced Study since 1999 after many years at Hebrew University. He's hosted many postdocs and visiting faculty and students who've gone onto great careers of their own.

Avi is my first co-author to win the Turing award. Our paper was just one link in a chain of papers, from Nisan-Wigderson to Impagliazzo-Wigderson showing how circuit bounds yield derandomization. Philosophically these results tell us randomness is just computation we cannot predict.

But Avi did so much more. NP has zero-knowledge proofs. Zig-Zag expanders that led to Reingold's SL = L. Monotone circuit lower-bounds using communication complexity. Upper and lower bound for matchingOptimal Extractors. And that's just the tip of the iceberg. 

Notably none of these papers are solely authored or even have much overlap in their author lists. Avi shared his wisdom with many, nearly 200 distinct co-authors according to DBLP. 

Beyond his research, Avi has been a great advocate for our field, advocating to the NSF as a founding member of the SIGACT Committee for the Advancement for Theoretical Computer Science, and on the Simons Foundation scientific board which led to Simons Fellows and the Simons Institute. Avi wrote a book placing computational complexity as a great mathematical discipline that just also happens to have great applications in so many different fields.

Congrats to Avi and this capstone to an incredible career and individual. 

Tuesday, April 09, 2024

Rance Cleaveland passed away on March 27, 2024. He will be missed

My friend and colleague Rance Cleaveland passed away on March 27, 2024 at the age of 62.  He was a professor at The University of Maryland at College Park in the Computer Science Department. He worked in Software Engineering. He did program verification and model checking. He had his own company, so he did both theoretical and practical work.

He joined UMCP in 2005. I had known him and some of his work before then so we got together for lunch about once a month to talk about the department(he was new to the dept. so I filled him in on things) and about computer science.He would probably be considered a theorist in Europe, though he was considered a Software Engineer in America.

The department's announcement is here

Below is

1) A note from Rance's Grad student Peter Fontana, who got his PhD in 2014.

2) An email that Jeroen Keiren sent to the concurrency mailinglist.

3) A picture of Peter, Jeroen, and Rance in that order left to right. 


Peter Fontana's Note:


I’m truly shocked and saddened to hear that Rance Cleaveland passed away. Rance advised me (as a Ph.D. student of his at UMCP) in model checking (also called formal verification).

Rance was an extremely kind advisor and extremely skilled in leadership and communication. He also had all of the swiftness, communication, and people understanding of a skilled manager. He always encouraged us to find the simplest example possible to illustrate a nuanced corner-case of a property we wanted to prove or disprove. The simplicity made complicated things clearer. He was also an extremely clear communicator and extremely skilled with people. Rance was always patient and kind, eager to guide rather than to chastise.  I will truly miss him.


Model checking involves the following:

(1) abstracting a programs as a state machine (automaton) with labels,

(2) writing a desirable property (such as “bad event X will never happen”) as a formula in a logic,

(3) using a computer to automatically show (over all possible cases) that the specified property is true or false.  This is model checking.

Theorists will note that this process of model checking is asking if state q of a machine satisfy a property phi, which is the model checking problem. If you are in the world of boolean formulas and propositions, the NP-complete satisfiability problem (SAT) asks: does there exists a boolean assignment q that s satisfy formula \phi? The analogous model checking problem is: given boolean assignment q (each proposition is either T or F), does q satisfy \phi? For boolean assignments, the model-checking problem is in P.

While Rance worked in a variety of areas related to formal verification that spanned process algebras, different logics, different automata types, and cyber-physical systems, with me we improved the art of timed automata model checking using a timed modal-mu calculus (timed logic). Timed automata and timed logics extend state machines by introducing a finite number of clocks. These clocks all advance (like time advancing) and can reset, and timed logics now have timing constraints (“within T time” is the most common constraint). We worked on extending the state of the art of what we could model-check on timed automata, both with theoretical proofs and by implementing a model checker (in C++) to model-check these richer timed properties. It turns out that this model checking work is decidable (model checking the simplest formulas in timed automata was previously shown to be PSPACE HARD).  I inherited work started by others, enhanced it, and passed it on; that work is currently being enhanced by others today. Our approach was novel in that we used a proof system of predicates and were able to model check more expressive formulas on timed automata with this enhanced system (See this paper). For details see my PhD thesis here.


Jeroen Keiren's message to the concurrency mailinglist

Dear colleagues,

It is with great sadness that I share the news of the sudden and unexpected passing of
our colleague Rance Cleaveland on 27 March 2024.

My thoughts are with his family, friends and loved ones during this sad time.

I am convinced that those of us that interacted with Rance throughout his career will remember him as a friendly person. He was always happy to discuss research, but also dwell on more personal topics or give his honest, yet in my experience always constructive, advice.

Rance obtained his PhD at Cornell, and held academic positions at Sussex, NC State and Stony Brook, before accepting his current position at the University of Maryland, College Park in 2005. UMD shared an obituary for Rance here.

Rance was not only interested in the theoretical foundations of our field, witnessed by over 150 scientific publications. He also had a strong focus on building the tools (for instance the Concurrency Workbench), and commercializing it (through his company Reactis).

Rance was also an active member of our community. He was one of the co-founders of TACAS, for which he was still serving on the steering committee, as well as one of the co-founders of the Springer journal Software Tools for Technology Transfer.

In the words of his wife “For Rance, the concurrency community was truly his intellectual home and he appreciated working with colleagues in Europe and the US - and around the world.”

I’m sorry to bring this news.

With kind regards,

Jeroen Keiren
Assistant professor, Formal System Analysis group
Eindhoven University of Technology, The Netherlands


A Picture of Peter, Jeroen, and Rance, left to right:

Wednesday, April 03, 2024

Answer to Question. MetaQuestion remains unsolved

 In a prior post I asked the following question:

find x,y,z positive natural numbers such that the following is true:

$$ \frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} = 4. $$

I first saw the question in a more fun way:

I did not put the answer in the post (should I have? That was the meta question.)

The question has an infinite number of (x,y,z) that work, so I'll just give the least one:

x= 154476802108746166441951315019919837485664325669565431700026634898253202035277999

y = 36875131794129999827197811565225474825492979968971970996283137471637224634055579

z = 4373612677928697257861252602371390152816537558161613618621437993378423467772036

1) For details on how you could have found the answer see here. Or watch a YouTube video on it here.

2) Did I really expect my readers to get this one? Note that I posted it on April Fools Day, though it is a legit problem with a legit answer.

3) The image that says that 95% of all people couldn't solve it---I wonder what their sample size was and where it was drawn from. I suspect that among mathematicians 99% or more can't solve it. 

4)  Comments on the comments I got:

a) Austin Buchanan says that Wolfram Alpha says NO SOLUTION. I wonder if Wolfram Alpha cannot handle numbers of this size.

b) Anonymous right after Austin had a comment that I MISREAD as saying that they found it using a python program. I asked that person to email me, and it turns out that NO-- they recalled where to look (on the web I assume).

c) Several commenters solved it by  looking at the web. Math Overflow and Quora had solutions. So did other places. This may make the meta question should a blogger post the solution a moot point for a well known problem. If you get a problem off the web its quite likely its well known, or at least well enough known, to have the answer also on the web. If you make up a problem yourself then its harder to tell.

5) I think its a very hard problem to solve unless you have the prior KNOWLEDGE to solve it, so it would not be a good math competition problem. 

6) The cute pictures of fruit in the presentation of the problem makes it LOOK like its a cute problem. It not. 

7) Only one comment on the meta question about should a blogger post the solution at the same time as the problem  (There were more comments about the unimportant question of whether 0 is a natural number.) The one comment says that a blogger SHOULD NOT - let the reader enjoy/agonize for a while. I agree.

8) Determining if a given math problem is interesting is a hard problem; however, that will be a topic for another blog. (Tip for young bloggers, if there are any (blogs are so 2010):  If you do ONE idea per blog then your blog can last longer.)

Monday, April 01, 2024

A Math Question and a Meta Question

 1) Question: find x,y,z natural numbers such that the following is true:

$$ \frac{x}{y+z} + \frac{y}{x+z} + \frac{z}{x+y} = 4. $$

I was first presented the problem a more fun way:

(NOTE- a commenter pointed out that ``Natural Numbers'' and `Positive Whole Values' are different since some people (and I AM one of them) include 0 as a natural. SO- to clarify, I want x and y and z to be naturals that are \(\ge 1\). )

2) Meta Question: When a blogger poses a question like that should they also post the answer? Have a pointer to the answer? Not have an answer? Pros and Cons:

a) If you do not list the answer at all (or post it in a few days) then people will not be tempted to look at the answer. They have to really work on it. (Maybe they can find the answer on the web). 

b) There are people whose curiosity far exceeds their ego. So they want to work on in for (say) 5 minutes and then LOOK AT THE ANSWER! I am one of those people. 

c) When you first look at a problem and work on it you are curious. If you have to wait a few days to get the answer then you may lose interest. 

I invite you to work on both the question and the meta question.  I will not be blocking people who post the answer in the comments, so if you want to work on it you might not want to look at the answers.

I will post the answer in a few days.