Sunday, December 15, 2024

Random Thoughts on AI (Human Generated)

 (I wrote this post without any AI help. OH- maybe not- I used spellcheck. Does that count? Lance claims he proofread it and found some typos to correct without any AI help.)

Random Thought on AI

I saw a great talk on AI recently by Bill Regli, who works in the field. 

Announcement of the talk: here

Video of the talk:  here

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1) One item Bill R mentioned ws that AI requires lots of Energy so
3-mile Island is being reopened. See here.

Later I recalled the song

        The Girl from 3-Mile Island

to the tune of

        The Girl from Ipanema.

The song is in my audio tape collection but that is not useful so I looked for it on the web. The copy on YouTube doesn't work; however, this website of songs about 3-mile island here included it.

In the 1990's I was in charge of the Dept Holiday Entertainment since I have an immense knowledge of, and collection of, novelty songs- many in CS and Math.

Today- My talents are no longer needed as anyone can Google Search and find stuff. I did a blog on that here. I still have SOME advantage since I know what's out there, but not as much. Indeed, AI can even write and sing songs. I blogged about that and pointed to one such song here.

SO, some people's talents and knowledge are becoming obsolete.  On the level of novelty songs I am actually HAPPY that things change- I can access so much stuff I could not before. But humans becoming obsolete is a serious issue of employment and self worth. Far more serious then MACHINES TAKE OVER THE WORLD scenarios.

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2) When technology made farming jobs go away, manufacturing jobs took their place. That was true in the LONG run, but in the SHORT run there were starving ex-farmers. The same may happen now.

(ADDED LATER; someone emailed me that Machines taking over farming and other things has caused standards of living to go up. YES, I agree- in the LONG run very good, but in the short run people did lose their livelihoods.)

Truck Drivers and Nurses may do better than Accountants and Lawyers:

Self Driving trucks are 10 years away and always will be.
Nurses need to have a bedside manner that AI doesn't (for now?).

One ADVANTAGE of AI is that if it makes white collar workers lose jobs the government might get serious about

Guaranteed Basic Income, and

Univ. Health care

(ADDED LATER: someone emailed me that there GBI is not the way to go. Okay, then I should rephase as when white collar workers lose their jobs then the problem of a social saftey net will suddently become important.) 

Similar: If global warming makes the Cayman Island sink then suddenly Global Warming will be an important problem to solve.

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3) An example of AI taking away jobs is the Writers Strike.

OLD WAY: There were 10 people writing Murder She Wrote Scripts.

NEW WAY: AN AI generates a first draft and only needs 2 people to polish it.

KEY: In a murder mystery the guilty person is an innocuous character you saw in the first 10 minutes or a celebrity guest star. Sometimes the innocuous character is the celebrity guest star.

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4) ChatGPT and school and cheating.

Calculator Scenario: We will allow students to use Chat GPT as we now allow calculators. Students are not as good at arithmetic, but we don't care.  Is Chat GPT similar?

Losing battle scenario: Ban Chat GPT

My solution which works--- for now: Ask questions that Chat GPT is not good at, allow chat GPT, insist the students understand their own work, and admit they used it. Works well in Grad courses and even senior courses. Might be hard in a Freshman courses.

Lance's Solution--- Stop giving out grades. See here

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5) Bill R said that we will always need humans who are better at judgment.

Maybe a computer has better judgment. I blogged on this here

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6) I asked two AI people at lunch if the AI revolution is just because of faster computers and hence is somewhat limited. They both said YES.

SO- could it be that we are worrying about nothing?

This also may be an issue with academia: if we hire lots of AI people because it's a hot area, it may cool off soon. Actually I thought the same thing about Quantum Computing, but I was wrong there.

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7) LLM's use LOTS of energy. If you get to ask one How do we solve global warming? they might say

First step: Turn me off!

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8) Scott did a great  blog post about the ways AI could go. See here.

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9) I recently emailed Lance a math question.

He emailed me the answer 5 minutes later.

I emailed that I was impressed

He emailed that he just asked  Chat GPT. He had not meant to fool me, he just assumed I would assume that. Like if you asked me what 13498*11991 was and I answered quickly you would assume I used a calculator. And if there is a complicated word in this post that is spelled correctly then you would assume I used spellcheck - and there is no embarrassment in that.

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10) If a painting is done with AI does any human get credit for it?

I always thought that people who forge paintings that look JUST LIKE (say) a van Gogh should be able to be honest about what they do and get good money since it LOOKS like a van Gogh who cares that it is NOT a van Gogh.  Same with AI- we should not care that a human was not involved.

IF an AI finds a cure for cancer, Great!

If an AI can write a TV series better than the human writers, Great!

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11) AI will force us to make moral choices. Here is a horrifying scenario:

Alice buys a self-driving car and is given some options, essentially the trolley problem:

If your car has to choose who to run over, what do you choose?

You have the option of picking by race, gender, age, who is better dressed, anything you want.

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12) Climate Change has become a political problem in that

Democrats think it IS a problem
Rep think it is NOT a problem

Which is a shame since free-market solutions that would normally appeal to Reps are not being done (e.g., a Carbon Tax). Indeed, we are doing the opposite- some states impose a tax on Hybrid cars


SO- how will AI go with politics? Scenarios

a) Dems are for regulation, Reps are against it. Elon Musk worries about AI and he is a powerful Rep so this might not happen.  Then again, he supports Reps, many of whom have BANNED E-cars in their states

b) AI-doomsayers want more regulation, AI-awesomers do not, and this cuts across party lines.

c) We will ignore the issue until it's too late.

If I was a betting man ...

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13) International cooperation on being careful with AI. Good luck with that.

My cynical view: International Treaties only work when there is nothing at stake

The Chem Weapons ban works because they are hard to use anyway.

The treaty on exploring Antarctica was working until people found stuff there they wanted. It is now falling apart

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Wednesday, December 11, 2024

It's Time to Stop Using Grades

We use grades to evaluate students and motivate them to learn. That works as long as grades remain a reasonably good measure of how well the student understands the material in a class. But Goodhart's law, "When a measure becomes a target, it ceases to be a good measure," cannot escape even this most basic of academic measurements. Grades become irrelevant or even worse, counterproductive, as chasing grades may undermine a student's ability to master the material. So perhaps it is time to retire the measure.

Grading became a weaker measure due to grade inflation and academic dishonesty. Let's do a short dive into both of these areas.

The average grade has increased about a full grade level since I went to college in the '80s, and now more than half of all grades given are A's. As college tuition increased, students started thinking of college more transactionally, expecting more from their college experience while putting less effort into classes. Administrators put more weight on student course surveys for faculty evaluation, and the easiest way to improve scores is to give higher grades. And repeat.

If everyone gets an A, no one gets an A. It just becomes harder to distinguish the strong students from the merely good.

Academic dishonesty goes back to the beginning of academics but has advanced dramatically with technology. In my fraternity, we had filing cabinets full of old homework and exams ostensibly to use as study guides. However, if a professor reused questions from year to year, one could gain an unfair advantage.

With the growth of the Internet, Chegg, and more recently large-language models, those looking for an edge never had it so good. ChatGPT-4o1 can answer nearly any undergraduate exam question in any field—it even got an easy A when I tested it with one of my undergraduate theory of computing finals.

AI becomes like steroids: those who don't use it find themselves at a disadvantage. If a pretty good student sees their peers using LLMs, they'll start using them as well, initially just as a learning aid. But there's a very fine line between using AI as a study guide and using AI to give you the answers. Many fall down a slippery slope, and this starts to undermine the mastery that comes with tackling problems on your own.

We can try and counter all this by returning to harsher grading and more heavily weighting in-person, no-tech exams, but these approaches cause other problems. Already we see companies and graduate schools devalue grades and focus on projects and research instead.

So let's acknowledge this endgame and just eliminate grades, maybe keeping only Pass and Fail for those who don't even show up. The ones who want to master the material can focus on doing so. Others can concentrate on working on projects. Still others can earn their way to a degree with little effort but also with little reward.

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Sunday, December 08, 2024

My comments on Lance's Favorite Theorems

In Lance's last post (see here) he listed his favorite theorems from 1965 to 2024.
There are roughly 60 Theorems. I mostly agree with his choices and omissions. I will point out where I don't.

I could make a comment on every single entry; however, that would be madness! Madness I say!

Instead, here are some random thoughts.  (Is that Random as in Random Restriction or Random as in Random Access Machine?  I leave that an exercise for the reader.)

1) 1965-1974

MANY BASIC RESULT WITH EASY PROOFS.
EXAMPLE:
The Cook-Levin Theorem. P, NP, and SAT is NPC

ANSWERS A QUESTION:
Ladner: Answers a very important question: YES, if P NE NP there are
intermediary sets. The set is constructed for the sole point of not being in P or NPC. Graph Isom and Factoring are natural candidates for being intermediary.

SEEMS TO HAVE BEEN FORGOTTEN:
Blum: Abstract Complexity Theory. Seems to not be taught anymore. I give a corollary for our young readers who might not know it:

There is a decidable set A such that If A is in DTIME(T(n)) then A is in DTIME((log T(n)). Hence A cannot be assigned a complexity. (The set A is constructed for the sole point of having this property. There are no natural examples or even candidates for sets that have this behavior.)

I might disagree with putting this on the list. It has not stood the test of time; however, it still seems important. 


II) 1975-1984.

This may be my favorite decade on the list; however, its been said
that everyone thinks that the best music was when they were a teenager.

EXAMPLES:

INTERESTING THEOREMS WITH INTERESTING PROOFS:
Everything on the list is in this category but I pick out three:

Baker-Gill-Solovay Oracles: The basic paper for my thesis work.

Furst-Saxe-Sipser Parity is not in constant depth.  A meaningful lower bound on a natural problem! Motivated by an Oracle open question (Sep PH from PSPACE) however, circuit complexity quickly became a field onto itself.  What is more interesting the circuit lower bound or the oracle-corollary? I would vote for the circuit lower bound. The issue was discussed here.

Valiant-Permanent. Perm is #P-complete is a theorem I've learned and forgotten many times. Scott much later gave a proof that may be more intuitive for some people (I am not one of them) see here.  The only theorem I've learned-and-forgotten more is the Hales-Jewitt Theorem.

 
III) 1985-1994.

A Decade of Surprises!

Barrington: Branching programs more powerful than we thought!

Toda: #P is more powerful then we thought!

LFKN: IP is more powerful than we thought! Bonus: used non-rel methods! (This result was not on the list but would have been if anybody except L or F or K or N had written the list.)

Nisan: Randomization is less powerful than we thought!

Lance did not have Factoring in Quantum P on the list for 1985-1994. It came out in 1994 towards the end of the year so it ended up missing both the 1985-1994 list and the 1995-2004 list. Reminds me of the Betty White awards, see here.  I do not think we disagree on the importance and merit of the result, though we disagree about altering the past- I would have included it in the post he did recently and explain that it was a late add to an old list.

IV) 1995-2004.

In 1990 a theorist told me that he could teach EVERYTHING known in complexity theory in a year-long graduate course.  Even then, that was not quite right, and may have really meant he could teach everything he thought was important. By 1995 this was no longer true. The PCP result alone would take a few months.

Theory begins to get really hard. Most of the papers before 1992 I have read and understood. Most of the papers after 1992 I have not, though I know what's in them. Sometimes not even that!

PCP: Many problems are hard to approximate.  Good to know, bad that its true. Proofs are hard!

Raz's Parallel Repetition: Really useful in later non-approx results (my favorite: Set Cover Lower bounds, see this survey here) but also Really Hard to read.

Most of the other papers are also hard and important.

AKS- Primality in P- not that hard. Indeed, surprising that it was not proven until 2002.

Lance did not include the work on Natural Proofs by Razborov and Rudich. He says why in a blog post here. I disagree with him- I would have put it in a top theorems list.

VI) 2005-2014.

Some upper bounds and some lower bounds. By now it was hard to have a surprising result since our intuitions were not so firm as to be surprised. (There is one exception in the 2015-2024 list.)

Reingold-Undirected Connectivity in Log Space: great result! I wish the proof was easier. I think people thought this would be true.

Lots of interesting lower bounds: Nash Equilibrium, Unique Game Conj, new PCP proof. None of which was surprising, though perhaps that we could proof things about these concepts is surprising.

JJUW-QIP=PSPACE. Really! I was shocked to find out that was true. No, I wasn't. I didn't understand QIP well enough. Was this surprising or not? Was the fact that this could be proven surprising or not?


VII) 2015-2024.

No real theme here though they all have hard proofs. I discuss a few.

Babai-Graph Isomorphism is the only result in this decade that I can explain to Darling. And she has a Masters Degree in SE so she knows stuff. (I recently told her the result that for all 2-colorings of R^6 there is a mono unit square  (see here). She was unimpressed.)

BZ- Dichotomy: Excellent result and explains the lack of natural intermediary problems.

CZ-Extracting Ramsey Graphs: An example of TCS helping to prove a result in math, though it also shows that the border between the two is thin. Obviously a favorite of mine.

JNVWY- MIP* = RE. This surprised people, including me.

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Wednesday, December 04, 2024

Favorite Theorems: The Complete List

Now in one place all of my sixty favorite theorems from the six decades of computational complexity (1965-2024).

2015-2024

1985-1994

To mark my first decade in computational complexity during my pre-blog days, I chose my first set of favorite theorems from that time period for an invited talk and paper (PDF) at the 1994 Foundations of Software Technology and Theoretical Computer Science (FST&TCS) conference in Madras (now Chennai), India. The links below go to the papers directly, except for Szelepcsényi’s, which I can't find online.
1975-1984 (From 2006)

1965-1974 (From 2005)


Will I do this again in ten years when I'm 70? Come back in 2034 and find out.
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Sunday, December 01, 2024

Conway's Trick for Divisibility. Asking its complexity is an odd question.

 (I got this material from a nice article by Arthur Benjamin here.)

 Conway suggested the following trick to determine if a number is divisible by each of the following: 

2,3,5,7,11,17,19,31

Note that

\( 152=2^3\times 19\)

\(153 =3^2 \times 17\)

\(154=2  \times 7 \times 11\)

\(155=5 \times 31\)

\(156=2^2  \times 13 \)

Here is the Div trick:

a) Input N

b) Divide N by 150 and note the remainder. So

 N=150q+r

r=N-150q 

Subtract q from r a few times: 

Note that

r-q = N-150q-q = N-151q

r-2q=N-152q

AH HA!- if 19 divides r-2q then 19 divides N. So divide r-2q by 19. (Note that r-2q is much smaller than N. Smaller enough to make this technique feasible? That is the question!)

r-3q=N-153q.

AH HA!- if 17 divides r-3q then 17 divides N. So Divide r-3q by 17.

r-4q=N-154q

AH HA- if 11 divides r-4q then 7 divides N. So Divide r-4q by7.

r-5q=N-155q

AH HA- if 31 divides r-5q then 31 divides N. So Divide r-5q by 31.

r-6q=N-156q

AH HA- if 13 divides r-6q then 13 divides N. So Divide r-6q by 13. 

Complexity with 1 division, 6 subtractions and 6 divisions of small numbers (r\le 150 and q\le N/150)

you find out the divisibility by 7,13,17,19,31.  For 2,3,5,11 there are well known tricks to use. OR you can test those as well by doing (for example) dividing r-4q=r-154 by 11.

Some Points

1) Is this method faster than just dividing N by the numbers (and using tricks for 2,3,5,11)? You would need to get into addition being faster than division, and look at the size of the numbers.

2) Is this method practical? For hand calculation YES. For computers it would be easy to code up but the main question of this post: is it better than just dividing N by numbers.

3) Are there larger runs of numbers that pick up more divisors? Yes. We present one. The order will look funny but we explain it later.

\(2000=2^4 \times 5^3 \) (you could skip this one, though dividing by 2000 is easier than by 2001)

\(2001=23\times 29\times 3\) (would divide N-2q by both 23 and 29)

\(2002=7\times 11\times 13\times 2\)

\(1998=37\times 54\)

\(2006=17\times 29\times 2\)

\(2010=67\times 30\)

\(2014=19\times 53\times 2\)

\(2013=61\times 33\)

\(2015=31\times 65\)

\(2009=41\times 49\)

\(2021=43\times 47\)

The order was suggested by Conway so that algorithm at every step adds or subtracts one of q, 2q, 4q, 6q, 8q, 12q. So after you get q you can compute these values. 

I leave it to the reader to count the number of divisions, subtractions, and size of the numbers involved.

4) For cracking RSA this technique is useless since RSA uses numbers of the form pq where p and q are large primes. For factoring randomly generated numbers I would be curious if this method is better than just dividing by numbers.

5) Project: find other sequences like those above that cover more prime factors.



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Monday, November 25, 2024

We Will All Write Like AI

Will our writing all converge to a generic AI style? 

Let's take a quick detour into LaTeX. Back in the late '80s, before LaTeX was the standard, there was TeX—a system with no default formatting, which meant everyone had their own unique style for papers. Then LaTeX arrived, and suddenly all our papers looked polished and professional. But the catch was, LaTeX only got you about 80% of the way there. The original manual even mentioned that you needed to add some extra TeX commands to really finish the job. Most of us didn’t bother, though, and soon all our papers had that same uniform look. It was efficient, and it was good enough. Microsoft Word ended up doing the same thing for everyone else—you could tweak it to be different, sure, but most people didn’t. It turns out most of us are just fine with "good enough."

I generally don't like to use large language models to write for me. But I do use them to refine my work, to catch errors or suggest improvements. The thing is, AI likes to take what I write and make it sound smoother, and I often think, "Wow, that really does sound better." But here’s the tricky part: it might be better, but it’s not mine. It’s the AI's voice. And still, if the stakes aren’t too high, sometimes I let the AI’s version slide. That’s the start of a slippery slope. Before you know it, we’re all letting AI make our writing a bit more generic, a bit more uniform. And eventually, we end up writing to match the AI’s preferred style.

For this blog post, I didn’t resist at all. Could you tell this is ChatGPT's style?
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Wednesday, November 20, 2024

For what d is the following true: For all 2-colorings of \(R^d\) has a mono unit square (Answering(?) the Question)

 In my last post (see here) I invited you to work on the following question:

Find a \(d\) such that

--There is a 2-coloring of \(R^d\) with no mono unit square.

--For all 2-colorings of \(R^{d+1}\) there is a mono unit square. 

Actually I should have phrased my question as What do we know about d?  

Here is what we know

a) \(d \ge 2\).  There is a 2-coloring of  \(R^2\) with no mono unit square. This is easy and I leave to you. 

b) \(d\le 5\). For all 2-colorings of \(R^6\) there is a mono unit square. I will give pointers to the relevant papers and to my slides later in this post.

c) \(d\le 4\). For all 2-colorings of \(R^5\) there is a mono unit square. This is by an observation about the proof for \(R^6\). It will be in the slides about \(R^6\).

d) \(d\le 3\). This is in a paper that the reader Dom emailed me a pointer to. Dom is better at Google Search than I am. The link is here.

MY SLIDES:

\(K_6\) is the complete graph on 6 vertices. We will be looking at 2-colorings of its edges

\(C_4\) is the cycle on 4 vertices. A mono \(C_4\) has all four edges the same color.

We need a result by Chvtal and Harary in this paper here.

Lemma: For all 2-colorings of the edges of \(K_6\) there is a mono \(C_4\).

The proof appears both in their paper,  here, and on slides I wrote here. 

Stefan Burr used this to prove the following theorem.

Thm: For all 2-colorings of \(R^6\) there is a mono unit square. 

The proof was appears (with credit given to Stefan Burr) in a paper by Erdos, Graham, Montgomery, Rothchild, Spencer, Straus, here, and on slides I wrote here.

Random Points

1) It is open what happens in \(R^3\). 

2) The proof for \(R^6\) uses very little geometry. Dom had a proof for \(R^6\) in a comment on my last post that used geometry. The proof for \(R^4\) uses geometry. 

3) An ill-defined open question: Find a proof that every 2-coloring of \(R^4\) has a mono unit square that does not use that much geometry and so I can make slides about it more easily.



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Sunday, November 17, 2024

For what d is the following true: for all 2-colorings of \(R^d\) there is a mono unit square (Asking the Question)

 In this post I give a question for you to think about. 

My next post will have the answer and the proof. 

1) The following are known and I have a set of slides about it here

a) For all 2-colorings of \(R^2\) there exists two points an inch apart that are the same color. (You can do this one.)

b) For all 3-colorings of \(R^2\) there exists two points an inch apart that are the same color. (You can do this one.)

c) For all 4-colorings of  \(R^2\) there exists two points an inch apart that are the same color. (You cannot do this one.) 

2) SO, lets look at other shapes

A unit square is  square with all sides of length 1.

Given a coloring of \(R^d\) a mono unit square is a unit square with all four corners the same color. 

a) There is a 2-coloring of \(R^2\) with no mono unit square. (You can do this one.)

b) What is the value of d such that 

-- There is a 2-coloring of  \(R^d\) with no mono unit square.

-- For all 2-colorings of \(R^{d+1}\) there is a mono unit square. 

My next post will tell you what is known about this problem.

Until then, you are invited to think about it and see what you can find out. Perhaps you will get a better result then what is known since you are untainted by conventional thinking. Perhaps not. 

Feel free to leave comments. However, if you don't want any hints then do not read the comments.



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