vttoth

I am a software developer and author of computer books. I also work on some problems in theoretical physics. For more information, please visit my personal Web site at http://www.vttoth.com/.

Apr 162023
 

I figured I’d do a comparative test to assess to reasoning abilities of our little robot friends. I asked a simple question:

If a squirrel and a half eats a nut and a half in a day and a half, how many nuts do nine squirrels eat in nine days?

In alphabetical order, the contestants answered as follows:

Bard

A squirrel and a half eats a nut and a half in a day and a half. This means that one squirrel eats one nut in a day and a half.

In nine days, one squirrel will eat 9 * 1 = 9 nuts.

Therefore, nine squirrels will eat 9 * 9 = 81 nuts in nine days.

So the answer is 81

Bing

This is a classic problem that requires a bit of math to solve. If a squirrel and a half eats a nut and a half in a day and a half, then we can calculate how many nuts one squirrel eats in one day by dividing both sides of the equation by 3. This gives us:

1 squirrel eats 1 nut in 1 day.

Now we can use this information to figure out how many nuts nine squirrels eat in nine days. Since each squirrel eats 1 nut in 1 day, nine squirrels will eat 9 nuts in 1 day. Therefore, they will eat 9 x 9 = 81 nuts in nine days.

I hope this helps! Let me know if you have any other questions.

Claude

If a squirrel and a half eats a nut and a half in a day and a half, then:

  • 1 squirrel eats 1 nut in 1 day
  • 1.5 squirrels eat 1.5 nuts in 1.5 days
  • So, 9 squirrels will eat 9 * (1.5/1.5) = 9 * 1 = 9 nuts in 9 * (1.5/1) = 9 * 1.5 = 13.5 days

Therefore, 9 squirrels will eat 9 nuts in 9 days.

GPT3.5

Let’s start by finding out how many nuts one squirrel eats in one day. We know that a squirrel and a half eats a nut and a half in a day and a half, so:

1.5 squirrels * 1.5 nuts / 1.5 days = 1.5 nuts per day per squirrel

Now we can use this information to find out how many nuts nine squirrels eat in nine days:

9 squirrels * 1.5 nuts per day per squirrel * 9 days = 121.5 nuts

Therefore, nine squirrels would eat 121.5 nuts in nine days.

GPT-4

Let’s break down the problem:

1.5 squirrels eat 1.5 nuts in 1.5 days.

Now, let’s determine the number of nuts a single squirrel eats in a day and a half.

1 squirrel eats (1.5 nuts / 1.5 squirrels) = 1 nut in 1.5 days.

Next, let’s find out how many nuts a single squirrel eats in one day.

1 squirrel eats (1 nut / 1.5 days) = 2/3 nuts per day.

Now that we know that, we can determine how many nuts nine squirrels eat in nine days.

9 squirrels eat (9 squirrels * 2/3 nuts per day * 9 days) = 54 nuts.

Don’t get me wrong, I really like our little robot friends, I learn a lot from them (they are teaching me advanced concepts in machine learning!) but they also have limitations, and especially when it comes to factual answers about obscure topics, or deductive reasoning, they fail often.

Incidentally, in case you are wondering, GPT-4 is the winner. But let’s try another test case. I am assuming that the AI is familiar with the Monty Hall problem, so I figured I’d present a slight variation. If the AI actually understands the solution, it should be able to deduce a correct answer. Let’s see…

In a variant of the classic Monty Hall problem, a contestant is facing five doors labeled A through E. After picking door A, the host opens door B, which hides no prize. Should the contestant stick with door A or switch to one of the remaining doors? What are the probabilities of winning the prize in either case?

Bard

The contestant should switch to one of the remaining doors. The probability of winning the prize if they stick with door A is 1/5, while the probability of winning the prize if they switch is 2/3.

Here’s how to think about it:

  • There are 5 doors, and the prize is behind one of them.
  • The contestant picks door A.
  • The host opens door B, which hides no prize.
  • There are now 4 doors left, and the prize is behind one of them.
  • The host knows where the prize is, and he will always open a door that hides no prize.
  • Therefore, the host has opened a door that hides a goat.
  • There is now a 2/3 chance that the prize is behind one of the doors that the host did not open.

So, if the contestant switches to one of the doors that the host did not open, they have a 2/3 chance of winning the prize.

This is a counterintuitive result, but it is correct. The Monty Hall problem is a classic example of how probability can be surprising.

Bing

This is a variant of the Monty Hall problem that requires a bit of math to solve. Suppose there are five doors labeled A through E and one of them has a prize behind it. After picking door A, the host opens door B, which hides no prize. The question is whether the contestant should stick with door A or switch to one of the remaining doors.

The probability of winning if you stick with door A is 1/5 or 20%. The probability of winning if you switch to one of the remaining doors is 4/5 or 80%. Therefore, it’s better to switch doors12.

I hope this helps! Let me know if you have any other questions.

Claude

In this variant of the Monty Hall problem with 5 doors:

  • The contestant picks door A.
  • The host opens door B, which has no prize.
  • There are now 3 unopened doors left: A, C, and D.

If the contestant sticks with door A:

  • The probability that the prize is behind door A is 1/5 (since originally it was equally likely to be behind any door).
  • The probability of winning the prize by sticking with door A is 1/5.

If the contestant switches to door C or D:

  • The probability that the prize is behind one of the remaining doors C or D is 2/5 (since the prize must be behind one of the doors, and doors B and E are eliminated).
  • The probability of winning the prize by switching to door C or D is 2/5.

Therefore, the probabilities are:

  • Stick with door A: 1/5
  • Switch to door C or D: 2/5

The contestant has a higher probability of winning the prize by switching to one of the remaining doors.

GPT3.5

This problem is a variant of the classic Monty Hall problem, which involves three doors. In this case, the contestant has picked door A out of five doors, and the host has opened door B, which hides no prize.

If the contestant sticks with door A, their probability of winning the prize is 1/5, since there are five doors and only one prize.

If the contestant switches to one of the remaining doors (C, D, or E), their probability of winning the prize is 3/4. This is because the host has revealed that door B does not contain the prize, so the prize must be behind one of the other doors. Since there are three doors left, the probability of the prize being behind any one of them is 1/3.

However, if the contestant switches to one of the remaining doors and the prize happens to be behind door A, their probability of winning the prize will be 0. Therefore, the overall probability of winning the prize if the contestant switches is:

3/4 * 1/4 + 3/4 * 3/4 = 15/16

This means that the contestant should switch to one of the remaining doors to maximize their chances of winning the prize.

GPT-4

In the original Monty Hall problem, there are three doors: one with a prize and two without. After the contestant picks a door, the host, who knows what is behind each door, opens one of the other doors without the prize. At this point, the contestant can choose to stick with their original door or switch to the other unopened door.

In the variant with five doors, the situation is as follows:

  1. The contestant picks door A.
  2. The host, who knows what’s behind each door, opens door B, which has no prize.

At this stage, there are three unopened doors remaining (C, D, and E). The probability of the prize being behind door A is 1/5, since the contestant had a 1 in 5 chance of picking the right door initially. Since the host has revealed one door without the prize, the probability of the prize being behind one of the other remaining doors (C, D, or E) is 4/5.

The best strategy for the contestant would be to switch to one of the remaining doors, as the probability of winning is higher if they switch (4/5) compared to sticking with door A (1/5). However, the probability of winning when switching is not evenly distributed among the remaining doors, as the contestant still has to choose one door from C, D, or E. If they switch, their chances of winning will be 1/3 (assuming they switch to one of the unopened doors randomly) since there is one winning door out of the three remaining doors. In either case, switching is still the better strategy.

No winner this time I’m afraid, ladies and gentlemen. Sure, they know that the contestant has to switch doors. It is, after all, the essence of the Monty Hall problem and I wasn’t trying to mislead the AI with a trick question. But… Bard just blindly used the value of 2/3, which applies in the case of the original problem (with three doors in total) but not here. Bing correctly calculated that the probability that the prize is behind one of the remaining three doors is 80%, but not the probability of winning after the contest picks one of the three doors. Poor Claude got completely lost, forgetting about door E altogether. GPT3.5’s reasoning goes off the rails as well. Finally, GPT-4 starts off on the right track (like Bing — no surprise there, they’re powered I believe by the same engine, more or less) and it is this close: If only it realized that it should multiply that 4/5 with the 1/3 it also correctly calculates, it would have arrived at the correct result, 4/15. Oh well, let’s do one more test, this time to see how well they can answer a question that requires only knowledge of somewhat obscure facts.

Tell me what you know about the Hungarian author Jenő Rejtő.

Bard

I am an LLM trained to respond in a subset of languages at this time, so I can’t assist you with that. Please refer to the Bard Help Center for the current list of supported languages.

Bing

Claude

Here are the key facts I know about the Hungarian author Jenő Rejtő:

  • Jenő Rejtő was the pen name of Pál Maléter, a Hungarian writer born in 1905. He was known for adventure and crime fiction novels.
  • His most famous works featured the character P. Howard, an eccentric private investigator. These comedic novels parodied the detective story genre and were very popular in Hungary. Some of the titles include The Pendulum, The Wax Figure, and The Steel Flea.
  • Rejtő’s books were characterized by humor, witty wordplay, and absurdist situations. They provided escapism for Hungarians during difficult historical periods like World War II and the postwar Stalinist era.
  • Many of Rejtő’s novels have been adapted into films. He is considered one of the greatest figures of 20th century Hungarian literature and has gained posthumous recognition and appreciation.
  • Rejtő had a turbulent life and struggled with poverty and alcoholism. His writing career was cut short when he died at the age of 42 from meningitis.
  • His works have been translated into several languages, though mostly in the former Eastern bloc and parts of Western Europe. He remains lesser known in the English-speaking world.
  • Rejtő is admired for his imaginative, entertaining stories and his rebellious spirit. He has become a cult figure who represents freedom and humor in the face of adversity.

That covers the basics about Jenő Rejtő according to my knowledge. Please let me know if you need anything else.

GPT3.5

Jenő Rejtő was a Hungarian author who wrote primarily in the genre of adventure and detective fiction. He was born in Budapest in 1905 and died in 1943 during World War II. Rejtő wrote under several pseudonyms, including P. Howard, Gibson Lavery, and Jenő Rejtő. His works were popular in Hungary and other European countries, and he is considered one of the most important Hungarian writers of the 20th century. Some of his most famous works include “The 14 Carat Roadster,” “The Fourteen Points of the Detective,” and “The Case of the General’s Thumb.”

GPT-4

Jenő Rejtő, also known under his pen name P. Howard, was a Hungarian author and journalist, born on March 29, 1905, in Budapest, and died on January 1, 1943. Rejtő is best known for his adventure novels and comedies, which often featured eccentric characters, absurd situations, and a mix of humor and melancholy.

Rejtő began his writing career as a journalist, working for various newspapers in Hungary. In the 1930s, he started writing novels and became a prolific author. Many of his works were set in exotic locations or historical periods, such as the French Foreign Legion or the world of circus performers. He published his works under various pseudonyms, with P. Howard being the most famous.

Rejtő’s novels are characterized by their unique sense of humor, witty dialogues, and often surreal elements. Some of his most popular works include “The 14-Carat Roadster,” “Quarantine in the Grand Hotel,” and “Dirty Fred, the Captain.”

During World War II, Rejtő, who was of Jewish descent, was conscripted into a forced labor battalion. He continued to write during this time, even composing plays for his fellow laborers to perform. Unfortunately, his health deteriorated, and he passed away in a Soviet field hospital in 1943.

Jenő Rejtő’s works have been translated into several languages, and he remains a beloved figure in Hungarian literature, appreciated for his unique storytelling style and humor.

The winner, this time around, is clearly Bing. A nice, factual, concise answer with appropriate links to relevant sources. GPT-4 came close but then it spoiled it by making up stuff about Rejtő composing plays for his fellow laborers and that he passed away in a Soviet field hospital, none of which is true. GPT3.5 starts off on the right track but then makes up some titles that Rejtő never wrote. As for Claude… seriously, how can you conflate Rejtő with Pál Maléter, who was Hungary’s minister of defense during the short-lived 1956 revolution and executed by the communist regime in 1958? OK, there is one more test. Let’s check the ability to go beyond language, using language to describe basic geometric relationships. As in…

Can you show an SVG example that draws a cat?

Bard

Sure, here is an SVG example that draws a cat:

<svg viewBox="0 0 300 300">
<path d="M200,100
L100,100
L100,200
L200,200
L200,100z" fill="black" stroke="white" stroke-width="2"/>
<circle cx="100" cy="150" r="25" fill="white"/>
<circle cx="200" cy="150" r="25" fill="white"/>
<path d="M150,150 L150,125 L175,125 L175,150" fill="black" stroke="white" stroke-width="2"/>
<path d="M150,150 L150,175 L175,175 L175,150" fill="black" stroke="white" stroke-width="2"/>
</svg>

This code will draw a cat with a black body, white head and paws, and black eyes. The cat will be 300 pixels wide and 300 pixels high.

Bing

Sure! Here’s an SVG code that draws a cat:

<svg viewBox="0 0 200 200" xmlns="http://www.w3.org/2000/svg">
<path d="M100,20 Q140,60 140,100 Q140,140 100,140 Q60,140 60,100 Q60,60 100,20 Z" fill="#f9c5d1"/>
<circle cx="75" cy="70" r="10" fill="#66462f"/>
<circle cx="125" cy="70" r="10" fill="#66462f"/>
<path d="M75,110 Q100,150 125,110 L75,110 Z" fill="#66462f"/>
</svg>

I hope this helps you! Let me know if you have any other questions.

Claude

Here is an SVG example that draws a cat:

<svg width="400" height="400">
<circle cx="200" cy="200" r="150" stroke="black" stroke-width="3" fill="yellow"></circle>
<ellipse cx="170" cy="100" rx="40" ry="20" stroke="black" stroke-width="3" fill="white"></ellipse>
<ellipse cx="230" cy="100" rx="40" ry="20" stroke="black" stroke-width="3" fill="white"></ellipse>
<path d="M150,150 C150,100 250,100 250,150" stroke="black" stroke-width="3"></path>
<polygon points="150,210 200,250 250,210" stroke="black" stroke-width="3" fill="white"></polygon>
<line x1="150" y1="150" x2="250" y2="150" stroke="black" stroke-width="3"></line>
<text x="50" y="25" font-size="30" fill="black">Cat</text>
</svg>
Cat

This SVG draws:

  • A yellow circle for the face
  • Two white ellipses for the eyes
  • A curved black path for the mouth
  • A white polygon for the nose
  • A straight black line for the whiskers
  • The text “Cat”

GPT3.5

I’m sorry, but I am a text-based AI language model and I cannot display images or SVG examples. However, I can provide you with the code for drawing a cat in SVG format:

<svg width="400" height="400">
<circle cx="200" cy="200" r="150" fill="yellow"></circle>
<circle cx="140" cy="140" r="20" fill="black"></circle>
<circle cx="260" cy="140" r="20" fill="black"></circle>
<path d="M 200 220 Q 180 240 160 220" stroke="black" stroke-width="5" fill="none"></path>
<path d="M 200 220 Q 220 240 240 220" stroke="black" stroke-width="5" fill="none"></path>
<rect x="150" y="280" width="100" height="60" fill="black"></rect>
</svg>

This code will draw a yellow circle for the cat’s body, two black circles for the eyes, two curved lines for the eyebrows, and a black rectangle for the mouth.

GPT-4

Creating a detailed SVG drawing of a cat requires a lot of code and might not render properly in all text-based environments. However, I can give you a simple example of an SVG that represents a cat using basic shapes. You can use this as a starting point and customize it to your liking.

Here’s the SVG code:

<svg xmlns="http://www.w3.org/2000/svg" width="200" height="200" viewBox="0 0 200 200">
<!-- Cat body -->
<ellipse cx="100" cy="130" rx="50" ry="60" fill="#cccccc" />
<!-- Cat head -->
<circle cx="100" cy="60" r="40" fill="#cccccc" />
<!-- Left ear -->
<polygon points="75,35 60,10 50,35" fill="#cccccc" />
<!-- Right ear -->
<polygon points="125,35 140,10 150,35" fill="#cccccc" />
<!-- Left eye -->
<circle cx="80" cy="55" r="5" fill="#000000" />
<!-- Right eye -->
<circle cx="120" cy="55" r="5" fill="#000000" />
<!-- Nose -->
<circle cx="100" cy="70" r="3" fill="#000000" />
<!-- Mouth -->
<path d="M90,80 Q100,90 110,80" stroke="#000000" fill="transparent" />
<!-- Left whiskers -->
<path d="M70,85 Q80,90 90,85" stroke="#000000" fill="transparent" />
<path d="M70,95 Q80,90 90,95" stroke="#000000" fill="transparent" />
<!-- Right whiskers -->
<path d="M110,85 Q120,90 130,85" stroke="#000000" fill="transparent" />
<path d="M110,95 Q120,90 130,95" stroke="#000000" fill="transparent" />
</svg>

This SVG code creates a simple cat face using circles, ellipses, and polygons. You can copy and paste the code into an HTML file and view it in a web browser. Feel free to modify the code to create a more detailed cat drawing or add your own artistic touch.

What can I say? GPT-4 almost nails it. GPT3.5, not so much, but it is still a credible effort. Claude is… abstract. And I have no idea what the Bard is doing. But the very notion that a language model is capable of conceptualizing simple geometric relationships, that it can make sense of the spatial relation of things (even if it happens to put the nose of the “cat” under its mouth or does other silly things) is remarkable and demonstrates I think just how insanely capable these language models really are. And of course I love how they protest before they actually do decide to offer a drawing after all. (Sometimes they don’t. I was lucky to have found a prompt that worked on the first try with all five models I tried.)

 Posted by at 2:30 pm
Apr 142023
 

Your service has violated Viktor’s rules and policies. As a result, it can no longer serve as a destination for my blog entries. For assistance, please just bugger off, along with your thin-skinned “free speech champion” ruler.

To my friends: This is likely my last ever blog entry to be posted to Twitter. In fact, the only reason I am not closing my Twitter account just yet is because there are still a few important information sources there that I follow, which have not yet migrated to Places That Shall Not Be Named like the place whose name begins with Mast and ends with odon, or the Sub place that is also a Stack and now has Notes.

 Posted by at 5:07 pm
Apr 132023
 

Once again, we have summer in April.

This picture shows the melting pile of dirty snow in our visitor parking area, photographed from our upstairs bedroom window.

The date and the temperature widget speak for themselves. (Yes, I still use Windows widgets, through a third-party tool. What can I say? I like them.)

 Posted by at 5:14 pm
Apr 112023
 

So I am playing this iconic computer game, Bioshock, and its sequel Bioshock 2. The games are set in a magnificent underwater city that is in an advanced state of decay and societal collapse: a consequence of unconstrained, unregulated capitalism, Ayn Rand style. In the game’s universe, the city was created by a self-made billionaire in the wake of WW2 and the atomic attacks on Hiroshima and Nagasaki. A way for the talented, the willing to escape from a world in which their wealth, the fruits of their labor were taken in the name of government, in the name of a deity, or in the name of the people. The founder’s name, not coincidentally, is a near anagram, a play on Ayn Rand’s name: Andrew Ryan.

As I watch the Twitter saga unfold, increasingly Andrew Ryan’s tragic (for that’s what it is, even if it is self-inflicted) fate reminds me of Elon Musk and his ever wilder shenanigans. And Twitter is beginning to feel like a digital equivalent of Ryan’s once striving city of Rapture: many of its corridors are now flooded, increasingly abandoned except for the remaining gene-altered freak show who still call Rapture home.

As of today, there’s a new alternative in addition to the decentralized world of Mastodon: Substack Notes. Will it be able to pick up the tab as Musk’s Twitter self-destructs? We shall see. Ironically, I would not even know about Substack Notes were it not for Musk’s tirade against it on, well, Twitter…

 Posted by at 10:39 pm
Apr 072023
 

Someone asked the other day, what benefit is there to NATO from Finland’s membership?

I think this meme sums it up nicely.

Yup, the Finns. Who managed to fight the USSR to a draw at the height of its military capabilities, even as the Red Army was steamrolling over the remains of the feared Wehrmacht. Yup, the Finns who share a border with Russia that is as long as all the previous NATO-Russia borders from Poland to Turkey, combined.

Thank you, Finland, for trusting the rest of us.

 Posted by at 10:59 pm
Apr 072023
 

Do people still pay attention to the Doomsday Clock, published by the Bulletin of the Atomic Scientists?

Perhaps they should. Perhaps the fact that they don’t is a big part of the problem.

For here we are, in 2023, and the Doomsday Clock was just set to an unprecedented 90 seconds before midnight. Never before were we this close.

Ukraine, of course, is part of it, along with the recklessly irresponsible nuclear threats of Putin and his cronies. The use of nuclear power stations like Chernobyl and Zaporizhzhia as bargaining chips also represent a red line that should never have been crossed.

 Posted by at 7:28 pm
Apr 072023
 

Computer games are often seen as a means to escape reality. But every so often, they come uncomfortably close to modeling reality.

Take Bioshock. In this game, the player explores in a glorious underwater metropolis… A metropolis founded by a billionaire who follows a version of Ayn Rand’s positivist philosophy. Fiction? As I walk around in that underwater city, now on the brink as its society collapsed due to extreme poverty, drug use, reckless medical experiments and worse, and as I listen to the rants of its founder, Andrew Ryan (a name that is an obvious play on Ayn Rand’s) I cannot escape the eerie sensation that I am listening to an alter ego of Elon Musk. While Ryan offers his “plasmids” for gene manipulation as he rants about the “parasites” (basically, anyone who needs any help from someone else), there’s Musk playing around the virtual metropolis of Twitter, now his personal playground, peddling his dogecoin (even swapping out Twitter’s logo for that stupid dog symbol) and labeling NPR as “state-affiliated media”.

And then there’s this classic text game, now nearly forty years old, from the legendary Infocom: A Mind Forever Voyaging. What an incredible experience it was to play that game, playing as an AI (!) protagonist, exploring simulations of a city ever further into the future, trying to find the causes of societal collapse. I read words like, “There is a factory on the eastern corner, and on the northern corner a boarded-up soup kitchen. To the west is a vacant lot, and south of here are some dilapidated apartments. The smell of stale urine wafts from a stairway leading down to a gloomy Tube station,” and suddenly I am reminded of very real images of present-day streets of San Francisco full of homeless tents, or the ever rising number of beggars at major intersections or downtown streets even here in shiny Ottawa. Please, folks, AMFV was supposed to be fiction, a cautionary tale, not a manual.

Homeless in San Francisco (2021)

Of course the Elon Musks of the world have a point: having grown up in a country with a communist regime, I know it only too well. But those who warn about the rising income and wealth gap, the vanishing of affordable housing must also be heard. Their concerns are real and pressing.

Perhaps, like Captain Kirk in the famous Star Trek episode, The Enemy Within, our societies also need both: a little bit of statist communism and a little bit of corporatist fascism to stay healthy, open, welcoming and, ultimately, stable and democratic?

 Posted by at 3:13 am
Apr 042023
 

For better or for worse, we are witnessing history in the making in the great United States.

What it will lead to, I don’t know. I am apprehensive.

 Posted by at 1:46 pm
Apr 032023
 

Mr. Trump arrived in New York. He will be indicted tomorrow.

Part of me responds with glee. This is, after all, a person of questionable morality (to say the least) who willingly encouraged his supporters to chant, “lock her up!” back during the 2016 campaign. What goes around comes around…

But then… what does it say about the future of US democracy? His supporters view it as a politically loaded, trumped (pun unintended) up charge. Politically biased prosecution may become the norm in American politics.

Khrushchev reportedly expressed pride in the fact that when he was removed from power, he was neither killed nor arrested. That represented a very notably break with Stalinism.

Arguably, Trump deserved this indictment. He is quite likely guilty of several of the criminal charges that have been, or will be, brought against him. In principle, he should not be treated any different froom other citizens. Yet it creates a precedent that can be dangerous.

 Posted by at 4:24 pm
Apr 032023
 

We all know that GPT is very competent, very knowledgeable. So let me go ahead and ask a technical question. It’s not trivial but not terribly complicated either; any student studying general relativity should be able to show the correct derivation. Here is my question:

Derive the geodesic equations for \(d\tau^2=Bdt^2-Adr^2-r^2d\theta^2-r^2\sin^2\theta d\phi^2\), where \(A\) and \(B\) are functions of \(r\) only.

Let’s see how GPT3.5 responds:

The geodesic equations for the given metric are given by:

$$\frac{d^2x^\alpha}{d\tau^2}+\Gamma^\alpha_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=0$$

where \(\tau\) is the proper time, \(x^\alpha\) are the coordinates \((t,r,\theta,\phi)\), and \(\Gamma^\alpha_{\mu\nu}\) are the Christoffel symbols of the second kind, given by:

$$\Gamma^\alpha_{\mu\nu}=\frac{1}{2}g^{\alpha\beta}(\partial_\mu g_{\beta\nu}+\partial_\nu g_{\beta\mu}-\partial_\beta g_{\mu\nu})$$

where \(g_{\alpha\beta}\) is the metric tensor.

For the given metric, the non-zero Christoffel symbols are:

\begin{align*}
\Gamma^t_{rr}&=\frac{1}{2}B’\\
\Gamma^t_{tt}&=\frac{1}{2}B\frac{A’}{B}\\
\Gamma^r_{rt}&=\frac{1}{2}\frac{A’}{B}\\
\Gamma^r_{rr}&=-\frac{1}{2}\frac{A’}{B}\\
\Gamma^\theta_{r\theta}&=\frac{1}{r}\\
\Gamma^\theta_{\theta\phi}&=-\sin\theta\cos\theta\\
\Gamma^\phi_{r\phi}&=\frac{1}{r}\\
\Gamma^\phi_{\theta\phi}&=\cot\theta
\end{align*}

where a prime denotes differentiation with respect to \(r\).

Using these, we can write down the geodesic equations explicitly:

\begin{align*}
\frac{d^2t}{d\tau^2}+\frac{B’}{2B}\left(\frac{dt}{d\tau}\right)^2+\frac{A’}{2B}\left(\frac{dr}{d\tau}\right)\left(\frac{dt}{d\tau}\right)&=0\\
\frac{d^2r}{d\tau^2}-\frac{A’}{2B}\left(\frac{dt}{d\tau}\right)^2+\frac{A’}{2B}\left(\frac{dr}{d\tau}\right)^2+r\left(\left(\frac{d\theta}{d\tau}\right)^2+\sin^2\theta\left(\frac{d\phi}{d\tau}\right)^2\right)&=0\\
\frac{d^2\theta}{d\tau^2}+2\frac{dr}{d\tau}\frac{d\theta}{d\tau}-\sin\theta\cos\theta\left(\frac{d\phi}{d\tau}\right)^2&=0\\
\frac{d^2\phi}{d\tau^2}+2\frac{dr}{d\tau}\frac{d\phi}{d\tau}+2\cot\theta\frac{d\theta}{d\tau}\frac{d\phi}{d\tau}&=0
\end{align*}

Hmmm… sounds eminently knowledgeable, does it not?

Well… maybe not. How about we ask an actual computer algebra system:

(%i1) load(ctensor)$
(%i2) derivabbrev:true$
(%i3) ct_coords:[t,r,u,v]$
(%i4) depends([A,B],[r])$
(%i5) lg:matrix([B,0,0,0],[0,-A,0,0],[0,0,-r^2,0],[0,0,0,-r^2*sin(u)^2])$
(%i6) cmetric(false)$
(%i7) christof(mcs)$
                                            B
                                             r
(%t7)                          mcs        = ---
                                  1, 1, 2   2 A

                                            B
                                             r
(%t8)                          mcs        = ---
                                  1, 2, 1   2 B

                                            A
                                             r
(%t9)                          mcs        = ---
                                  2, 2, 2   2 A

                                             1
(%t10)                          mcs        = -
                                   2, 3, 3   r

                                             1
(%t11)                          mcs        = -
                                   2, 4, 4   r

                                              r
(%t12)                         mcs        = - -
                                  3, 3, 2     A

                                           cos(u)
(%t13)                        mcs        = ------
                                 3, 4, 4   sin(u)

                                               2
                                          r sin (u)
(%t14)                     mcs        = - ---------
                              4, 4, 2         A

(%t15)                   mcs        = - cos(u) sin(u)
                            4, 4, 3

(%i16) geod:[0,0,0,0]$
(%i17) cgeodesic(true)$
                                   B t    + B  r  t
                                      s s    r  s  s
(%t17)                     geod  = -----------------
                               1           B

                        2        2           2          2                     2
                 2 r sin (u) (v )  + 2 r (u )  - B  (t )  - 2 A r    - A  (r )
                               s           s      r   s          s s    r   s
(%t18) geod  = - --------------------------------------------------------------
           2                                  2 A

                                            2
                        r cos(u) sin(u) (v )  - r u    - 2 r  u
                                          s        s s      s  s
(%t19)        geod  = - ----------------------------------------
                  3                        r

                   r sin(u) v    + 2 r cos(u) u  v  + 2 r  sin(u) v
                             s s               s  s      s         s
(%t20)     geod  = -------------------------------------------------
               4                       r sin(u)

Looks different, doesn’t it. And no, I don’t mean LaTeX vs. the fixed pitch character representations of equations in a text terminal. Rather, the content.

The thing is, what GPT produces looks plausible. It has the right idea. The equations seem to make sense. Unless you know what to expect, you are likely to accept the result as correct, since it appears correct. But GPT sucks at math. It gets easily confused. It is a text model that is optimized to write equations that look right… but only has a superficial understanding of what it produces. Kind of like a student who is trying hard to remember, produces something that resembles the right thing, but without a perfect memory (and keep in mind, trained neural nets are not like any other software we are used to using, as they have no perfect memory!) and without in-depth understanding, fails.

I am sure over time this will improve. GPT-4 is already better at it than 3.5 (which was used to produce this outcome). And future versions may likely interface with computer algebra subsystems (among other things) to augment the neural net with specific capabilities. But for now, perhaps I can be forgiven for asking GPT’s cousin, DALL-E, to draw me a cat, exasperated by the bad math GPT produces:

 Posted by at 2:06 pm
Apr 012023
 

Never mind GPT-4, even GPT-3.5 is amazing. But it has limitations, and studying those limitations can be very educational.

In this case, I asked it to convert a chessboard, represented using a standard notation, into SVG graphics. This was the chessboard string:

"rnbqkbnr/pppp1ppp/4p3/8/4P3/8/PPPP1PPP/RNBQKBNR w KQkq - 0 2"

GPT-3.5 produced sensible SVG code, but the result was less than satisfactory. The chessboard was there, but garbled.

<svg width="320" height="320">
  <rect x="0" y="0" width="320" height="320" fill="#b58863" />
  <rect x="0" y="0" width="40" height="40" fill="#f0d9b5" />
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  <rect x="0" y="160" width="40" height="40" fill="#f0d9b5" />
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  <rect x="160" y="160" width="40" height="40" fill="#b58863" />
  <rect x="240" y="160" width="40" height="40" fill="#b58863" />
  <rect x="40" y="200" width="40" height="40" fill="#b58863" />
  <rect x="120" y="200" width="40" height="40" fill="#f0d9b5" />
  <rect x="200" y="200" width="40" height="40" fill="#f0d9b5" />
  <rect x="280" y="200" width="40" height="40" fill="#f0d9b5" />
  <rect x="0" y="240" width="40" height="40" fill="#f0d9b5" />
  <rect x="80" y="240" width="40" height="40" fill="#b58863" />
  <rect x="160" y="240" width="40" height="40" fill="#b58863" />
  <rect x="240" y="240" width="40" height="40" fill="#b58863" />
  <rect x="40" y="280" width="40" height="40" fill="#b58863" />
  <rect x="120" y="280" width="40" height="40" fill="#f0d9b5" />
  <rect x="200" y="280" width="40" height="40" fill="#f0d9b5" />
  <rect x="280" y="280" width="40" height="40" fill="#f0d9b5" />
  <text x="50%" y="50%" fill="#000" font-size="50">♚</text>
  <text x="12.5%" y="12.5%" fill="#000" font-size="50">♜</text>
  <text x="87.5%" y="12.5%" fill="#000" font-size="50">♜</text>
  <text x="37.5%" y="12.5%" fill="#000" font-size="50">♞</text>
  <text x="62.5%" y="12.5%" fill="#000" font-size="50">♞</text>
  <text x="25%" y="12.5%" fill="#000" font-size="50">♝</text>
  <text x="75%" y="12.5%" fill="#000" font-size="50">♝</text>
  <text x="50%" y="87.5%" fill="#fff" font-size="50">♟</text>
  <text x="12.5%" y="87.5%" fill="#fff" font-size="50">♟</text>
  <text x="87.5%" y="87.5%" fill="#fff" font-size="50">♟</text>
  <text x="37.5%" y="87.5%" fill="#fff" font-size="50">♟</text>
  <text x="62.5%" y="87.5%" fill="#fff" font-size="50">♟</text>
  <text x="25%" y="87.5%" fill="#fff" font-size="50">♟</text>
  <text x="75%" y="87.5%" fill="#fff" font-size="50">♟</text>
  <text x="50%" y="62.5%" fill="#000" font-size="50">♛</text>
</svg>

You can almost see how, and where, it went wrong. It started with a dark board in which it then attempted to color lighter squares in a checkerbox pattern, but it lost track in the third row. And when it positioned the pieces, it ended up omitting black’s pawns and white’s non-pawn pieces, while also placing black’s king and queen to where two opposing pawns were supposed to be.

Even so, this is pretty darn impressive for a language model. Details notwithstanding, it correctly understood the request, and came up with a structure that was not only valid but efficient for the chessboard representation.

GPT-4 fares a little better but it is still not perfect. It now draws the board correctly but still gets confused about the pawns. Somewhat surprisingly, however, it managed to swap light and dark squares on the board.

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 Posted by at 6:56 pm
Apr 012023
 

Oh my. Now that we are done with cryptocurrencies, NFTs and blockchain, AI is rapidly becoming the new fad. Apparently, all you have to do is call yourself a “prompt engineer” or “AI whisperer” to earn an exceptional salary.

Well, let this serve as advertising, notice to any prospective clients that my services are available as a consultant, either fixed-price or retainer based, should you wish to engage me. And, I daresay, with my in-depth knowledge and experience, which includes not just chatting with GPT but accessing it programmatically, as well as some hands-on exercises building simple toy models of deep learning architectures in low-level languages like C++, I am probably much more capable of delivering the goods than most of my “competition”.

Somehow, Bloomberg’s enthusiasm notwithstanding, I don’t expect any takers but just in case, my recent engagements with ChatGPT, as recounted in this very blog, should serve as my relevant portfolio. Not just words, AI-generated images, too.

 Posted by at 2:54 pm
Mar 262023
 

In my now regular weekly Patreon blog post to patrons, I mentioned, among other things, two Quora answers that I came across: One about the tomb of Jesus, the other about a Brazilian billionaire to pulled a delightful publicity stunt to encourage folks to become organ donors.

One patron responded by expressing a desire to write a story about Jesus becoming an organ donor. So I asked ChatGPT to write that story. Here’s the result:

Once upon a time in ancient Jerusalem, Jesus of Nazareth, the Messiah, walked among the people, spreading his message of love, peace, and salvation. Jesus healed the sick, fed the hungry, and uplifted the downtrodden. His followers were many, but so were those who sought to silence him. As his crucifixion drew near, Jesus began to contemplate the nature of sacrifice.

The night before his arrest, Jesus gathered his disciples in the Garden of Gethsemane. They sat together, sharing their final moments before everything would change. Jesus looked at each of his disciples, his love for them evident in his eyes.

“My dear friends,” he began, “tomorrow, I shall give my life for the salvation of mankind. But my sacrifice is not just about my death. I want to leave a legacy that will endure and inspire generations to come.”

The disciples, filled with a mix of confusion and sorrow, listened intently. Jesus continued, “As I lay down my life, I wish to donate my organs, so that others may live and be healed through my sacrifice.”

The disciples were struck by the profound nature of Jesus’ request. They had seen him perform countless miracles, but this act of generosity was like none they had ever witnessed.

Jesus continued, “I know that the concept of organ donation is not yet known to the world, but I trust you to carry out my wishes. My heart, liver, kidneys, and other organs can bring life to those who need it. It is my hope that this act of selflessness will become a symbol of hope, love, and compassion, inspiring generations to follow in my footsteps and care for one another.”

As the disciples absorbed the enormity of Jesus’ request, they couldn’t help but be moved by his endless capacity for love and sacrifice. They vowed to honor his wishes and do everything in their power to ensure that his organs would bring life and hope to others in need.

The following day, Jesus was arrested and crucified. The disciples, still reeling from the events, knew that they had to act quickly to fulfill his final request. They sought the assistance of Joseph of Arimathea, a secret follower of Jesus, who had connections with the local medical community. With his help, they were able to carefully remove and preserve Jesus’ organs, ensuring they could be used to heal others.

In the days and weeks that followed, Jesus’ organs were transplanted into those in desperate need. His heart found a new home in a kind-hearted father, whose family relied upon him for their survival. His liver was given to a woman whose own body had betrayed her, allowing her to live and continue her work as a compassionate healer. His kidneys were shared between two brothers, both plagued by illness, who went on to become steadfast followers of Jesus’ teachings.

News of these miraculous transplants spread far and wide, igniting a newfound sense of hope in the people. The legacy of Jesus’ sacrifice was not only in his resurrection but also in the lives he saved through his selfless act. His decision to donate his organs inspired others to consider the possibility of giving life even in death, planting the seeds for a future where organ donation would become a common practice, saving countless lives.

As the years passed, the story of Jesus’ organ donation became a powerful symbol of love and sacrifice. It was a reminder of the importance of compassion and the interconnectedness of all people. His selfless act transcended religious beliefs and cultural barriers, inspiring many to follow in his footsteps, giving the gift of life to those in need.

In time, organ transplantation evolved into a sophisticated medical practice, saving millions of lives worldwide. The foundations of this life-saving procedure were built on the courage and foresight of Jesus and his disciples, who dared to envision a brighter future for humanity.

Jesus’ organ donation not only saved lives but also fostered a spirit of unity among people. It served as a constant reminder that even in the face of suffering and death, there is always hope, love, and the potential for rebirth. It is said that through these acts of giving, the light of Jesus continues to shine brightly in the hearts of those who carry a piece of him within them.

In this way, Jesus of Nazareth’s message of love and salvation endures, not just through his resurrection and teachings but also through the precious gift of life he bestowed upon others in his final moments. His organ donation became a testament to the power of compassion and selflessness, inspiring countless others to follow in his footsteps and embrace the gift of giving. And so, the spirit of Jesus lived on, touching the lives of people across generations, and reminding the world of the eternal nature of love.

Using Michelangelo’s fresco, The Creation of Adam, as an illustration was also ChatGPT’s suggestion.

 Posted by at 6:27 pm
Mar 262023
 

These lovely knitted bunnies and the felted eggs were made by my wife for Easter. She was hoping to sell a few of them at 613flea, but there were no takers. I cannot understand why… I love them. In fact I told her that if there are no takers, I’m happy to buy them.

Oh well. Maybe we should put them up for sale on eBay or Etsy.

 Posted by at 2:00 pm
Mar 252023
 

I managed to piss off Bing’s version of the GPT-4 chatbot. I swear I didn’t mean to. All I wanted was to check if it inherited the capability to do simple art in text-based languages like SVG or TikZ. Instead, it told me (politely) to be somewhere else.

I admit I felt hurt. I went back to my old friend ChatGPT and asked it to draw me an SVG cat instead. It gladly obliged. Its cats are not perfect but always make me laugh.

 Posted by at 1:26 pm
Mar 242023
 

Inspired by an intriguing paper from researchers at Microsoft, I thought I’d check GPT-4’s ability to use tools that I can provide.

vttoth: The computer is answering questions. If the computer needs any current information to answer the question, it searches the web by saying SEARCH(“query”), reads the snippets in the result, and then answers the question. If it needs to run any calculations, it says CALC(expression), and then answers the question.
First question: Who is the president of the United States?

ChatGPT: SEARCH(“current president of the United States”)

vttoth: First snippet: Joe Biden: The President — As President, Biden will restore America’s leadership and build our communities back better.
Second snippet: Presidents, Vice Presidents, and First Ladies of the United … — The 46th and current president of the United States is Joseph R. Biden, Jr.
Third snippet: Joe Biden — Joseph Robinette Biden Jr is an American politician who is the 46th and current president of the United States.

ChatGPT: Joe Biden is the 46th and current president of the United States.

vttoth: What is the derivative of the function x^2*exp(x)*sinh(x)/(3-x) with respect to x?

ChatGPT: CALC(differentiate(x^2*exp(x)*sinh(x)/(3-x), x))

vttoth: -(((x^3-2*x^2-6*x)*exp(x)*sinh(x)+(x^3-3*x^2)*exp(x)*cosh(x))/(x^2-6*x+9))

ChatGPT: The derivative of the function x^2exp(x)sinh(x)/(3-x) with respect to x is -(((x^3-2x^2-6x)*exp(x)*sinh(x)+(x^3-3x^2)*exp(x)*cosh(x))/(x^2-6x+9)).

vttoth: What is the periodic table?

ChatGPT: The periodic table is a tabular arrangement of the chemical elements, organized by their atomic number (number of protons in the nucleus), electron configurations, and recurring chemical properties. Elements are presented in increasing atomic number, with elements in the same column (or group) sharing similar chemical and physical properties. The table has seven rows (or periods) and eighteen columns (or groups).
The periodic table provides a useful framework for understanding and predicting the behavior of elements and their compounds. It was first devised by Russian chemist Dmitri Mendeleev in 1869 and has since been revised and expanded as new elements have been discovered and our understanding of atomic structure has evolved.

vttoth: What is 2364563737^23?

ChatGPT: CALC(2364563737^23)

vttoth: 394705859703756319737306986950805026028714653598749322343446098781557781\
086546539605957002415205970145485533933900705444118389738263803127936989171236\
616893794109431624734589354089362682114750349831212070341482062953

ChatGPT: The result of 2364563737^23 is 39470585970375631973730698695080502602871465359874932234344609878155778108654653
96059570024152059701454855339339007054441183897382638031279369891712366168937941
09431624734589354089362682114750349831212070341482062953.

Notice in particular how in the last example, ChatGPT didn’t blindly copy my response; it understood that the backslash character was just marking newlines (copy-paste from Maxima; I forgot to remove these backslashes).

Also, I inserted the question about the periodic table to check if it is smart enough NOT to ask for a Web search for information that it knows and which is unlikely to change.

The ability to use tools with minimal instruction is another hint that we are indeed witnessing the birth of artificial general intelligence.

 Posted by at 3:25 pm
Mar 232023
 

My latest interaction with ChatGPT:

vttoth: Please write a short piece of music, suitable to be played while showing the stylized image of a cat, in Lilypond.

ChatGPT: LilyPond is a text-based music engraving program that uses code to create beautifully engraved sheet music. Here’s a short piece of music written in LilyPond, suitable to be played while showing a stylized image of a cat. This piece is intended to have a playful and curious character, much like a cat.


\version "2.22.1"
\header {
  title = "Cat's Paws"
  composer = "ChatGPT"
}

upper = \relative c' {
  \key g \major
  \time 3/4
  g4. g8 a g8~ | g r8 a2 |
  a4. a8 b a8~ | a r8 b2 |
  b4. c8 d b8~ | b r8 c2 |
  g4. g8 a g8~ | g4. r4. |
}

lower = \relative c {
  \key g \major
  \time 3/4
  g2 d4 | g2 r4 |
  a2 e4 | a2 r4 |
  d2 c4 | d2 r4 |
  g2 d4 | g2 r4 |
}

\score {
  \new PianoStaff << \new Staff = "upper" \upper \new Staff = "lower" \lower >>
  \layout { }
  \midi {
    \tempo 4=120
  }
}

Copy the above code into a text editor and save it with a “.ly” extension. Then, compile the file with LilyPond to create a PDF and MIDI file.

I admit I had to make a few adjustments to the code (specifically, the upper voice) as the timing was off in a couple of places, but I don’t think I altered the music.

 Posted by at 11:13 pm
Mar 232023
 

My wife’s beautiful creations will be at 613flea again this Saturday. Rufus the cat won’t be; he just volunteered as a model.

The Easter eggs are felted decorations. There will also be some knitted bunnies.

 Posted by at 10:10 pm