Turing Machines - How Computer Science Was Created By Accident
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- Опубликовано: 3 фев 2020
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I always feel the sads thinking about how much Turing contributed to humanity and how little humanity gave him in return
it wasnt "humanity", it was homophobic christians. Lets not paint all of humanity with one brush and hide the christians amongst them.
@dodeca hedron We shouldn't really be blaming any religion based on what happened to Alan Turing. It's more towards society. After all, even in the Islamic society, we are prohibited from being homosexual but we are never taught to put hate on those who commit it. Think of it like the act of premarital sex. Some may think it is a sin but some say it's their right to obtain abstinence or not. It's a matter of our own choice for whether we can accept the person regardless of their sins. We are against the act, not the person who did it. Let's just be grateful for what Turing has done for our generation. (◍•ᴗ•◍)✧*。
@dodeca hedron Christianity did not birth science. Up until quite recently in our history, the church had actively suppressed any scientific advancements which went against their false doctrine.
While the modern church does a lot of good, it was not always this way.
And Christianity didn't do anything on anyone's behalf lmao, everything the church did was to increase their own power and wealth, just like every other empire throughout history.
So while I agree that they shouldn't be completely demonized, they should not be praised either.
Also, Christianity did not start or even popularise the idea of looking of the poor and orphaned. That and most of what you wrote is false, you shouldn't spew unfounded statements just because you read it somewhere.
Comparing science to religion makes no sense, since the one is a collection of stories and rules someone wants you to live by, and the other is just a tool.
dodeca hedron It was conservative Christian morals that led to Turing’s treatment. Not society as a whole. Islam has its own problems, but when speaking of Turing, Christian morality of “thou shalt not” (as opposed to “does this action cause harm?”)was the cause (because England was legally a Christian nation), not any societal look into resources.
In fairness to humanity, most of humanity was not even aware of his contributions during his lifetime. Many were secret and the rest were esoteric. It's a little like finding out fifty years later that your crazy uncle really was protecting the planet from devastation.
dodeca hedron:
Neither christianity nor christians birthed science. Christians may ultimately have figured out that it was to their advantage to _advance_ science. But the Greek philosophers were engaged in science several hundred years before christianity existed.
I tried to stay in Hilbert's Hotel, but they kept changing my room.
I majored Computer Science. This is my professor's first slide in Professional Software Development class: Computer Science is not Science, Software Engineering is not Engineering. He was quick right. Computer Science is mathematics, which has nothing to do with science. All engineers require a engineer license, software engineers don't need one.
A job board run by the state of Minnesota has 23 top level categories. One of them is "Computers and Mathematical".
When Edsger W. Dijkstra was filling out paperwork for a marriage license, he was asked for his profession. He wrote down "Computer programmer." A clerk rejected that form, saying there was no such profession. He changed it to "Applied mathematician."
Couple of fun facts:
- Lambda calculus is still used a _lot._ In fact, its the basis of many modern programming languages. Lambda calculus is more symbolic and tends to be easier for humans to deal with, while Turing machines are well... machines, and easier to implement as hardware. So the equivalence between the two means we can build compilers to translate from (relatively) human-readable languages based on lambda calculus to the list of 1s and 0s that make up Turing's model of computation.
- Quantum computers are _not_ more powerful than Turing machines. They can perform certain classes of computations in polynomial time that a classic computer requires exponential time for, but in terms of decideability (which is what math-y types are usually talking about when they say "powerful,") they are exactly equivalent. That's why its possible to build simulators for quantum computers that run on a non-quantum computer. They just can take a really long time to run (depending on what operations you asked your your "quantum" computer to perform.)
Of course, quantum computing is still extremely new and its possible someone will invent a (computationally useful) quantum gate that absolutely cannot be simulated on a classic computer no matter how much time you give it. So quantum computing still has the possibility of becoming more powerful than a Turing machine, but under our current formulations it is not.
Yes, I think quantum computing adds a "dimension" to conventional 1D (one spatial, plus time) computing that could yield its own new paradigm. Thus new ideas that can be proven that currently cannot be proved nor disproved. The fact that we have an inkling of something about what can happen with quantum computing, is a strong clue there is something in our own cognitive constitution that can be connected with once we open the quantum gate. That is to say, we probably have quantum element to our own mental cognition, biological brains probably have quantum functions... and maybe others we haven't even started on yet. Imagine when we figure out how to compute in 4 spatial dimensions plus time, meaningfully. :)
@@braveecologic2030 > "conventional 1D computing"
I'm not sure what you mean by "1D". Computers are real-world devices that exist in 3 dimensions. Maybe you mean the fact that we tend to lay out memory as a 1D array? In which case, there is nothing fundamental about that. We can (and have) built 2D-addressable memory. Not sure if we've ever tried to build 3D-addressable memory but we could in principle (its mostly a manufacturing issue with trying to build electronic structure "inside" a cube). The only reason we don't do that more often is that its not terribly efficient. Its easy enough to compute ((z*ymax)+y)*xmax+x to convert a 3D coordinate into a 1D array index. Far easier than the engineering involved in making the physical array addressable in higher dimensions. (Also, even though we set it up as 1D-addressable, all modern memory chips are physically constructed as a 2D structure on a wafer.. and the wafer is technically 3D but we try to make them as thin as possible and ignore the 3rd dimension due to the manufacturing difficulties noted above).
> compute in 4 spatial dimensions plus time, meaningfully
We can, and do, compute things in 3+1 dimensions regularly. Some physicists compute in 10+1 (ie: M-theory) and mathematicians even do computations on infinite-dimensional spaces at times.
"Meaningfully" is rather in the eye of the beholder though. The computer (quantum or classical) doesn't really care how many dimensions you're using. Higher dimensional work tends to take longer, but computers don't get tired either. The only concept of "meaningful" that's actually well, meaningful, is how obvious any result is to a human's intuition and sadly, its a biological limitation that prevents us from understanding anything higher than 3+1 dimensions - no amount of computing power will ever abridge that limitation.
(Well maybe. I guess if we ever get to a point where we can directly interface computer technology with our own brains to the point that we're literally "thinking" on the computer chip, then perhaps it would be possible for our intuition to be extended beyond what pure biology allows.. but we're talking technology that may not even be possible and certainly not within the next few centuries).
@@altrag I'm talking about step by step computing (the process) being 1 dimensional. Conventional spacetime can be considered in 5 dimensions, the additional spatial dimension being akin to inward outward and can be readily visualised by viewing a solid object in the centre of your mind and seeing all it's outer surfaces at once, you can then use time to go inside the object for the reverse perspective. I talked about that in my book, Sustainability and how to be a meaningful species in the Cosmos. Taking those perspectives and developing computational systems, processes and algorithms that leverage that, could be pretty cool.
@@braveecologic2030 Apologies. I had my suspicions but tried to respond as if you were using any form of logic based in reality. I'll leave it to you to figure out how quantum computers (or anything else) should work in whatever woowoo world you believe you're living in.
@@altrag There's not really a need to be insulting. I understand current technology and did you know, throughout history there have been people who use their minds to think of new things. Give it a go. While you're at it, don't insult me.
11:18 "Turing realized that internal state tables could be encoded as 1s and 0s..." That could use some additional exposition.
yeah but the vid was already 17 mins
@@upandatom Point taken. Maybe in a follow-up video -- the whole "hard wired / single purpose" computer vs. the "programmable / general purpose" computer dichotomy.
@@upandatom tbh I was most excited about tgis part and was disapointed. I hope you make a follow up video because your videos are an amazing balance of cute animations and awesome content
That's the entire area of Information Theory, its a pretty big sub-area of Computing Science.
Now go read Shannon
Up and Atom
you make 15 minutes feel like 6.... wonderful video, by the way.
You really do have a skills for teaching. This is by far the best introduction to Turing Machine explanation I have ever seen. I can see you work hard to deliver quality content. Great Job !!
Thank you :)
I second that!
@@tolex3 I third that. Or should I say I 11 that.
Excellent video as always. I always kinda felt that my college could have spent more time on computer history.
Oddly enough, when I was going for my degree in Computer Science, Turing barely got a mention, whereas Lovelace and Babbage were covered in great detail. Lady Lovelace was brilliant as she was writing code for a computer that didn't even exist yet. What's more, it was proven that her code worked.
Professor Alan Turing will be remembered forever for his contribution♥️✨
15:00 : Quantum computers can't actually compute anything that a Turing machine can't compute; they're important because they can compute some things _faster_ than any device based on classical physics.
Depends on the number of qbits although exponential quantum computers have only just over taken the classical computer.
Also we are yet to design a reliable method for excluding errors.
Still a long way off and even longer before you or i are sat in front of one at home which your grandchildren's grandchildren wont be experiencing either.
A Quantum Computer can produce an infinite truly random stream of bits. This can't be computedby a classical Turing machine. The model equivalent to a quantum computer is the probabilistic Turing machine, the one with a perfect random number generator. This is significant.
@@annaclarafenyo8185 Can you pls. explain? We can understand extremely chaotic systems / events which may look like truly random events. However they are deterministic and mathematically precise and repeatable and not possible to be considered as truly random. Such systems' components should obey rules of Physics. Such things can not be called random in the true sense of the word. I think it is a giant question with implications for many a phenomenon such as consciousness, free will, which leads - mind or matter, is mind an emergent property of matter just as mass and charge etc....
@@kpk331 Your chaotic system model isn't classical computation, but deterministic real number systems. This is not a physical model of computation, you have to convert it to a computer program. When you do this, you end up truncating the real numbers.
If you have a deterministic system described by a differential equation (even a chaotic one), one which is simulated on a computer, and you start at a computable initial condition, and you evolve for a computable amount of time, you end up at a computable position.
A truly random real number, an infinite stream of random 0s and 1s, is uncomputable with probability 1. This is why the model of random numbers different from classical computation.
@@jackburton8352 : As of now... a quantum "computer" cannot be truly called a "turing complete" machine, since it is relying on probabilities. For it to be completed, then means that, the quantum bits, and the complete remanifestation of molecules can be created at will, by will. IF that happens, then it can be classified as complete. Right now, we cannot regenerate something from nothing. We have not even begin to create complete machines to do that yet. So far, we have generally recreated and used computers to count. That is all that we have achieved. Can you create an apple purely from pure molecules and atoms? The answer is a "no". So no... no quantum machines exist as yet. We merely have "quantum calculators" (this is a different thing). To truly call something a quantum machine, based on Turing's Completeness' Theory. Then it means that we should be able to make something out of its smallest components. e.g. atoms.
Really nice vid. Love the location and the way you filmed it. Tom Groenestyn's animations are just amazing! Everything is so well timed and spot on. The dialog is soooo interesting! You guys produce exactly the kind of stuff that gets me thinking and improves my knowledge and makes me want to learn more. Thank you, thank you :)
Fabulous! I appreciate how you take the time to insert additional info, like Turing's relationship to Alonzo and the future potential of quantum computing. Keep up the good work!
Absolutely amazing work.
I imagine how hard you may have worked on these animations which are so informative and funny at the same time. Thank you so much for this.
I have a PhD in computer engineering, and I approve of this video. 😁 Seriously, Jade did an amazing job.
I have a lowly Bachelor of Science in Comp Sci but also approve.
I have a phD in spotting bullshit. Seriously... shut up.👍🏻
Carl G can you provide a link to your graduating thesis?
but what was your focus because computer engineering plays better with electrical engineering far more than it does with Computer Science.. I am about to graduate with my B.S. in Computer Science
?are you sure your PhD is in computer engineering??
It's great to see you delving into computer science topics. I've been using them in my classes and my students love it! Keep up the good work! Wish to see more of them in the future 😁
I surprisingly never knew about Turing machines before. I really love your animations, and they're great ways to teach visually.
The birth of Computer Science as it drops to the ground beneath the Turing Machine was for me a real highpoint. Especially the quivering...I would also like to suggest that other disciplines have fundamental disagreements just as challenging as the ones in Math. I continue to really enjoy where your mind is taking us, Jade!
Jade, your animations never fail to make me keel over with laughter. Learning something is just an added bonus. Keep up the great work!
I think it’s the other way around..
A great video which is very helpful in understanding the subject "Theory of Computation' , the birth of computers and 'Computer Science' on the whole, . Perhaps this is the best video I have come across on RUclips. Thanks.
I thought I already knew this topic well, but this video gave me a new clarity of understanding of it. Great explanation!
Computer science is quite new to me and i am glad you decided to explore this more from now on.
This story was great! Probably i enjoyed watching this video the most😊
Feeling excited for upcoming videos✌️
thanks!
The is amazing, Jade has explained the Turing machines is such a succinct manner.
I am amazed by the clarity and conciseness of your video. A true pleasure even for the ones already fluent in this topic.
Thanks a lot for sharing this, this has inspired me more to study deeper all the explained topics. Loved your video.
Thanks, Jade. It gets even more interesting when I realize that I am watching this video using a Turing machine. And that it was made using a series of Turing machines. And also presented using Turing machines.
This was one of best explanations I've seen. Thank you so much!
You are the best math/science teacher on the internet! Your explanations are always understandable, well-motivated, and (drum roll, please) accurate! Keep up the good work!
Jade, your teaching is first class. Very engaging and, of greatest importance, just simply so enjoyable to watch. I hope you continue with this stuff!
crystal clear, amazing talent for explaining complex science based subjects, you go deeper than most of the others creators
This is the video that made me understand the Turing Machine. Thank you!
I really liked the explanation that shows how you could get from human computers to the idea of a Turing machine. That really helped - I've not seen it explained that way before and it really helped my understanding! I also found it exciting to learn that the goals of consistency, completeness, and decidability went back to Hilbert - these are things that we're often concerned about in the design of programming languages, and it's fun to learn that it goes back that far!
Nice to see you giving a shout out to the lambda calculus at the end - I was hoping you would! It's turned out to be pretty influential now in the field of programming language design these days, and is forming the basis of current efforts to formalise mathematics using computers (using type theory as opposed to set theory). :) :)
I just realised those Dilbert jokes... lol.. It is a parody of Hilbert. LOL...... God...
Okay so I've started binge watching your vids.. I'm retaining more knowledge from your format that I ever did in school! Thank you.
Ordered an audible on your rec..
Wish you were state side 🥰.
Teaching is your calling I'd say.. take care, stay safe in these harder times.
Lake Havasu-
Az, USA.
I recommend the tree method of solving recurrence relations as a topic. It shows where the formulas in Big-O notation come from.
Good idea! But only works for linear recurrences.
Just discovered this channel & I'm loving the content as well as animations
Excellent video Jade! Keep up the great work!
Hardware wise every computer can be described by the Von Neumann architecture, software wise everything a computer does cna be described by a touring machine. The simplicity of these models (which open up an endless world of uses) is beautiful.
The FSM example provides a very concise insight into the Turing machine - this is brilliant 🎉
You are great at explaining things, I saw the same concept at various other channels, finally now I really know what was happening
very clear explanation. i learned how the turing machine works and what is an algoritm
Rewatching, and I really love this channel!
I sit back now and imagine how they did find all this hard but profound concepts at that period of history and that too so quickly is beyond me. Truly marvellous. Because they were ground breaking.
For the record, the human "computers" did not just perform rote calculations, they often had math degrees, or at least a talent for math, and part of their job was developing efficient algorithms to produce quick and accurate computations based on the requirements and formulas presented to them by engineers and scientists. (These algorithms they developed are why many of them could compute trajectories or orbits in near real time.) Many of the procedures and tricks they developed formed the basis of computer software in later years. (A lot of "computers" ended up programming computers.) "Rise of the Rocket Girls" by Nathalia Holt is a fun read on the computer team at JPL in the 40s to 60s for anyone interested in the subject.
I often use a Turing Machine based module in my Eurorack synthesizer case, and now I finally learned the reasoning behind this, so thanks.
Amazing!
You're the best UwU
And you're animations are also the best!!
This puts me into contemplative mode... Just pondering the beauty of those ideas, ideas are more beautiful than their realizations to me. And Computer Science is a beautiful idea.
VERY thumbs-up for this one, Jade! Wonderful for a wide age range. Thanks again!
Excellent break down of the issue! I had an undergrad professor that was a student of Church's, but had no idea that he was Turing's advisor as well. You've taught this philosophy PhD something new.
Thanks so much! this is the first explanation that made it 'click'. Including the state table was very informative
Enjoying your channel. Keep up the good work!
Hey, I love your videos! I have been showing them to a lot of people to try to get them interested in different topics and now a lot of my friends like math more. I think Godel's incompleteness theorems are really interesting and could grab a lot of people's interest if there was a video that made it easier to understand while still showing people how interesting it is.
From about 0:51 - God I love that music! It''s like a warm "Everything will be explained" hug, in sound somehow.
Great video. Finally someone in youtube explains Turing machines with grammars and languages
Actually, the model of computing that caught on was the von Neumann model. The von Neumann architecture, with its concept of a CPU, with registers & ALU, reading its instructions from memory, the same memory where data resides (loaded from I/O ports if necessary), allowing programs to be reprogrammable as easily as writing the result of 2+2, was even easier to understand than the Turing model.
You girl, are amazing! thank you for breaking it down and making it easy and fun to understand. You do an amazing job every time!
A truly wonderful explanation!
This is an incredible video. It is really intuitive and I can easily recommend this video to someone who is interested in theoretical computer science. Great video! 👍
Just subbed yesterday, from now on I'm a member of the early squad.
As usual it's a very interesting video.
Congratulations Jade!!
Thanks for making this video. I enjoyed it.
A very nice presentation of a pivotal moment in our shared history.
Interesting. I've heard of Turing Machines, but Havn't really heard anyone talk about them before. Thank you.
And i just want to say that Jade is a great teacher, she explains every topic very elegantly.
Very cool, easy-to-understand, informative video presentation! Tibees (Toby Hendy) did a great video on Alan Turing too.
Well made video, Turing work on chemical morphogenesis is really interesting also!
Hi Jade! Great video! There is another story in early computer science that startled me: The accidental invention of the first compiler (compiling a subset of Bertram Russel's Principia Mathematica into numbers and calculations) by Kurt Gödel when he proved his incompleteness theorems. This so much reminded me of modern day compilers...
Would you like to tell the story of Gödel's first compiler? It is another story that defeats the Hilbert Programme.
You are like the teacher that every person wish they had when they were students. Clear with context.
I love, love, LOVE history of science. Not just what the discovery is but how it came to be. LOVE it!!!
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'Cause an Up & Atom party
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Wow, I am amazed by your breadth of knowledge! How come you explained so well having a physics background?
well, wonderful video and explanation, thanks !!!
I would like to see a video which goes more in depth about formal systems, and the subsequent proof that a sufficiently powerful one is necessary incomplete by nature by Godel. I think it is an interesting alternative perspective of how to answer Hilbert's questions about mathematics.
very informative video jade, so much of knowledge in so little time, really amazing, please keep it up....I have a request, can you please make a video on randomized algorithms? i think i am in need of that....
Thanks for video. You have inspired me. Your other videos seem very interesting, i subscribed.
excelent explanation of the turing machine. Maybe the best.
Great video! Please make one on Incompleteness theorems too.
The concept of "Effective" procedure touches on your "P=NP" video; related themes.
The 2 most influential people of the computer age were Alan Turing and Tommy Flowers. We owe them both so much for their contributions to our current day technology. Thanks for posing such an interesting video.
Von neuman
Hi, I really enjoyed your video. You might be interested to know that in the 1990s, Hava Siegelmann proved mathematically that there is a range of computation from Turing and one end to the most powerful form, which she called Super-Turing computation. She published a really interesting book in 1998 on the subject, “Neural Networks and Analog Computation: Beyond the Turing Limit. Siegelmann is a really brilliant mathematician and computer scientist. While “Hypercomputation” is essentially a vague group of conjectures by philosophers, Super-Turing computation is a mathematical fact and is currently the basis for a new generation of computation being spearheaded by DARPA (the government research agency). Siegelmann, in fact, is the head of the L2M (Lifelong Learning Machine) program there. Her project has taken the first steps toward computational systems that alter their programs in real time based on input from the environment. Her book is an excellent place to start. You might want to do a video on Siegelmann and the L2M program.
Her work is incorrect. It is based on the false idea that atomic systems like in the brain can do "analog computation". They can't, because of the Landauer bound. To work with a bit of data at temperature, to keep it from thermalizing, you need a certain amount of energy, roughly kT log 2 per bit, where k is Boltzmann's constant. There cannot be analog computations in brains, as the real valued quantities in brains can only be read out to finite precision by the biological mechanism, and the rest of the precision is lost to thermal and quantum jitter.
I am not that familiar with Siegelmann's work, but as a former mad scientist at both NASA and DARPA, I do know that Turing machines can simulate Neural Networks (been there done that) AND that Neural Networks can simulate Turing Machines (most obviously in Turing's head, but I think it's been mathematically proven, at least for integer weights). The big question is whether it is a 1-1 correspondence?
I suspect that being able to solve the generalized Halting Problem requires something more powerful than a Turning Machine. However, SonarQube today can solve the Halting Problem on your laptop for many instances of Java code. Can it's abstraction model be extended to "understand" code samples well enough to solve *ALL* classes? I simply don't know. I don't even know how many possible classes exist. I do know that I can't run them all, and my day job does not pay me do this kind of abstraction. I suspect and fear that it's turtles all the way down-i.e. just as SonarQube can solve the Halting Problem for some classes of code samples, adding another layer of indirection may solve more classes of code samples, but others will remain no matter how many times you abstract the relevant features of the abstracted code.
The question I would have (especially for Siegelmann) is what additional abilities emerge in Neural Networks when weights are not just integers, but Rational numbers? Or Real, or Transcendental numbers? The number of states is obviously going to get ridiculously larger, but so what? After all, math allows us to work with big numbers fairly easily. Ignoring efficiencies, is there anything that Neural Networks (like our brains) can do that Turing machines can't? In a finite universe, in which efficiencies do matter, I suspect so. However, our brains are only made of quanticized atoms, so why can't machines (in the worst case), just simulate them?
8:26 "equal" would be a better word here, "even" may mean divisible by 2.
that is how I interpreted it at first. I was thinking why she was using such a weird example b4 realizing that we r looking for equal 1s and 0s
To be fair, the context makes it clear enough. If somebody said to me they had an even number of black cards and red cards, I would instinctively assume that they had the same number of each, not that they had, say, 8 red cards and 6 black cards.
@@X22GJP - a few weeks of having to deal with programs for a living and I guarantee you will have lost all appetite for such assumptions ;)
@@X22GJP Maybe that's a Commonwealth usage? As an American, I would assume that the number of black cards and the number of red cards were each divisible by two.
No
Great video. Some day, I hope that you'll take a look at sewing looms and adding machines. These are mechanical computers at their heart.
I think this is the best video about understanding turin machine
Hey Jade! This was a really interesting video and I'm looking forward to anymore CS videos you decide to do in the future. ;)
hey you are a brilliant teacher , turing machine never made sense to me before this video , thanks
Now in 2022 I learned of the Google Quantum Computer that physicists used to determine that wormholes can exist in space.
I love these videos. I am getting much better educated than I ever could in school. You have a talent for making the complex simple!
I didn't know Turing Machines were that amazing! Very cool!
Your crazy hyper melodramatic cartoons just keep getting better. They are like cognitive punctuation, and the little laugh I get becomes the touch point in my memory. Keep it up, and bring back wild eyes Turing ASAP!
For me, it is 'If A Turing Machine Can't Compute It, It's Not Computable'. Had a lot of fun studying this course as grad student.
What a fantastic explanation!
This makes me wonder what are the foundational axioms of problem-solving if we are to treat it as a formal system?
Great video, as always! 👏
4:20
I never thought I'd see an animation of Hilbert simping over Euclid lmao
I love the fact that, contrary to many youtubers, you seem to film each of your video in a completely random location.
Seems like you haven't watched Tom Scott videos then
Hardly random, just different
@@martiddy With Tom Scott the place is usually related to the story.
@@X22GJP It is random in the sens that she is talking about Turing Machines from a shed or a garage with an electrical box as a background.
Brilliant video. Thanks!
Another great video Jade. You deserve more than 150k subscribers - have you considered joint videos with other channels like Mathologer, Numberphile, Standup Maths, Veritasium, Eddie Woo, Braincraft or The Australian Academy of Science ? (All run by aussies, as it happens). You could also consider Minute Physics, Computerphile, Steve Mould, Smarter Every Day, 3 Blue 1 Brown, Scishow, Chris Hawkes, Physics Girl, Domain of Science etc.
yeah you're right I should network more :(
It's amazing how many crucial inventions where made accidentally by someone just goofing around, penicillin coming to mind also. Excellent angle and great video! Good to see you're still going strong.
Love the animations BTW. 😆
Jade always has the clearest explanations!
I love everything about this episode.
The best video I have watched on the topic!
I've done that course, decades ago. Turing machines - you hear about them once, you remember forever. Lambda calculus: you cram it into your brain, it escapes under high pressure as soon as the final exam is over.
This is your best work, Jade.
Always watch full promotion at the end...out of sheer respect for the creator and the thought put into not bombarding with promotion even before the video. Great content..!!
Me: reads title
Also me: now I am not sad about realizing I am a mistake
If Microsoft implemented a Turing Machine they'd load the program into high memory and only allow a finite space for the data and Steve Balmer would come out on stage shouting at how lucky we were to have 640 kb and what a great feature this was.