Lambda Calculus!
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- čas přidán 28. 06. 2022
- TRUTTLE1 DISCORD: / discord
(It's now called the Bale of Esoturtles because why not.)
Have you ever wanted to have a programming language/mathematical system that literally just took functions and applied other functions to them? Well too bad, because that's what Lambda Calculus is! And it's Turing complete, so shut up about it being limited.
LINKS:
A Lambda Calculus interpreter called Lambster: lambster.dev/
Alligator Eggs: metatoys.org/alligator/
Some scenes generated with 3Blue1Brown's Manim: github.com/3b1b/manim
MUSIC:
"Jr Troopa Theme" from Paper Mario 64
"Bit Shift" by Kevin MacLeod
"Monody" by TheFatRat
"The Last Dungeon - Encore" from Wonder Boy: The Dragon's Trap - Věda a technologie
Based on the exclamation mark in the title, we can conclude that the lambda calculus is fact an unintentional esolang.
That was the intent. It is Turing complete but pretty different from most (but not all) mainstream programming languages.
@@PefectPiePlace2 "That was the intent" was describing my intent on putting this video in my esolangs playlist, not the intent of Church himself.
@@PefectPiePlace2 I was also planning on eventually making videos about esolangs such as Unlambda or Grass, but those both use Lambda Calculus.
It has some other interesting properties.
For example, mathematicians tend to be scared of paradoxes, and go to a lot of trouble to constrain their theories to rule out paradoxes. But λ-calculus can actually express a paradox as an expression that you can do manipulations on and draw logical conclusions from, and reality doesn’t suddenly collapse around your ears.
@@Truttle1ya gotta love truttle1 responding to a ghost
"but its a card game" this line read was perfect
This is hands-down the best explanation of lambda calculus I've ever heard. Good job!
functional programming is applied category theory, where a coconut is just a nut
Now this is interesting. Do you know any book or article that I can read on the subject?
@@diogosimao there is a youtube video series explaining category theory if you're interested
@@aravindmuthu5748 yes!
I am a junior in college, this single video has helped me more then any lecture i have experienced in this semester. Thank you
I am also a junior in college lol
And when you implement an evaluation algorithm, you get LISP.
Make it with better syntax and you get Haskell
LISP syntax is what it is because it is homoiconic.
@@talkysassis make it ugly and you get Erlang
8:20 it should be `(a successor) b` not `a (successor b)` [which technically = (b+1)^a] I can't believe this wasn't immediately apparent in the extremely clear and human readable syntax of lambda calculus smh
Oh wow now it makes sense lol
It looks like in the very next scene (when "add" was expanded into its definition) it was fixed to have removed all parentheses. So I guess it was just the graphic at that timestamp you provided
Genuinely glad to see you are covering more computer science related stuff. I've been fascinated with lamda calculus and how it can be used to do math.
Very sorry about missing the premiere, anyone who forgives me will be sent a 50% discount on their next purchase of Tux Cola.
I forgive, now where that tux cola
This whole paradigm could be represented with the new esolang I'm calling "Threadr".
You know those fancy thread first/last functions (-> and ->> respectively) in Clojure? Those are the only two things that are allowed in the language other than basic math and lambda definitions/application!
sounds like the qi racket library
never heard of them cause i never touched clojure. could you elaborate?
Guess i'll have to stay up until 2am to watch this master piece
There are so many fun things in terms of Turing completeness!
Microsoft PowerPoint, HTML and CSS (if used together), SUBLEQ, and heck there was a sigbovik paper that was Turing complete.
PowerPoint is my favorite game engine
I will need to watch this x times and play with it for a long time to wrap my head around this.
Ok, X = SUCCESSOR 100
You can turn any algebraic datatype into a lambda calculus representation: values of the type are represented as functions that perform one level of pattern matching. For example if you have a standard functional singly-linked list type, then your pattern matching needs to know what to do with a cons, and what to do with an empty list, so it takes two functions (which I'll call f and x). In that case, the list [1,2,3] is the function: (^f. ^x. f 1 (f 2 (f 3 x))), and the empty list is (^f. ^x. x) i.e. the same as false and Church zero.
The ADT for a boolean is just a choice between true and false, and translates to the same as the ones you describe in the video, assuming you put true first and false second.
This suggests some other ways to do numbers in lambda calculus, other than Church numerals. E.g. you can make a linked list of booleans (forward or backward), or put 2^n booleans at the leaves of a perfectly-balanced binary tree to make a fixed-size number, or make a giant tuple of 64 booleans.
that's very neat, never seen that before!
Fun fact, lambdas can be interpreted as single method objects.
Your list lambdas are like objects with a single 'fold' method that visits the list.
Using selector functions like \x.\y.x you can select between multiple "methods".
My head hurts
I dunno if anyone has ever used this one esoteric programming language called Microsoft PowerPoint 🤔
I think that esoteric programming language called Doom deserves more attention than PowerPoint
I knew this would be a great video the moment I saw the title. I love this channel!
8:46 multiplication is MUCH more clever if you know how to do it. just take lambda a, lambda b, lambda f, lambda x: a (b f) x, which is behaviorally identical to the bluebird combinator: lambda f, lambda g, lambda x: f (g x).
exponentiation is just applying one number to the other, even simpler!
I spent the past couple months with pure functional programming (lambda calculus and similar) as a special interest so *happy noises*
i did not understand a single word except "turing machine"
I’m so excited for this vid! Will be watching tonight :)
You are better than Church at explaining this. I don't know he felt the need to remove the intuition behind everything he wrote, compare that to Turin who is actually fun to read.
Fun fact, Church's students agree that he was really boring as a teacher, just barely reading his books and copying the proofs
very good video, amazing explanation, and engaging humor! :D
I found this video at the end of my semester and it's AMAZING
good video man,the links are so good .I was doing SICP but had to stop due to something ,i so wish i completed it.
3:53 Doesn't carl know that a turing machine is a card game?
Have you heard about Concatenative Calculus? It's like lambda calculus (actually more like combinatorial calculus), but juxtaposition denotes composition instead of application and instead of Polish Notation it results in RPN. It's also way easier to pass and return multiple values and extend with non-pure functionality like I/O than functional programming.
Making composition the main thing of the system makes so much sense: unlike application it's an associative operation and a composition of a list of functions is just a list of transformations to be done in order, which is how people normally think about algorithms. Its associativity also makes it extremely easy to factor out frequently occuring sets of commands to new named functions.
And have I mentioned how simple concatenative interpreters/compilers are? Because of this most concatenative languages actually expose tools to play with their internals so metaprogramming comes naturaly to anyone familiar with a language.
very interesting, but I was doubtful when I read this. now after checking out Dawn, and untyped multistack concatenative calculus; I have to say... I am quite disappointed. it surely looks like an interesting way to implement a concatenative language; but from the blog posts they've shown; it looks like a much worse language to use (excluding ecosystem) than some FP language like Haskell. it's not quite a "this language has syntax i dont like!", it's more like it is needlessly verbose and difficult to understand. Check out the last blog post and see the difference. Plus, the ease of IO being implemented is one-sided to functional programming, as there is no added difficulty. the difficulty would increase if you specify *pure* functional programming, but even then i believe it is much easier than doing so for UMCC. but its a promising language to say the least.
@@ribosomerocker "Foundations of Dawn" only presents the theoretical basis of concatenative programming. I don't really get why the multistack part is there and I know other, more representative and actually implemented languages that better showcase what CC is capable of. Take a look at Joy or Factor instead.
@@aleksandersabak Ah. I am a big fan of both Joy and Factor, but I've never heard the "concatenative calculus" name. Very interesting.
@@I_would_like_to_buy_an_E Their site is very helpful. They also have a Discord server if you'd like to witness conversation there. But generally I just used their site and made some projects.
Man, this is so fucking abstract but so fucking cool. Love it
The insane viewcount to like ratio tells me this is the video I needed! I am 5minutes in and even if the rest of the video is a black screen with white noise, this will still be a masterpiece.
im scared
That's completely reasonable.
Just saying dude, I absolutely love your videos. Please keep it up!
Gonna try this is python lmao
WHY DO I LOVE THIS SO MUCH
i've been working on a modified lambda calculus that completely sidesteps the need for renaming variables, and it's by producing a very different restriction: all function definitions must be pure juxtapositions (such as: lambda a b c d = a (b d) (c d)). you can recover all lambda calculus behavior by using placeholder variables wherever you would want a constant to appear in your expression, such as: (lambda p a b c = p a (p b c)) pair; this produces a tree-like pair nesting from three provided arguments.
de bruijn beat you to the race with a much more elegant (and obviously turing-complete) solution
he died in 2012 at the age of 93 so he probably beat you to the race by a _lot_
oh wait, what you're describing is combinator calculus, which doesn't avoid that, because every lambda calculus expression can be converted to it.
3:24 personal attack incoming
I still come back to this video a lot
Same
Church numerals is isomorphic to the set theoretic definition of numbers.
me when new truttle1 video
Here for the premiere!
". . . instead of what you're supposed to be doing" fine! I'll go do my work then.
No idea what this video was about, but I enjoyed it.
Nice vid!
CZcams... the place where some guy explains to you in 5 sentences what your professor couldnt in multiple hours of lectures.
2:05
This is not applying a function to another function, this is applying a function to the result of a function.
That's literally the definition of composition of two functions.
@@anthonyisom7468 Aye, kinda. But these are two different things. When you compose two functions, the composition operator *does* take functions as inputs, but f is not taking g as its input, it's taking the *result* of g as its input.
@@MCLooyverse Since the result of g is also a function, you're still composing two functions.
@@BrunodeSouzaLino Huh? The result of g is a number (real, rational, integer, it hasn't been specified), not a function.
@@MCLooyverse The result of any number in lambda calculus is a function. The only thing that exists in lambda calculus are functions.
Can you cover dlang templates; I wrote an appendable list in it
even tho i have seen these things before, this simplified explanation still baffled me
First time i feel confident that i actually understand lambda calculus
nice trailer lol
I UNDERSTRAND NOTHING
@@Blue-Maned_Hawk OKAY? WHY DID YOU TELL HIM AND WHY ARE WE YELLING
can you make a vid about JS F*ck? It's a pretty fun esolang that's also pretty popular.
Lambda Calculus: Turing complete before Turing complete was a thing.
still somehow less cursed than pure prolog (i.e first-order logic/horn clauses)
This is excellent (thumbs up!). But it is very frustrating the way the steps in reductions replace each other instead of being on the screen at the same time.
Ah doctor freeman -scientist
best trailer
Since you already made add/multiply functions, it’s also possible to create exponentiation, tetration, pentation, hexation, …
really nice
Can you please please please do a video on plankalkül sir
truttle posted :P
That Curry Tangente ist Just an adhd mood
Basically IF() function from excel
did you change the color of the one?
I had to reduce my speed to .25x to understand your example about the NOT function, but it's clear now
lambda calculus the yes
1:40,
Fun Fact: There is actually a distinction between Function and Lambda Function,
Functions map from one Set to another Set
Lambda Functions map from one Set to another Set, but are self-preserving:
So:
(x) -> { y } can be a lambda function, but generalized Functions are more abstract and would include things like:
x = a Set that returns another Set y
Another thing is (x)->{ y } is technically always Surjective, unless you were to consider things like null-outputs otherwise.
And not all Functions are Surjective.. therefore FunctionalInterfaces are ~just 1 type of function.
If the Function maps from one Set to the same Set, then it is called an Operation,, and if it has a symbol, it is called an Operator
If a mapping were to map from any Object to another Object (those objects aren’t necessarily Sets), then we refer to the mapping as a Morphism. There are different properties that morphisms could have also
* lambda functions are technically different from FunctionalInterfaces (I heard),, but it’s probably a small difference. ie FunctionalInterfaces can be void... etc
2:45,
*All functions are self-preserving technically. Ignore that contrast. One of the only distinctions between a Function and LambdaFunction would be the Surjectivity:
L x.xxy (z) implies that for any input x to produce 1 output - therefore the function is 1:1. Since not all functions are 1:1,, this is one thing that lambda functions have specified naturally
Methods behave similarly. If I wanted a system that had Functions, it would probably be harder to code than a system that just worked with LambdaFunctions - bc I would just use methods,, rather than creating some kind of complex datastructure
The comment about “self-preserving” came from me over-analyzing the Bound Variable.
When I see a Bound Variable, I see a Generic Type,, which allows me to maintain the behavior of the variable passed into it -> self preservation…
All functions are self-preserving naturally, because all Morphisms are self-preserving by definition. Well… one of the definitions (it’s a topic that is widely studied by different mathematicians,, and that property might be “up to interpretation”)
what about the Akkerman function
How to simulate floating point numbers in lambda calculus?
three church numerals
The highs are too high and the lows too low. Equalize sound a bit more including music. really good video!!! :))
isn’t this just math haskell
What about fixpoint
2:10 Technically this isn't applying a function to another function; this is just the *composition* of two functions. You're applying f to the number g(x). Other than that, great video!
So basically numbers are constant functions?
Nice AWS Lambda image
what about "monad is a monoid of endofunctors"
Don’t feel bad that your spice tolerance is low. Once, my cousin said something along the lines of, “My spice tolerance is low, and this food isn’t even spicy to me.” Despite my better judgement, I tried the food, and my mouth burned.
How would you do subtraction/division?
Assuming church numerals, successor funcion `suc`, pair constructor `pair` with deconstructors `first` and `second`.
Subtraction:
>0 = λx.x(λy.true)(false)
pre = λx.second (x(λy.pair (suc (first y))(first y))(pair 0 0))
sub = λxy.y pre x
Division:
Y = λf.(λx.f(xx))(λx.f(xx))
lt = λxy.>0 (sub y x)
div = Y λfxy.(lt x y)(0)(suc (f (sub x y) y))
mod = Y λfxy.(lt x y)(x)(f (sub x y) y)
The only idea I have come up with to do subtraction is to make a f' that cancels out f.
E.g f(f'(x)) = x = f'(f(x))
f(f(f'(x))) = f(x)
@@photophone5574 Just like addition is based on a successor function that takes a number and returns a number one higher, subtraction on Church numerals is based on a predecessor function that takes a number and returns a number one lower (or zero in case of zero, because Church numerals don't handle negatives).
To construct a predecessor function you need a pair function that will store two values and access them. These functions can be defined as follows:
pair = λabx.xab
first = λp.p true
second = λp. p false
The pair constructor takes two values to store and the third value: a function that will extract one of those values. Church booleans work well as extractors, that's why deconstructors "first" and "second" take a pair and provide it with "true" and "false" as decontstructors.
When we have pairs we can start calculating predecessor. The idea is to start with zero, and increment it N times, but after every increment keep the result of the previous increment. The result of the secon-do-last increment will be the number N-1 that we are looking for. We start with a pair of two zeros (the second one is a placeholder for -1) and keep replacing this pair with an increment of that pair:
pre = λx.second (x (λp.pair (suc (first p)) (first p)) (pair 0 0))
Finally with a function that can decrement a number we can just apply it N times to subtract n, so:
sub = λab.b pre a
@@photophone5574 @Spicy Cat you can also write a predecesor function using a box function, which is similar to a pair function but it only holds one value
using the box function makes it possible to ignore one of the applications of f, which results in the predecesor of the number
I'm going to use L instead of lambda
box = Lx.Lf.f x
unbox = Lb. b (Lx.x)
addToBox = Lb. box (b succ)
alwaysZero = Lf. zero
pred = Ln. unbox (n addToBox alwaysZero)
for example for pred 2 you apply the function addToBox 2 times to alwaysZero which ignores the first addition and just returns a boxed zero, then addToBox is applied to the boxed zero resulting in a boxed one
first application:
addToBox (alwaysZero) -> box (alwaysZero succ) -> box zero
second application:
addToBox (box zero) -> box ((box zero) succ) -> box (succ zero) -> box one
and then you unwrap the result with unbox which yields the result of one
this makes it so that pred 0 = 0
using this way of thinking, you can write a similar function that does the same thing in a single term, though its much more confusing that way:
Ln.Lf.Lx. n (Lg.Lh. h (g f)) (Lu.x) (Lu.u)
where Lg.Lh. h (g f) acts a bit like addToBox, Lu.x acts like alwaysZero and Lu.u acts like the final unbox
also division is trickier I'm pretty sure you'd need the Y combinator for that unless there's something I'm missing
plz dark mod
WHY DO THESE CHARACTERS CANT STOP MOVING WHILE THEY TALK THATS INSANEEE DUDE
nice channel
Funny. I still have no idea what you're talking about. For some reason I think it would be better if you would leave ALL the steps on the screen at the some time so that I can follow along sequentially.
b-but truttle, you didn't show how to do hello world in lambda calculus xd
It does no have I/O
Okay but what is a Monad??
Yoo it's the rust maniac
@@Nick-lx4fo yoo
my brain hurt
b-but wherr's the truth machine.....
I actually did make one, but it wasn’t really interesting.
turing machinen’t
I think I would argue that function composition is not the same as applying functions to functions, though I guess you could argue that the composition operator is some form of lift from a function f: A -> B, to a morphism (f o) taking the category of arrows into A to the category of arrows into B.
Also not sure I agree with "everything that's Turing complete is a programming language"
Honestly a decent overview of Church encoding otherwise
Since the only thing that exists in Lambda Calculus are functions, the result of of both operations is the same, regardless of what they may or may not mean in another language.
2:47 Evil Rush be like
I have learned lambda calculus in combination with semantics and logical programming. And then logic.
And mulitiplication defined exponentiation!
omg i rmemeber the fucntion machines
That lip-sync looks so cursed
How?
Haskell be like:
Is redstone Turing complete?
You could totally simulate a finite-state turing machine so yeah. People have also built entire programmable Computers in it
The Discord invite in the description doesn't work.
Oh my god, you've turned on phone verification‽ What the fuck‽
I can define a turing machine off the top of my head, but it's not pretty and involves a heterogenous 7-tuple. (Starting State, All States, Accepting States, Input Alphabet, Initial Tape Contents, Tape Alphabet, State-Tape Transition Function)
The state transition function can be treated as an opaque 2-parameter function that returms a triple, you'd store the left tape and the right tape in conslists, and recurse while forwarding the two tape sides and the next state.
Interesting to see a programming language that predates computers.
holy crap Carolina Reapers are hotter than pepper spray.
1:34 mmh desmos so based
Awesome video
hrrrnnnggg there were a lot of obfuscates feet in this one.
Why haven’t I blocked you yet?
@@Truttle1 because I'm a valued subscriber :3
YAYYYYY
just put the work in a column on the screen, don't flash each line for a millisecond for chrissakes. like, do you want me to actually read your work??
3(4x - 3) + 2
2:10 no thats not what plugging one function into another means. What you have described is function composition.
a function f:A->B is defined as a set of ordered pairs AxB, such that no two elements of f have the same first value (roughly speaking). We define f(x) to essentially mean "find whichever ordered pair has x as the first element, and return the second element". For example, if f={(3, 2), (4, 3), (5, 4)} then to find f(4), we find the ordered pair which has 4 as the first element (namely, (4, 3)) and return the second element. Thus f(4) is 3. There's a lot more nuance but it's not too important for this explanation.
If we have a function f:B->C, and a function g:A->B, we define fog:A->C to be the function represented by f(g(x)). Roughly speaking, we plug in x to g, then plug in this value to f. This is what you describe at 2:10. This is completely different from f(g) which is what plugging one function into another actually means. For this we just use our above definition, where find the ordered pair which has g as it's first value, and return the second value.
Let's use an example to illustrate. Let's say that g = {(1, 2), (2,3), (3,4), ...}. This function can be represented with the equation g(n)=n+1 when n is a natural number. Now let's say that f = {(g, -1), (2, 1), (3, 2), (4, 3), ...}. We can represent this function with the equation f(n)=n-1 when n is a natural number. we say that f(g(x)) = g(x)-1 = n+1-1 = n, when n is a natural number. This is not what plugging one function into another means though, this is just function composition. f(g) = -1, since that's the corresponding value in our function.
The result is still a function though.