Lambda calculus meets machine learning. ~ João Marcos Flach. https://lume.ufrgs.br/bitstream/handle/10183/264011/001175975.pdf #LambdaCalculus #MachineLearning #NeuralNetwork

Hindley and Seldin had an exercise that says to find two lambda terms P, Q which do not have β normal forms but PQ does.

It's not the hardest exercise once you have the idea of passing stuff around in the usual Ω fashion until you can shuffle a constant function onto the front, but I think it could be a fun #codegolf challenge for a lazy afternoon.

My first attempt was
\[ P= (\lambda xy.y (\lambda x yz.z)(x x))(\lambda xy.y (\lambda x yz.z)(x x)) \] \[Q = (\lambda x y z. y (x x)z)(\lambda x y z. y (x x)z)\] #lambdacalculus

About functional programming (A fun way to reason). ~ Mary Paskhaver. https://medium.com/geekculture/use-lambda-calculus-to-think-about-functional-programming-4734f80e7714 #FunctionalProgramming #LambdaCalculus

If it β-reduces to 2 ≤ 1 in stlc , its type ought to be
#lambdacalculus

IN λx.x(λy.y) x, y are Both x and y are bound and not free
#lambdacalculus

Type systems for the OO programmer. ~ Marc Etter. https://eprints.ost.ch/id/eprint/1110/1/Type%20Systems%20for%20the%20OO%20Programmer.pdf #Java #OOP #Types #LambdaCalculus #FunctionalProgramming

Propositions As Types • 1
• https://inquiryintoinquiry.com/2013/01/29/propositions-as-types-1/

One of my favorite mathematical tricks — it almost seems too tricky to be true — is the #PropositionsAsTypesAnalogy. Moreover, I see hints the 2-part analogy can be extended to a 3-part analogy, as follows.

\[\text{proof hint : proof : proposition ~::~ untyped term : typed term : type}\]

See my notes on #PropositionsAsTypes for more information.
• https://oeis.org/wiki/Propositions_As_Types_Analogy

#CurryHowardIsomorphism #LambdaCalculus #ProofTheory #TypeTheory

Did you know you could use DNA to store your lambda calculus programs?

https://github.com/jevko/writing/blob/main/2023-01-25-programming-with-dna.md

#dna #lambdacalculus #programming

I stumbled upon one guy proposing a new foundation for computer mathematics called "Abstraction logic". As a newbie, I found it far more more comprehensible than usual any material on intuitionistic type theory versions.

But there's no social proof of it.

Any #logic #typetheory #lambdacalculus #foundation of #mathematics #automatedreasoning #prooftheory people who are willing to have a look to weight on it?

Links:
https://arxiv.org/pdf/2207.05610.pdf
https://youtu.be/LbFKSaPhBSA
https://obua.com/publications/philosophy-of-abstraction-logic/2/

From time to time I become interested in lambda calculus again. Finally started to maintain a repository with notes about lambda calculus in it: https://notabug.org/ZelphirKaltstahl/lambda-calculus-notes

It is not much yet, but maybe I will add more notes in the future.

#lambda #lambdacalculus

#functionalprogramming

Many people's vote for most #beautiful construct in #math is \[e^{i\pi}+1=0 \].

Yeah, maybe. But I think a close contender (if you include the #CS realm) is \[ \lambda b.\lambda e.eb \].

This serves as the complete #Church encoding of #exponentiation in #LambdaCalculus, driving home subtler points about #function #mapping and ordered pairs and the primacy of exponentiation over add/mult, both of which have uglier #LC representations.

https://en.wikipedia.org/wiki/Lambda_calculus#Arithmetic_in_lambda_calculus

@jsiehler Man, I love #Mathematica... and when I see expressions like this, it reminds me of my #MUSH/#MUX #code days... #procedural languages feel so hacky to me now. 🙃

Here's an old implementation of factorial using more-or-less #LambdaCalculus:

fact := k |-> k[p |-> p[a |-> b |-> g |-> g[f |-> x |-> f[a[f][x]]][f |-> a[b[f]] ]]][g |-> g[h |-> h][h |-> h]][a |-> b |-> b]

Inspired by @rosactrl background picture I once again took the guided
tour by Eric Normand
through the paper Can Programming Be Liberated from the von Neumann Style?

Reminded me of the question raised by @glebec in his (here again) recommended presentation #lambdacalculus #functionalcafe

Especially this slide

Did you know that some unary encodings for numbers in lambda calculus have been worked out without the use of laughing gas?

https://xtao.org/blog/lambda-numbers.html

#lambda #calculus #lambdacalculus #encoding #numbers #unary

https://woodrush.github.io/blog/lambdalisp.html

#lisp #scheme #lambdacalculus

Introducing the LAST programming language:

https://xtao.org/blog/last-intro.html

#last #programming #language #lambdacalculus #lambda #calculus #binary

"A #TuringMachine, after all, is not (and was never meant to be) a #ProgrammingLanguage! The #LambdaCalculus is both a programming language in itself and the foundation of all other functional languages." #Haskell

https://boarders.github.io/posts/halting1.html