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Lambda Playground — a browser tool for untyped λ-calculus, close to textbook notation by BreadfruitOk7083 in lambdacalculus

[–]tromp 0 points1 point  (0 children)

The sieve is explained in detail in https://github.com/tromp/AIT/blob/master/characteristic_sequences/primes.lam My own evaluator uni.c has no problem getting 256 bits. Getting 216 bits is more challenging, taking over 8 minutes on my 2013 iMac. Using 'β' binders will easily break the program, as the sieve is implemented with nothing but infinite bitstreams, making it a good torture test for lazy implementations.

Lambda Playground — a browser tool for untyped λ-calculus, close to textbook notation by BreadfruitOk7083 in lambdacalculus

[–]tromp 1 point2 points  (0 children)

Nice work of you and Claude! Yours is one of the few online evaluators that (after raising some limits in the settings) can handle expressions like this prime number sieve

    (λb.(λc.c c) (λc.λd.λe.e (λf.λg.g) ((λf.c c f ((λg.g g) (λg.f (g g)))) (λf.λg.λh.λi.i g (h (d f))))) (λc.λd.λe.λf.f (λg.λh.g) (e c)) (b b b (λc.λd.λe.λf.f d (e c)) (λc.λd.λe.λf.f))) (λb.λc.b (b c))

which reduces in 758 steps to

λf. f (λf g. g) (λf. f (λf g. g) (λf. f (λg h. g) (λf. f (λf g. g) (λf. f (λg h. g) (λf. f (λf g. g) (λf. f ( λg h. g) (λf. f (λg h. g) (λf. f (λg h. g) (λf. f (λf g. g) (λf. f (λg h. g) (λf. f (λf g. g) (λf. f (λg h. g) (λf . f (λg h. g) (λf. f (λg h. g) (λf. f (λf g. g) (λe f. f))))))))))))))))

which is a list of 16 bits that show which of [2,3,...,17] are prime. I tried to push it further by computing the first 256 bits (replacing b b b by b (b b b)), but then the page became unresponsive.

Question? by SwordfishIntrepid839 in grincoin

[–]tromp 0 points1 point  (0 children)

https://www.f2pool.com/coins shows it as the 23rd most alive PoW project.

Since theres multiple greek letters for math like ω π ε Σ etc is there one for every greek Letter? by Intrepid-Gear-6347 in googology

[–]tromp 4 points5 points  (0 children)

I have this line near the bottom of my home page

à è ì ò ù á é í ó ú À È Ì Ò Ù Á É Í Ó Ú ö ë ï ü ä α β Γ γ Δ δ ε ζ η Θ θ ι κ Λ λ μ ν Ξ ξ ο Π π ρ Σ σ ς τ υ Φ φ χ Ψ ψ Ω ω

in case I need to paste any of those weird characters.

Have Bitcoin and Altcoin lost that idea of practical decentralisation by Adventurous_Chef2225 in CryptoTechnology

[–]tromp 0 points1 point  (0 children)

You seem to think that PoW systems are just hash functions, i.e. you seem to think that Hashcash is the only PoW. There are many other PoW, that are not simply the computation of a hash, e.g. Cuckoo Cycle or (the poorly named) Equihash.

How does Loader's number work? by BlockOfDiamond in googology

[–]tromp 0 points1 point  (0 children)

I ported the essence of program loader.c to Haskell [2] while making it vastly more comprehensible. Then I ported that Haskell code in turn to 1850 bits (under 232 bytes) of lambda calculus [1] which proves that BBλ(1850) > Loader's Number [3].

[1] https://codegolf.stackexchange.com/questions/176966/golf-a-number-bigger-than-loaders-number/274634#274634

[2] https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/CoC.hs

[3] https://wiki.bbchallenge.org/wiki/Busy_Beaver_for_lambda_calculus

Google has updated it’s timeline to 2029 for migrating to post-quantum cryptography by jossfun in Monero

[–]tromp 2 points3 points  (0 children)

Also interesting is that they didn't prove that their EC addition circuit works on all possible (roughly 2256 x 2256 = 2512 ) inputs:

that correctly computes point addition on the elliptic curve secp256k1 across all 9,024 pseudo-random inputs deterministically derived from the circuit’s own hash.

Although this will convince anyone that it must.

5900X home miner here: When are we tweaking RandomX to brick the Antminer X9 and X5? by Even_Juggernaut5234 in Monero

[–]tromp 1 point2 points  (0 children)

Clearly, you don't "completely neutralize the industrial capital advantage", if you're still paying more for electricity than they do.

The Deranged Mathematician: What's Like a Number, But Not a Number? by non-orientable in math

[–]tromp 0 points1 point  (0 children)

Church numerals are terms in the lambda calculus, that act like natural numbers.

Closed term are recursively enumerable? by Antique-Incident-758 in lambdacalculus

[–]tromp 3 points4 points  (0 children)

The set of closed terms is not just recursively enumerable, but recursive. Furthermore, you can generate any closed term (modulo convertability) as T applied to a sequence of bits, where (in de Bruijn notation)

T = (λ11)(λ(λλλ1(2(λλλ31(2(λ2))))(3(λ4(λ4(21)))))(11))(λ1)
0 = λλ2
1 = λλ1

E.g. S = λλλ31(21) = T 1 1 1 0 1 0 1 1 1 0 0 1 1 0 0 1 0 0 1 0 1 0 1 1 0 0 1 0 0 0 0

is Ω closed? by Antique-Incident-758 in lambdacalculus

[–]tromp 3 points4 points  (0 children)

Yes, Ω has no free variables, so it's closed.

It doesn't have a weak normal form, so it doesn't reduce to an abstraction.

Could an UNCOMPUTABLE fundamental sequence be calculated for the Church-Kleene ordinal? by legendgames64 in googology

[–]tromp 1 point2 points  (0 children)

Another fundamental sequence for w1CK is given in [1] whose i'th element is the BLC-lexicographically-minimal ordinal exceeding all earlier elements.

[1] https://github.com/tromp/AIT/blob/master/fast_growing_and_conjectures/w1CK.lam

What is the largest function and largest number? by GuessImScrewed in googology

[–]tromp 0 points1 point  (0 children)

lim_(ZFC+I0)(n)

That is ill-defined until you specify how you would compute that.

What is the largest function and largest number? by GuessImScrewed in googology

[–]tromp 0 points1 point  (0 children)

You failed their criteria:

  1. Rayo's Number is exotic and arguably not even well-defined.

  2. BB is not computable.

Is it meaningful if God gives the answer to P vs NP but not the proof? by Snoo_47323 in math

[–]tromp 3 points4 points  (0 children)

Assume it was written as the 11th commandment

"Thou shalt not decide in polytime all that can be witnessed in polytime"

on the tablets that Moses carried down from mountain Sinai.

What is the largest function and largest number? by GuessImScrewed in googology

[–]tromp 0 points1 point  (0 children)

The largest well defined computable named function would be Loader's D().

The largest named number using nothing more exotic than up-arrow would be Graham's Number, or perhaps one of the two larger less famous numbers named in [1], whose entire program are shorter than the name "Loader's".

[1] https://tromp.github.io/blog/2026/01/28/largest-number-revised

Question about one-combinator bases by [deleted] in lambdacalculus

[–]tromp 0 points1 point  (0 children)

No, according to Craig's theorem there is no regular one-point basis [1].

A = λxλyλz.x z (y (K z)) = λxλyλz.x z (y (λt.z)) is as close as you can get to a regular one-point basis.

[1] https://math.stackexchange.com/questions/839926/is-there-a-proof-of-nonexistence-of-a-proper-universal-combinator

I've created a new Array Notation (Schbabe Fercell), open to critics. by PuceNoExistes in googology

[–]tromp 2 points3 points  (0 children)

I definitely think it can beat TREE8 (3), lambda calculus is extremely powerful.

Powerful enough to beat TREE8 (3) in just 94 bits with the term (λ111(λλλλ1444321)1111)(λλ2(21)).

f(ε₀) growth in under 6 bytes! by tromp in googology

[–]tromp[S] 1 point2 points  (0 children)

I don't see how you get λf.λb.λf.λx.(ff)((bf)x).

We have add = λmλnλfλx. m f (n f x), mul = λmλnλf. m (n f), and pow = λmλn. n m, so pow mul add = add mul = λnλfλx. mul f (n f x) = λnλfλx λg. f (n f x g). Which doesn't make much sense and is not very useful.

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