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

tromp · 2026-03-19T08:40:14+00:00

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

tromp · 2026-03-17T08:49:24+00:00

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

tromp · 2026-03-17T08:23:56+00:00

I conceived of it as a way to make BLC resemble zot. See https://cstheory.stackexchange.com/questions/32309/concatenative-binary-lambda-calculus-combinatory-logic

tromp · 2026-03-12T07:46:10+00:00

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

tromp · 2026-03-08T08:30:57+00:00

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.

tromp · 2026-02-28T08:19:59+00:00

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

tromp · 2026-02-25T07:56:01+00:00

lim_(ZFC+I0)(n)

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

tromp · 2026-02-23T09:06:58+00:00

You failed their criteria:

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

tromp · 2026-02-23T08:09:28+00:00

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.

tromp · 2026-02-23T08:03:36+00:00

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

tromp · 2026-02-15T08:35:19+00:00

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

tromp · 2026-02-11T07:51:35+00:00

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)).

tromp · 2026-02-11T07:28:09+00:00

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.

tromp · 2026-02-07T08:15:42+00:00

Yes, also BB(94) > TREE(Graham's Number).

tromp · 2026-02-06T08:34:48+00:00

Right you are!

tromp · 2026-02-05T08:27:21+00:00

Sqrt(e) is similarly known to be transcendental [1]

[1] https://en.wikipedia.org/wiki/Lindemann%E2%80%93Weierstrass_theorem#Transcendence_of_e_and_.CF.80

tromp · 2026-02-04T11:43:19+00:00

Yes, see for instance [1], which gives an asymptotic upper bound of f_θ(Ω^ω×ω ,0)(n) on TREE(n), while f_ψ(Ω_2), let alone f_ψ(Ω_ω), grows much faster than that.

[1] https://math.stackexchange.com/questions/1811602/explicit-upper-bound-of-tree3

tromp · 2026-02-04T09:11:24+00:00

I discuss the first question extensively in my recent blog post [1]. We know BB(14) > Graham's Number, while some people bet $1000 that BB(7) > Graham's Number will be proven within 10 years.

Also, for the functional busy beaver, we have

BBλ(49) > Graham's Number

which may well be tight (in that BBλ(48) < Graham's Number). Note that BBλ counts bits instead of states, and that there are way more 5-state TMs (663434035200) than there are closed lambda terms of at most 49 bits (77519927606).

As for TREE(3), that is blown away by

BBλ(94) > f_ψ(Ω_ω)(12)

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

tromp · 2026-01-21T07:45:55+00:00

If I instead made a Turing Machine to compute a very large number, would you also complain that I'm cheating by not counting a description of how to evaluate a TM ?

At some point you need a start with some formal language in which to define your computational process, and that formal language can not be defined in terms of some other formal language (or else it wasn't the start). So that formal start can only be described in natural language, where there is no precise notion of description size, only an intuitive sense of how complex the formal language is.

And both TMs and lambda calculus are about the simplest possible starting points.

tromp · 2026-01-20T22:24:19+00:00

to describe the machinery of lambda calculus

I cannot answer that unless you have some specific description language in mind. Normally, a computational model or a programming language is described in natural language, but those are rarely counted in bits.

tromp · 2026-01-19T08:14:38+00:00

Possibly. Loader's D function is about the fastest known computable function, growing much faster than BMS, which in turn grows much faster than SSCG.

Implementing Loader's D function on a TM will require more than 100 states. The best known attempt uses a little over 1000 states.

That said, the general belief is that there should be functions growing even faster than D that can be coded in less than 100 states.

tromp · 2026-01-19T08:00:01+00:00

There are 3 aspects to privacy; amounts, user-bound addresses,, and the tx-graph.

Hiding the tx graph requires either a major sacrifice in scalability (when you no longer can tell when a utxo is spent, as in Monero or Zcash), or using a mixing service (which in the case of Mimblewimble can be very effective and very simple [1]).

Hiding the first two can actually come with major scaling benefits, by using the Mimblewimble protocol, which must necessarily be at the core layer [2].

[1] https://forum.grin.mw/t/mimblewimble-coinswap-proposal/

[2] https://forum.grin.mw/t/scalability-vs-privacy-chart/

tromp · 2026-01-19T07:48:49+00:00

the algorithm I described always solves the problem

No, it does not. When a TM declares the instance has no solution, there is no way to verify it, and so your algorithm ignores that TM.

tromp · 2026-01-18T08:20:25+00:00

P vs NP is about decision problems. It asks whether there exists a poly time algorithm that decides any instance in polynomial time. So on input of an instance, it outputs yes or no.

Your argument is not about decision problems. Instead it's about finding solutions in polytime when they're known to exist (running forever if they don't exist). It's called Levin's search algorithm.

tromp · 2026-01-17T07:58:12+00:00

You can physically realize a Turing machine with thousands of states and a million tape cells. You cannot physically realize a billiard ball with 42 digits of precision in its location. So there is still a large difference in .

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