P vs. NP and the Difficulty of Computation: A ruliological approach | Hacker News Reader

P vs. NP and the Difficulty of Computation: A ruliological approach

writings.stephenwolfram.com

94 pointstzury38 comments

Can someone tell me what I am missing here?

This seems to suffer from a finite-size effect. Wolfram's machines have a tiny state space (s ≤ 4, k ≤ 3). For some class of NP problems, this will be insufficient to encode complex algorithms and is low dimensional enough that it is unlikely to be able to encode hard instances ("worst case") of the problem class. The solution space simply cannot support them.

In this regime, hard problem classes only have easy solutions, think random k-SAT below the satisfiability threshold, where algorithms like FIX (Coja-Oghlan) approximate the decision problem in polynomial time. In random k-SAT, the "hardness" cannot emerge away from the phase transition and by analogy (watch my hand wave in the wind so free) I can imagine that they would not exist at small scales. Almost like the opposite of the overlap gap property.

Wolfram's implicit counter-claim seems to be that the density of irreducibility among small machines approximates the density in the infinite limit (...or something? Via his "Principle of Computational Equivalence"), but I'm not following that argument. I am sure someone has brought this up to him! I just don't understand his response. Is there some way of characterizing / capturing the complexity floor of a given problem (For an NP-hard Problem P the reduced space needs to be at least as big as S to, WHP, describe a few hard instances)?

The cynic is me says those interesting but ultimately barren long-form articles are just content marketing for Mathematica software.

No lol, Stephen Wolfram is more invested in his writings than he is in Mathematica. He genuinely believes he’s going to revolutionize math and physics.

He’s smarter than your average nutjob, but he’s still a bit of a crank.

I think you have it wrong. Wolfram's claim is that for a wide array of small (s,k) (including s <= 4, k <= 3), there's complex behavior and a profusion of (provably?) Turing machine equivalent (TME) machines. At the end of the article, Wolfram talks about awarding a prize in 2007 for a proof that (s=2,k=3) was TME.

The `s` stands for states and `k` for colors, without talking at all about tape length. One way to say "principle of computational equivalence" is that "if it looks complex, it probably is". That is, TME is the norm, rather than the exception.

If true, this probably means that you can make up for the clunky computation power of small (s,k) by conditioning large swathes of input tape to overcome the limitation. That is, you have unfettered access to the input tape and, with just a sprinkle of TME, you can eeke out computation by fiddling with the input tape to get the (s,k) machine to run how you want.

So, if finite sized scaling effects were actually in effect, it would only work in Wolfram's favor. If there's a profusion of small TME (s,k), one would probably expect computation to only get easier as (s,k) increases.

I think you also have the random k-SAT business wrong. There's this idea that "complexity happens at the edge of chaos" and I think this is pretty much clearly wrong.

Random k-SAT is, from what I understand, effectively almost surely polynomial time solveable. Below the critical threshold, it's easy to determine in the negative if the instance is unsolvable (I'm not sure if DPLL works, but I think something does?). Above the threshold, when it's almost surely solveable, I think something as simple as walksat will work. Near, or even "at", the threshold, my understanding is that something like survey propagation effectively solves this [0].

k-SAT is a little clunky to work in, so you might take issue with my take on it being solveable but if you take something like Hamiltonian cycle on (Erdos-Renyi) random graphs, the Hamiltonian cycle has a phase transition, just like k-SAT (and a host of other NP-Complete problems) but does have a provably an almost sure polynomial time algorithm to determine Hamiltonicity, even at the critical threshold [1].

There's some recent work with trying to choose "random" k-SAT instances with different distributions, and I think that's more hopeful at being able to find difficult random instances, but I'm not sure there's actually been a lot of work in that area [2].

[0] https://arxiv.org/abs/cs/0212002

[1] https://www.math.cmu.edu/~af1p/Texfiles/AFFHCIRG.pdf

[2] https://arxiv.org/abs/1706.08431

I find that the "ruliological approach" is very similar to feasible mathematics by Jiatu Li (https://eccc.weizmann.ac.il/report/2025/086/). In the last section before the Personal Notes, "In effect, we’re seeing that theoretical computer science can be done not only “purely theoretically”—say with methods from traditional mathematics—but also “empirically”, finding results and developing intuition by doing explicit computational experiments and enumerations." Where regular mathematics is "purely theoretical" and "empirically" is what Jiatu Li also describes in his paper sometimes referred to reverse mathematics like from Quanta magazine.

I appreciated the great explanation of space complexity and it eludicated why some scientific authors don't include it in their analysis of algorithms. However, Wolfram found that "by successively investigating both larger inputs and longer runtimes, one can develop reasonable confidence that—at least most of the time—one is correctly identifying both cases that lead to halting, and ones that do not." There are exceptions like Machine 600720 that have exceptionally long runtimes but I gain a much better understanding about an algorithm if I'm provided the space complexity. It's still an open question in pure theory but it could be understood from empirical results.

Most of the difficulty arises from conflating discreet and continuous values which a la dirichlet function cannot be integrated whereas discrete values do not physically exist in real computation. Both the physical boundary of the transistor is a probabalistic field as is obviously the analog signal it (probabilistically) discretizes. When you ask ill posed questions in math you get to debate the answer for decades.

Someone should tell Stephen Wolfram about the bbchallenge wiki (bb for busy beaver): https://wiki.bbchallenge.org/wiki/Main_Page

The man is a savant. He knows.

To me, this reads like a profusion of empirical experiments without any cohesive direction or desire towards deeper understanding.

Yah Stephen Wolfram is too often grandiose thereby missing the hard edges.

But in this case, given how hard P=NP is, it might create wiggle room for progress.

Ideally it would have gone on and said in view of lemma/proof/conjecture X, sampling enumerated programs might shine light on ... no doubt that'd be better.

But here I'm inclined to let it slide if it's a new attack vector.

Is this for programmers? Serious question.

This is so tangentially related to the P vs NP problem that the title is basically pure clickbait. Remove every sentence relating to polynomial anything and the information content of the write-up doesn't change at all.

It reads like slop. It’s repetitive, abstract and adds essentially nothing beyond him babbling about himself.

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