Pretty cool given computing progress over the last 20 years:
1. CPUs sped up 40-60x
2. GPU FLOPS/$ increased ~10,000x
1. https://www.cpubenchmark.net/year-on-year.html
2. https://en.wikipedia.org/wiki/FLOPS#Hardware_costs
You can then of course spend months tuning and optimizing the hell out of your code to squeeze a further 10x performance.
If you're really crazy like Sarek, you can also formally verify these implementations: https://sarsko.github.io/_pages/SarekSkot%C3%A5m_thesis.pdf
You mean like a metric? Not all SAT instances have a natural metric, maybe a large fraction could be relaxed to MAXSAT, but it seems parallel solvers are not very popular in that setting either, i.e., there is no parallel track and parallel solvers are disqualified [1] from participating.
I'm not sure I follow how metrics are related to parallelism though. A naive strategy would be to try every assignment in parallel on 2^n processors. There are plenty of better strategies for search space splitting [2] and stochastic local search [3] algorithms that seem amenable to parallelization.
[1]: https://maxsat-evaluations.github.io/2023/rules.html
[2]: http://sat.inesc-id.pt/~ruben/papers/martins-ictai10-talk.pd...
[3]: https://link.springer.com/article/10.1023/A:1006350622830
I'm no expert but my thesis advisor is cited in the OP article, so I'm just guessing but it's an interesting question. It's not for the lack of trying in the part of SAT researchers; after the hardware advances with SAT solvers implementation in the early 2000s, they would've looked at parallelism and concurrency as well. But with continuous optimization (like with training machine learning), there's gradient descent which tells you where to guess next. But with Boolean logic there's nothing like that, the needle is either in the next bale of hay, or it isn't.
I can see how continuity accelerates the search for solutions, but I see that more as an orthogonal property, not a limitation of parallelism. Even if your domain is continuous, accelerating gradient descent is nontrivial, and many discrete optimizations problems benefit from parallelism.
But, notice that a shard whose k preset bits are in direct conflict should take zero time -- it's smarter to detect a conflict and terminate than carry on computing all 2^(n-k) invalid assignments. Each such conflicting shard decreases the benefit of your parallelization -- a depth-first solver wouldn't spend almost any time on that particular branch, but you've wasted a whole core on it.
The real question is, what if your algorithm is smarter than brute force? Such algorithms may spend considerable time preprocessing, which is not trivially parallelized, resulting in many cores duplicating tons of work with very little effective speedup.
Now apparently neither of us read the article, because indeed there are a couple sections where it discusses parallelization. You might want to look at those remarks. For example, there's current research using GPUs and tensor processors to parallelize SAT.
But going back further I think there's a historical context, and the article alludes to why it was not in fashion the preceding decade. Parallel SAT was looked at for a long time, but that really depended on progress in parallel hardware (the transition from Moore's law having to end i.e. unicore to multicore). So in the 2000s when consumer-level multicore CPUs arose, one would imagine that would've been a turning point for running parallel SAT algorithms. But historically, it was in 2001 that sequential solvers also had their implementation breakthrough, i.e. the Chaff solvers were a 10-100x sequential improvement, and experimentally beating the parallel implementations of the time. It was a big deal. So in terms of fashion there was a lot of interest specifically in sequential solvers at the time. Hence the ebb and flow of research focus.
And it makes sense as a justification. From a bird's eye view, if, n >> k, then the subproblems are still large NP-complete problems. Which is a reasonable rationale for firstly investigating the best theory/algorithms for a sequential solver in the first place, one that would be run at the core of every parallel hardware node. As the article points out, Knuth says to have a good algorithm first, before you resort to throwing ever more hardware at the problem. (In some papers, SAT algorithms that are complete as well as deterministic are also offered as theoretical justifications, and the article also touches on why requiring these properties makes parallelization more complex.)
What the new GPUs/Tensors are doing are massively splitting up the space, perhaps so that n !>> k, and this alternative approach is again mentioned in the concluding paragraph. Clearly that seems like a promising area of future research to the authors. And who knows, maybe one day, there's some new breakthrough, that for all intents and purposes makes P = NP.
I guess if I were to summarize your argument, (current) solvers do not use GPUs because there are not many algorithms that can leverage the additional compute (and maybe there is some theoretical barrier?). On the other hand, the continuous optimization folks had neural networks for more than half a century before they figured out how to make them run fast, and were also ridiculed for pursing what was widely believed to be a dead-end. So I would be careful about falling for the "smart people tried really hard for a while" argument.
I think that unlike the story with neural nets, there wasn't some faction trying to say that parallelizing SAT is a bad idea. In fact the state-of-the-art sequential solvers use techniques to exploit properties of CPU caches to get their speedup breakthrough; that in itself was an important lesson within the research community to pay close attention to hardware capabilities. My impression is that ever since that success within their discipline, SAT specialists became interested not just in algorithms/formal theory but also hardware and implementation, and in today's case that would include having a close look at GPUs/TPUs for the possibility of new algorithms.
Maybe that's as much a cause as an effect. What's the incentive to create and improve parallel solvers if there's no place that evaluates and rewards your work ?
[1] https://www.cs.utexas.edu/~marijn/publications/cube.pdf
The thing is much more deeper than what you described. You need to choose which variables you will assign, that's first. Then you have to assign them, and it is important to do as carefully because assignment may produce inbalanced search space.
Splitting on one variable when there are hundredths of thousands of them is very, very inefficient.
They are not friends with bottlenecks like clause queues used in contemporary stochastic SAT solvers.
They can be good at something like survey and/or belief propagation [1].
[1] https://simons.berkeley.edu/talks/belief-survey-propagation
The problem is that survey/belief propagatin is effective in the area of random CNF problems and it is not a complete solution technique. It may compute a solution but it cannot provide you with the evidence that there is no soolution.
It shares short conflict clauses between parallel solvers and achieves superlinear speedup in some cases, e.g., 4 parallel solvers solve faster than one forth of the single solver soolution time.
Short conflict clauses are rare so there is little communication between solvers required.
CryptoMiniSAT: https://github.com/msoos/cryptominisat
Author's goal to have solver that is good in computing range from single CPU up to cluster. Judging from CryptoMiniSAT successes, he has mostly reached the goal.
Anyway, I believe it's similar to LP - we still use simplex method in practice, despite the fact that LP has a polynomial algorithm.
What likely happens in SAT is even though there actually is (as I believe) a nice polynomial algorithm on the order of O(n^4) or so (which involves repeated solving of linear systems), the CDCL still wins in practical problems simply because in most cases, the upfront cost of solving the equations is higher than trying to find a first solution.
However, not all hope is lost. You could actually presolve a more general problem, and then just rely on propagation to deal with a more specific instance. For instance, for integer factorization, you could presolve a general system for integers of certain bit size, and then if you wanted to factorize a specific integer, you would just plugin constants into the presolved instance. That would eliminate most of the upfront cost and bring it on par with CDCL.
Unfortunately, DIMACS is not all that good format to represent SAT, because it is not composable like this - it doesn't allow for representation of presolved instances. But composability might be a win for a different types of algorithms, which do presolving.
The 2XSAT reduction is 100% correct (I have been using it to solve small integer factoring problems), the deduction theorem proof still has a small flaw which I think is fixable, and the third part is an ongoing research which is trucking along. I will get there eventually, but if more people would be looking at this approach, then we (humanity) could arrive there faster, that's all.
It is a VERY fun work. Code entirely written by Nicolas Prevot, a magician of CUDA. Paper link here: https://comp.nus.edu.sg/~meel/Papers/sat21-psm.pdf
I still hold that GPGPUs can't do CDCL. However, they may be helpful in some of its sub-parts, or may be able to run another algorithm for solving SAT that we haven't invented yet.
Just my 2 cents,
Mate
Unit propagation is P complete.
The state of the art with this is about 2x the speedup with an unbounded number of threads.
That's not bad, but is not very useful because in most use cases of SAT you have multiple distinct problems solvable in parallel anyway.
It's not a deal breaker, but we really need some unprecedented advance in mathematics to negate this or to solve SAT without this communication.
So much that you can probably get a turing award by making a SAT solver that can be 100-1000x faster on a GPU than on a CPU.
Also that would lead to solving a bunch of NP complete problems 1000x faster, so you can see why that would be a big deal.
However, that thought never strikes me because I haven't gotten used to thinking about problems as SAT reducible. But since the thought never strikes me, I also don't get many opportunities to get used to that!
How does one get started with incorporating SAT (or SMT) solving as a natural part of going about one's day?
Another thing is that the best kind of problem for SAT solvers are problems that ask a yes/no question. Is there a way to satisfy all the constraints, and if so what are the variable assignments that do so? It's also possible to use SAT solvers for other kinds of problems (e.g. find the solution with the best score) but it's more advanced.
The lazy approach to SMT was a huge step forward, but itself spawned a whole lineage of refinements, alternatives and changed the state of automated reasoning entirely.
A lot of work today seems to be going into theory combinations so that you can efficiently answer problems for example involve arrays and integers in a manner that has dependencies between the two.
'Solving' (it's an undecidable problem) SMT would open up a bunch of new possibilities..
In what sense? People often use SMT (Satisfiability modulo theories) but don't state _which_ theories. I often SMT-solve with decidable theories, like Presburger Arithmetic.
There’s the auxiliary problem that decidable but slow can often be equivalent to undecidable.
The SAT Competition [1] is a good place to find state of the art.
https://github.com/msoos/minisat-v1.14
It's an early version of MiniSat by Niklas Eén and Niklas Sörensson. You can get the original ZIP from minisat.se, but it's easier to read from GitHub. Enjoy!
Yes, it is a very well written summary of what has been going on with SAT solvers in the last decades. I personally think most of this (r)evolution can be attributed to miniSAT. A relatively easy to read (at least compared to other theorem provers), free software implementation of a SAT solver, with instantiations if I remember correctly. Since variables only have two possible valuations (true/false) in boolean logic, they can be instantiated and the problem can be split up, and sometimes greatly simplified in doing so. I.e. we can replace each occurrence of variable 'a' with 'true' to make a new smaller problem, and then replace each occurrence variable 'a' with 'false' to make another smaller problem. I.e. split 1 problem into 2 simpler problems, and solve them in parallel etc.
I personally also develop a SAT solver called CryptoMiniSat, but the above two are easier to read :)
I don't have a background in SAT solving but the "clause learning" in CDCL looks a lot like what we call a Resolution-derivation in Resolution theorem-proving. In short, given a conjunction of two (propositional or first order) clauses (¬φ ∨ ψ) ∧ (φ ∨ χ) Resolution eliminates the contradiction in the "complementary pair" (¬φ, φ) and derives the new clause ψ ∨ χ. If that's what CDCL folk call learning then that's really interesting to me because most of Inductive Logic Programming is an effort to use Resolution for inductive inference i.e. machine learning, but in logic. I guess I should check the article's bibliography. They reference s a textbook that might be useful ("Handbook of Satisfiability". Though one of the authors is an author of the article so that may not bode well for the textbook).
What would be the gold standard reference for CDCL? Something that would be taught at an undergraduate class, say?
Edit: after reading a bit on the net it seems like clause learning in CDCL is basically Resolution-derivation by input Resolution. Cool beans.
>> Resolution, a method for combining pairs of constraints to make additional valid constraints, also leads to a polynomial time solution.
"Constraints" being a slightly odd-sounding way to put it, to me anyway, but I guess that's because I don't have the relevant background to think of clauses that way.
The progress made in various NP-Complete problems, SAT especially, shows that the vast majority of problem instances tend to be solvable, with the worst-case complexity problems being relatively rare but having much greater complexity cost.
10-15 years ago, the basic idea was "P≠NP because otherwise computers could do crazy shit." Well, looks like they can do crazy shit!
As motivation for why P != NP, one can think of it as a finite restatement of the Halting Problem [1] in the form of asking whether a (polynomially sized) Turing machine has an input that will halt in K steps.
That's a really poor motivation. Analogically, you can think of the NP question as solving a polynomial equation over Z_2. What you're arguing is that because for Z the problem is unsolvable (due to DPRM theorem), then for Z_2, we have to try all the solutions.
That argument fundamentally ignores that the source of unsolvability is the infinite nature of Z.
For the record, I believe that P=NP (see https://news.ycombinator.com/item?id=35729626). I wish people spend more time looking for polynomial algorithm using logic approach, as I am doing (trying to extend the 2SAT and XORSAT proof to 2XSAT). This approach could give new algorithms that actually compose and can be paralelized (DIMACS is a really awkward representation).
I would argue that computation is a much more natural setting for thinking about these ideas. Whether is recursively enumerable sets, Diophantine equations or set theory doesn't fundamentally matter so much, as they're all equivalent in some way. I wouldn't argue that you, or anyone else, shouldn't think of it in that way, but in my opinion, for many people, myself included, casting it through a computational lens makes more sense.
Not to harp on a small point but I would argue the diagonalization proof of the Halting problem relies on there being different orders of infinity, not that Z itself is infinite. Since Godel's incompleteness, we know that the continuum hypothesis can be true or false depending on which (consistent) axioms you choose, so I would argue that it's not so much which axioms we choose for how we think about infinities that's fundamental but the fact that consistent systems of sufficient complexity must be incomplete.
I'm firmly in the P!=NP camp but I commend you for putting your beliefs out there and actually doing the work. I do believe people throw up their hands when they hear "NP-Complete" when they absolutely shouldn't. The path from problems being "easy" (polynomial) to "hard" (exponential) is not very well understood and even if a problem is, in general, NP-Complete doesn't mean that a particular ensemble of problems being drawn from couldn't be solved by some clever polynomial time algorithm.
And of course, P ?= NP still remains an open problem so we still don't understand something pretty fundamental.
I am philosophically very close to finitism, and I think from the perspective of the Skolem's paradox it's not clear whether different infinities are not just a "mirage", so to speak. The distinction between finite and infinite seems to be much more "real" to me.
> but the fact that consistent systems of sufficient complexity must be incomplete
The problem is that if P!=NP, then it would seem to imply (from what I have seen) that there is a large class of finite deduction systems which are incomplete, no matter what rules of inference you pick. This to me is a lot more counterintuitive than simply accepting that finite and infinite are just very, very different worlds, and our intuition easily fails when we transfer results from one to another.
And as I describe in that other thread, with the 2XSAT being NP-complete, the idea that you have two finite sets in Z_2^n that are each relatively easy to describe (instance of 2SAT and instance of XORSAT), but their intersection (which is in 2XSAT) is somehow "difficult" to describe is just.. really really hard to take seriously. Of course it's just another intuition, but I don't see how anybody who believes that P!=NP can look at this and not start doubting it.
> I do believe people throw up their hands when they hear "NP-Complete" when they absolutely shouldn't.
I very much agree. I was originally agnostic on whether P=NP (now I am tilted to believe that P=NP and there is actually an efficient algorithm). I always felt that assuming P=NP ("assuming" in the philosophical not mathematical sense) and looking for a polynomial algorithm (what would the polynomial algorithm have to look like, for example, it would have to represent exponential number of solutions to look at them at once) is a better way to progress on the question rather than assuming P!=NP. (Kinda like if we assume that the world is knowable, we are more motivated to know it, rather than if we assume it's unknowable.)
It's also better in practice because if you actually try to construct a polynomial algorithm, and it fails, you will get a much more concrete idea what the obstacle is (a specific set of instances where it fails). I think because of the natural proofs, if you assume P!=NP, then you believe that progressing this way is essentially impossible. So I think that's the reason why people give up, because if you believe P!=NP, resolving it this way is hopeless, and at the same time, nobody has any better idea. However, if you ever so slightly believe that P=NP, and humans are just not smart enough to figure it out in 50 years (remember, it took several centuries to figure out Gaussian elimination, which seems completely obvious in retrospect), then you're not constrained with natural proof limitations and there is no reason for hopelessness.
I think I'm also in the same camp (finitism). My view is that infinities are kind of "verbs" rather than "nouns", even if we use them as if they were nouns for short-hand. I guess another way to say that is infinities have algorithms associated with them, where we kind of "turn the crank" and get more output.
> And as I describe in that other thread, with the 2XSAT being NP-complete, the idea that you have two finite sets in Z_2^n that are each relatively easy to describe (instance of 2SAT and instance of XORSAT), but their intersection (which is in 2XSAT) is somehow "difficult" to describe is just.. really really hard to take seriously. Of course it's just another intuition, but I don't see how anybody who believes that P!=NP can look at this and not start doubting it.
I don't understand the hesitation here. Once you have a fundamental gate (NOR, NAND, etc.) you have universal computation. In my opinion, 2-SAT is just barely in P because of the extraordinary restriction that each clause have 2 variables. As soon as you relax that condition, it's easy to do arbitrary computation.
> ... if you assume P!=NP, then you believe that progressing this way is essentially impossible.
I think we're in agreement here. My personal view is that it's not so much better to assume P=NP as it is to understand that there are different ensembles of problem space to choose from and even if the larger "space" is NP-Complete, the ensemble might only produce "almost sure" polynomial solvable instances. This is the case for Hamiltonian Cycle problems on Erdos-Renyi random graphs, for example.
> ... then you're not constrained with natural proof limitations and there is no reason for hopelessness.
I do wish I understood the natural proof limitation a bit better. I suspect there's a loophole somewhere or, at the very least, pointing us to another region of proof technique, but I just don't understand it well enough to know what's going on.
> ... now I am tilted to believe that P=NP and there is actually an efficient algorithm ...
I guess I don't have much hope of convincing you otherwise but I would like to point out again the "finite Halting Problem" interpretation. For an arbitrary program F(inp,K) (input `inp` and runs for K steps as a parameter), you're saying that the following can effectively be short circuited from an exponential search to a polynomial one:
for inp in [0 .. 2^{n-1}] do
if F(inp,K) then return true
done
return false
I think your intuition is wrong here. The number of variables in the clause is not the reason why NP is difficult. XORSAT also has arbitrary number of variables per clauses, and is doable in P. Furthermore, the 2XSAT reduction shows that you can represent any 3-SAT clause only with 2-SAT and XORSAT clauses.
> To my eyes, if this can be short-circuited, it's essentially saying that for arbitrary functions, there's no actual reason to do computation as it can be short-circuited. In other words, you're saying one can do better than raw simulation in every single case. While potentially true, it seems like a pretty bold statement.
I don't think that's what P=NP would imply. Remember, if P=NP, the algorithm is polynomial in the worst case, it can still be true that for a portion (even a majority) of actual instances, the simulation is faster than applying the algorithm. (Also it's possible that for some instances, functions that already have been properly "inverted", the decision algorithm and the simulation are equivalent, this is similar to linear equations, once you have factorized the matrix to the upper echelon form, then application of the matrix is the same process as solving it.) What the result will say that you don't need to be exponential in the worst case.
XORSAT is not NP-Complete because a chain of XOR boolean functions can't be chained to create arbitrary computation. 2SAT is restrictive because, even though it has a "fundamental" gate (NAND/NOR), the "splay" of variable interaction is so limited. In some sense, 2SAT and XORSAT, each individually, are conspiring to prevent general purpose computation.
> I don't think that's what P=NP would imply. ... What the result will say that you don't need to be exponential in the worst case.
If P=NP, then the program I listed above can be short circuited from exponential search to polynomial.
This is, to my knowledge, a previously unknown reduction and the reason why I now strongly believe that P=NP.
You're basically saying "Look at this polynomial problem and this other polynomial problem, if we combine them in a natural way, suddenly it becomes exponential?". My point is that your intuition is wrong. For example, XOR by itself is not Turing machine equivalent, nor is NOT, nor is AND and OR by themselves, but combine some subset and suddenly you get NAND, NOR etc. which gives arbitrary computation.
One of the interpretations of the Halting Problem is that one can do no better than raw simulation. The intuition, by extension, is that for NP-Complete problems, one can also do no better than raw simulation (search the space for solutions). If we could do better than simulation, if there were a "short circuit" to the computation, then the program I listed above, which looks suspiciously like running arbitrary program with some time and space bounds, would not need to run in exponential time.
Arbitrary computation is the norm, not the exception. 2SAT and XORSAT are the exception because of their restrictions putting them in polynomial time and their inability to do arbitrary computation. If their fragility is disturbed, even by adding just a whiff of freedom in the form of more gates or arrangement, then they suddenly allow for arbitrary computation.
You believe your addition is just a slight nudge so the topple to arbitrary computation and, potentially, exponential time seems drastic, but you have a bias on what's "natural". For example, you could have decided to add some portion of 3 variable clauses to the 2SAT instance to see how quickly it blows up (if at all). Or you could have combined some different proportion of XOR clauses to the 2SAT instance to see how quickly it becomes intractable (or if it stays tractable). Which is more natural? Which is the smaller nudge? Why is your nudge not a shove?
FYI, there's a lot of work done for NP-Complete phase transitions. When I said this is poorly understood, I mean it. For all intents and purposes, "random" 3SAT (each clause chooses which variable uniformly at random, with equal probability of being negated) is, for all intents and purposes, solvable, even for millions of variables. As far as I know, there isn't a good proposal or understanding for a "natural" and "random" 3SAT ensemble that remains difficult as the scale increases.
Yes, but that's not the only thing I am saying. Look more closely at what 2XSAT really is - it is basically a 2SAT transformed on an affine subspace of Z_2^n. You claimed that the complexity of 3SAT comes from 3 variables per clause, and not only two, as in 2SAT, but here I am showing an NP-complete class that has the same property as 2SAT!
We can also restate 2XSAT reduction in this way - the only clause you need to have to be NP-complete is a clause that looks like (A XOR B) OR C (for arbitrary literals A,B,C). As you can notice, this clause already has the property of 2SAT clause (it forbids 2 combinations of A,B,C out of 8) rather than of 3SAT clause (which forbids 1 combination of A,B,C out of 8).
I think you're actually assuming the conclusion, if you think that because something gives rise to "universal computation", then it must be difficult. But that's possibly silently assuming P!=NP.
Still, the main point that 2XSAT reduction brings (at least to me) is that it hints that there is a clever generalization of both algorithms for 2SAT and XORSAT that runs in P (and if it doesn't, it will at least explain more clearly what the issue is). So that's my strategy, I am generalizing the proof of 2SAT to handle what I call generalized literals, XORs of any number of variables, i.e. linear equations, rather than simple literals.
> Or you could have combined some different proportion of XOR clauses to the 2SAT instance to see how quickly it becomes intractable (or if it stays tractable).
I don't understand this paragraph. I am not sure how I would measure whether a particular instance of 2XSAT is tractable or not. (I think they are all tractable, but I am still working on it.) I think the "hard" but satisfiable instances in 2XSAT have a really small dimension of affine subspace defined by XORSAT portion, and weak enough 2SAT portion to disallow all solutions.
> FYI, there's a lot of work done for NP-Complete phase transitions.
I am really not into phase transitions, but I think 2XSAT being NP-complete could explain them pretty nicely. I think most of the observed behavior comes from XORSAT portion dominating hard but satisfiable instances. AFAICT XORSAT exhibits similar phase transition, where there is only relatively small number of low-dimensional solutions to a random instance of XORSAT - either they have many solutions or the system becomes unsolvable. The only difficulty in analysing this is that 2SAT portion of 2XSAT, if it is dense enough, can also contribute to XORSAT portion (by forcing some variables to be constant or equal).
https://www.researchgate.net/profile/Lavallee-Ivan/publicati...
If you have too few constraints it's easy to find a solution to the problem. If there are too many constraints it's also easy to show there are no solutions. But in the middle there's a sweet spot in the ratio of constraints to variables that's when the problem becomes fiendishly difficult.
I think you've fundamentally misunderstood the meaning of the P!=NP problem. In very simplified terms, it's not about what a computer can do, but about how long it takes to do something in the worst case.
All an LLM does is guess the next token based on the previous tokens and its training weights. For it to give you a solution to a large SAT problem it'll have to spit out a million character long binary string.
The likelyhood most of that string will be entirely hallucinated is very high. LLMs are not magic.
Deep Learning is already used internally in some SAT solvers to heuristically pick in which direction the search should go.
Everyone is downvoting you because that's not the mathematical explanation, but it's absolutely the informal explanation that the 'pop science' ones were saying.
That was the informal explanation given to the computer scientists. But that's not so convincing to non computer scientists and it's not the informal 'pop science' explanation.
An example (not the only one) of the kind of 'pop science' explanation I mean, is Impagliazzo's Five Worlds (https://gwern.net/doc/cs/cryptography/1995-impagliazzo.pdf). One of those hypothetical worlds he called 'Algorithmica' and it's where P = NP. One of the amazing and outlandish things that could be accomplished in such an exotic world would be the following feat: "Thus, a computer could be taught to recognize and parse grammatically correct English just by having sufficiently many examples of correct and incorrect English statements, without needing any specialized knowledge of grammar or English."
It's not so wild to me, to think that if someone's understanding of P vs. NP was from that kind of pop science article, then they would think we should start considering more seriously that we are in the Algorithmica (P = NP) world where such feats are possible!
In particular, the hand-waving is the phrase 'sufficiently many examples.' This may be impossible to provide even if only a googol (10^100) examples are needed, because of mass and energy limitations of the universe - there's only so much you can do with 10^80 atoms, and the speed of light is too slow.
"Only a googol" because Graham's Number is mind-boggingly huge - https://en.wikipedia.org/wiki/Graham%27s_number
The Wikipedia entry for "galactic algorithm" at https://en.wikipedia.org/wiki/Galactic_algorithm even mentions SAT: "a hypothetical large but polynomial O(n^2^100) algorithm for the Boolean satisfiability problem, although unusable in practice, would settle the P versus NP problem".
Algorithmica may therefore be very little different than our world, other than that people no longer debate if P = NP.
I most certainly am not.
