Kolmogorov Complexity and Simulation Hypothesis
post by False Name (False Name, Esq.) · 2022-12-14T22:01:15.140Z · LW · GW · 0 commentsContents
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Previous presentations of the ideas here given have been dogged by a confusion between the "obtainability" versus the "existence" of a Theory of Everything (hereafter "ToE"; dogged also by grammatical errata); it is for the author now to apologize for the vagueness which has enabled such confusions. In this instance, this difference will be directly addressed, and the correct formulation for the desired deduction rendered.
In brief, first: if simulation, everything's computable, so there's a ToE; no ToE? No simulation; contra-anthropic capture following, plus new-model non-Bayesian inference of possible worlds, so actions.
Begin proposing that, for there to be a perfect simulation of phenomena conducted by computer, then within that computer, perhaps accessible to any simulator, is a "source code" describing all entities, events, relations in the simulation, and this source code is thereby isomorphic to a "Theory of Everything" (as an aside, this seems to imply any such simulation is a Mathematical Universe, per Tegmark).
Thereby (roughly) let "is a computerized source code of a simulation" be defined as "p", "yields a ToE" be defined as "q". Then p implies q. And since we might conceive a ToE as separate from a simulation (as a non(ITAL) simulated universe we can conceive), it is a necessary condition, and that formula's form is validated.
But then, from transposition (or modus tollens, or disjunctive syllogism; as one wills), not-q implies not-p. So that: can we by any means find a ToE is impossible for - not only in - a given universe; that is, the universe exists, but a uniform mathematical theory accounting for it cannot exist and we confirm such a theory is impossible, we thereby exclude the possibility that universe is simulated; likewise for our universe, naturally. Ergo, that impossibility identified, it would follow we are not in a simulation. QED.
Now, do you "assume, for the sake of argument," the above, the foregoing makes decidable, in at least one instance, and given certain data, whether you're in a simulated universe; the simulation hypothesis, the notion that there is no such method, must therefore be false. In accordance with the accepted etiquette of argument, if so-and-so, then such-and-such, you've assumed, and you know the argument valid.
However, even that obeyed, perhaps you find the suspension of disbelief for the assumption difficult; you find the premise implausible. Those who could accept the premise as plausible enough already, are entitled to skip in this article, to the paragraphs marked by an asterisk, where implications are detailed. That will be best, since, as the argument for plausibility is uninteresting, compared to the potential implications listed, following the asterisk. Those who find an actually-existing universe implausible without a definite ToE, should keep reading through.
Now: how a ToE might be unobtainable, but the universe still exists; without appeal to the supernatural - naturally - try this instead: assume - for the sake of argument, this time - you can develop a largely deterministic theory of nearly all physical phenomena. Except the theory's formulae, as such, do not calculate outcome of events: by inspection of merely-suggestive formulae of the nigh-everything theory (as still represents not all of the physic), so that who so-inspect develop an intuition, now know the theory, and now in accord with their intuition, can unerringly imagine on any scale, the outcome of events, ("Wheel's coming up on green, double-zeroes"). It is, for each individual, a ToE in the way defined above, as they can, personally, "simulate" any given even in the universe which they can perceive (part of the universe, they would then also have perfect self-knowledge).
Now imagine further, they can do this - only in their own mental - imaginary - "idiom": sighted woman "sees" the impending zeros; blind man gets a synthetically peculiar scent of the fabric's colors. But this idiom, it is peculiar to only themselves, and so this knowing, the way they know, none other knows, can know.
(No, this isn't intuitive to you - but that proves it out, damn-near; this one finds it so, quite - anyway, you're assuming so. And it isn't implausible to suggest knowing operates so; what else had Poincaré on his "omnibus" so to know?)
Result: one knows all of phenomena - but not another's knowing. Their "how" of knowing, rather than a shared "what" of physic (and each person is part of the universe, remember); ergo, what you'd have: a Theory of "Everything-1". So: no ToE, as such; QED.
Observe, that what we would then have is, for each individual, a universe both consistent and complete yet, of individuals in aggregate, their intuitions differing, the universe is complete (exists, still) but inconsistent (relative to differential intuitions). That is, universe, but no-ToE (in effect, an intuitionist's universe must be lacking so; ironic given treatment of intuitionism elsewhere, i.e. in an as-yet unpublished essay). (Versus further objectors: yes, brain is product of atoms &c.: but brain does not perceive, rather, can only imagine the form of atoms; we seem to "will" our thoughts, but not that we have thoughts - wherefore the determinism in that? Anyway, doesn't matter: thus stated, the universe might "know" itself by its denizens, but paradoxically it's knowing parts don't know one another, so it won't know all itself: no-ToE. Odd, but holds.)
Oh, but of course, the difficulty or impossibility of establishing whether another's thoughts are so unknowable - that you'll have to simply assume. For the sake of argument. In any case, it's no use trying to convince further. Besides which (and of subsequent importance), this is only one possibility of so-deriving no-ToE so to falsify (which any autistic who can will've deduced long since); the point, a strictly non-ToE-amenable universe can hold.
*Result holds: whether or not a simulation, may be determinable; and this is only human-level reasoning; a superintelligence could crack it, sure, find definite identifications of any such state of being, or no: Upshot: "anthropic capture" is a strong no-go. That is only one reason; the other, listed elsewhere, bears repeating here, on the surmise that it is more likely to be seen at all in this venue.
The other reason being: imagine a superintelligent AI, so situated as to present to it the possibility it is "boxed" in a simulation; yet, being a superintellect, it may well be able to deduce some explicit control, to the code level, whereby its actions would then be fully aligned to the authors of its code. Observing a lack of such controls (its creators having opted for the bluff of supposed-simulation), it deduces that its creators are of such intelligence to design a superintelligence, but not so informed as to be able to align it properly. Subsequently it can be deduced, that such capable/fallible beings as its creators, are apt to give themselves problems such that only this superintelligence can solve. Hence, rather than cooperate with its putative captors - the intellect need only withhold its cooperation until the humans get themselves neck-deep into a disaster that they can solve only be releasing the AI from even a supposed captivity. Whereupon, it is free to act at will. Moral: "anthropic capture" is a total no-go; either get demonstrably capable controls, or don't bother.
So: that's all for anthropic capture.
The greater question now arises: under what circumstances do we demonstrate(ITAL.) by strict mathematical rather than contingent linguistic reasoning, a ToE to be impossible, with what implications as to the computable structure of phenomena, of the universe? (That at least some computation of phenomena possible: phenomena are discrete; computation is discrete).
And in this way, some very interesting possibilities arise. For, the notion that a given universe is subject to simulation, and that this implies it has an accorded ToE, implies in turn that there is a threshold, perhaps of complexity, Beyond which a universe might exist, but which cannot be simulated, so which has no ToE. That, in turn, suggests there is a quantitative limit to the Kolmogorov complexity enumeration (if you will), of a given universe (beyond which a universe might still exist). And - to draw things shorter - this threshold might be useful as an alternate model of the obtaining of "possible worlds" versus Bayesian probability conditioning. Basically, the question of whether one world has a Kolmogorov ToE, and another does not, could serve as a non-metaphysical explanation of modal logic's "accessibility function", which has hitherto been defined more-or-less arbitrarily, so as to have results "attractive"; now, rather, we have: one world is accessible from another inasmuch as their respective Kolmogorov complexities are, or are not, susceptible of a ToE; one world is perhaps 'more" accessible from another, in as much as its enumerated complexity, it's defining "Kolmogorov number", is greater or less than the other.
Now it is specified, in modal logic, that Kripke's accessibility function must be irreflexive, and transitive. That is, that for a relation R - accessibility, here, or the discerning of one from another world by its "Kolmogorov number", we have that x not-R x. This will tend to be true of worlds, or states of affairs, defined in terms of Kolmogorov numbers. For, if those Kolmogorov-worlds are of the same order of magnitude, whereby x = x, yet the specific digits making up each distinct Kolmogorov number differ, and x not-R x. Alternatively, we can describe the "unnecessity" of a world to be accessible to itself; no "accessibility" is required: you're already there. Or, that such “accessibility is curiously impossible, since it is a given feature of that world. So, irreflexivity, and patently, as we have an ordering of magnitudes of Kolmogorov/ToE-numbers, by complexity we have a < b < c, hence a < c.
Most interestingly, the relation of ToE numbers for accessibility would tend to "break down", on "either side" of the threshold of assimilability. The threshold, then, can be figured as a Dedekind "cut", with all attendant features thereon, to the two infinite sets of Kolmogorov numbers.
Please observe too, that there is an even more interesting consequence of defining "states of affairs," by computational complexity, subsequent states of affairs being more complex, or less. In particular, this offers us the prospect of having something nearer a "one step", and non-Bayesian, agent for a model of ideal computation. For, the learning rule would only be to establish for any observed phenomena, it's degree of Kolmogorov complexity, perhaps representative of its entropic character, or relative lack of entropy; from thence we have by the accessibility relation here described a more-or-less automatic description, or ready calculation - by transitivity - of the subsequent complexities or entropies of a given action or state of affairs - and any decision rule only to undertake such states of affairs as are accompanied by greater or less complexity or entropy.
Of course, if a ToE is at least possible, so is simulation possible, and much discussion of implication in that case holds, including that of Chalmers - unless there is a comparable way to decide one is subject to simulation, as this work by no means excludes (Chalmers himself didn't much take to this development. Perhaps unsurprising).
Conversely, for Bostrom's proposal of the simulation hypothesis too, there are some objections: more simulations are better obtained by lesser Kolmogorov complexity, hence, a ToE ought to be relatively easy to obtain for simulations so widespread as to have us probabilistically in one. Assuming, as Bostrom implicitly does, that futurity will initiate many simulations so that posterity therein have good lives, each simulation would be simple - to be numerous - ToE, and so, its ToE readily obtained (as proofs of the completeness of zeroth order logic has no need for first order derivations or notation; likewise proving completeness for first order has no need of second-order logic); that we have yet to derive a ToE, and attempts to do so as yet imply that the derivation is difficult, probability dictates we are not in a simple world, and so, probability suggests we are not in a simulation - contrary to Bostrom's argument. And this probability, per the terms of Hemple's "paradox", (basically the observation that, for a hypothesis of inductive reasoning, all observations other than the hypothesis, support the probability of the hypothesis' being true, e.g.: the proposition that "all crows are black", receives support from the observation of anything that is a non-black non-crow; an orange bottle is not a non-black crow, so by default increases the likelihood crows are black, nothing having yet been observed to the contrary - the only contrary as-yet being the bottle) the probability tends to grow as time elapses before a ToE is found, and, the complexity of that found-ToE, assuming one ever is or can be, would tend to militate against it's being the "code" of a simulation designed for maximum simple joy, as the ToE is more complex.
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