I present a step-by-step argument in philosophy of mind. The main conclusion is that it is probably possible for conscious homomorphically encrypted digital minds to exist. This has surprising implications: it demonstrates a case where "mind exceeds physics" (epistemically), which implies the disjunction "mind exceeds reality" or "reality exceeds physics". The main new parts of the discussion consist of (a) an argument that, if digital computers are conscious, so are homomorphically encrypted versions of them (steps 7-9); (b) speculation on the ontological consequences of homomorphically encrypted consciousness, in the form of a trilemma (steps 10-11).

Step 1. Physics

Let P be the set of possible physics states of the universe, according to "the true physics". I am assuming that the intellectual project of physics has an idealized completion, which discovers a theory integrating all potentially accessible physical information. The theory will tend to be microscopic (although not necessarily strictly) and lawful (also not necessarily strictly). It need not integrate all real information, as some such information might not be accessible (e.g. in the case of the simulation hypothesis).

Rejecting this step: fundamental skepticism about even idealized forms of the intellectual project of physics; various religious/spiritual beliefs.

Step 2. Mind

Let M be the set of possible mental states of minds in the universe. Note, an element of M specifies something like a set or multiset of minds, as the universe could contain multiple minds. We don't need M to be a complete theory of mind (specifying color qualia and so on); the main concern is doxastic facts, about beliefs of different agents. For example, I believe there is a wall behind me; this is a doxastic mental fact. This step makes no commitment to reductionism or non-reductionism. (Color qualia raise a number of semantic issues extraneous to this discussion; it is sufficient for now to consider mental states to be quotiented over any functionally equivalent color inversion/rotations, as these make no doxastic differences.)

Rejecting this step: eliminativism, especially eliminative physicalism.

Step 3. Reality

Let R be the set of possible reality states, according to "the true reality theory". To motivate the idea, physics (P) only includes physical facts that could in principle be determined from the contents of our universe. There would remain basic ambiguities about the substrate, such as multiverse theories, or whether our universe exists in a computer simulation. R represents "the true theory of reality", whatever that is; it is meant to include enough information to determine all that is real. For example, if physicalism is strictly true, then $R=P$, or is at least isomorphic. Solomonoff induction, and similarly the speed prior, posit that reality consists of an input to a universal Turing machine (specifying some other Turing machine and its input), and its execution trajectory, producing digital subjective experience.

Let $f:R\to P$ specify the universe's physical state as a function of the reality state. Let $g:R\to M$ specify the universe's mental state as a function of the reality state. These presumably exist under the above assumptions, because physics and mind are both aspects of reality, though these need not be efficiently computable functions. (The general structure of physics and mind being aspects of reality is inspired by neutral monism, though it does not necessitate neutral monism.)

Rejecting this step: fundamental doubt about the existence of a reality on which mind and physics supervene; incompatibilism between reality of mind and of physics.

Step 4. Natural supervenience

Similar to David Chalmers's concept in The Conscious Mind. Informally, every possible physical state has a unique corresponding mental state. Formally:

$\forall (p:P),(\exists r:R,f(r)=p)\to (\exists !(m:M),\forall (r_2:R),(f\left(r_2\right)=p)\to (g\left(r_2\right)=m\left)\right)$

Here $\exists !$ means "there exists a unique".

Assuming ZFC and natural supervenience, there exists the mapping function $h:P\to M$ commuting ($h\circ f=g$), though again, h need not be efficiently computable.

Natural supervenience is necessary for it to be meaningful to refer to the mental properties corresponding to some physical entity. For example, to ask about the mental state corresponding to a physical dog. Natural supervenience makes no strong claim about physics "causing" mind; it is rather a claim of constant conjunction, in the sense of Hume. We are not ruling out, for example, physics and mind being always consistent due to a common cause.

Rejecting this step: Interaction dualism. "Antenna theory". Belief in P-zombies as not just logically possible, but really possible in this universe. Belief in influence of extra-physical entities, such as ghosts or deities, on consciousness.

Step 5. Digital consciousness

Assume it is possible for a digital computer running a program to be conscious. We don't need to make strong assumptions about "abstract algorithms being conscious" here, just that realistic physical computers that run some program (such as a brain emulation) contain consciousness. This topic has been discussed to death, but to briefly say why I think digital computer consciousness is possible:

• The mind not being digitally simulable in a behaviorist manner (accepting normal levels of stochasticity/noise) would imply hypercomputation in physics, which is dubious.
• Chalmers's fading qualia argument implies that, if a brain is gradually transformed into a behaviorally equivalent simulation, and the simulation is not conscious, then qualia must fade either gradually or suddenly; both are problematic.
• Having knowledge that no digital computer can be conscious would imply we have knowledge of ultimate reality $r:R$, specifically, that we do not exist in a digital computer simulation. While I don't accept the simulation hypothesis as likely, it seems presumptuous to reject it on philosophy of mind grounds.

Rejecting this step: Brains as hypercomputers; or physical substrate dependence, e.g. only organic matter can be conscious.

Step 6. Real-physics fully homomorphic encryption is possible

Fully homomorphic encryption allows running a computation in an encrypted manner, producing an encrypted output; knowing the physical state of the computer and the output, without knowing the key, is insufficient to determine details of the computation or its output in physical polynomial time. Physical polynomial time is polynomial time with respect to the computing power of physics, BQP according to standard theories of quantum computation. Homomorphic encryption is not proven to work (since P != NP is not proven). However, quantum-resistant homomorphic encryption, e.g. based on lattices, is an active area of research, and is generally believed to be possible. This assumption says that (a) quantum-resistant homomorphic encryption is possible and (b) quantum-resistance is enough; physics doesn't have more computing power than quantum. Or alternatively, non-quantum FHE is possible, and quantum computers are impossible. Or alternatively, the physical universe's computation is more powerful than quantum, and yet FHE resisting it is still possible.

Rejecting this step: Belief that the physical universe has enough computing power to break any FHE scheme in polynomial time. Non-standard computational complexity theory (e.g. P = NP), cryptography, or physics.

Step 7. Homomorphically encrypted consciousness is possible

(Original thought experiment proposed by Scott Aaronson.)

Assume that a conscious digital computer can be homomorphically encrypted, and still be conscious, if the decryption key is available nearby. Since the key is nearby, the homomorphic encryption does not practically obscure anything. It functions more as a virtualization layer, similar to a virtual machine. If we already accept digital computer consciousness as possible, we need to tolerate some virtualization, so why not this kind?

An intuition backing this assumption is "can't get something from nothing". If we decrypt the output, we get the results that we would have gotten from running a conscious computation (perhaps including the entire brain emulation state trajectory in the output), so we by default assume consciousness happened in the process. We got the results without any fancy brain lesioning (to remove the seat of consciousness while preserving functional behavior), just a virtualization step.

As a concrete example, consider if someone using brain emulations as workers in a corporation decided to homomorphically encrypt the emulation (and later decrypt the results with a key on hand), to get the results of the work, without any subjective experience of work. It would seem dubious to claim that no consciousness happened in the course of the work (which could even include, for example, writing papers about consciousness), due to the homomorphic encryption layer.

As with digital consciousness, if we knew that homomorphically encrypted computations (with a nearby decryption key) were not conscious, then we would know something about ultimate reality, namely that we are not in a homomorphically encrypted simulation.

Rejecting this step: Picky quasi-functionalism. Enough multiple realizability to get digital computer consciousness, but not enough to get homomorphically encrypted consciousness, even if the decryption key is right there.

Step 8. Moving the key further away doesn't change things

Now that the homomorphically encrypted conscious mind is separated from the key, consider moving the key 1 centimeter further away. We assume this doesn't change the consciousness of the system, as long as the key is no more than 1 light-year away, so that it is in principle possible to retrieve the key. We can iterate to move the key 1 light-year away in small steps, without changing the consciousness of the overall system.

As an intuition, suppose the contrary that the computation with the nearby key was conscious, but not with the far-away key. We run the computation, still encrypted, to completion, while the key is far away. Then we bring the key back and decrypt it. It seems we "got something from nothing" here: we got the results of a conscious computation with no corresponding consciousness, and no fancy brain lesioning, just a virtualization layer with extra steps.

Rejecting this step: Either a discrete jump where moving the key 1 cm removes consciousness (yet consciousness can be brought back by moving the key back 1cm?), or a continuous gradation of diminished consciousness across distance, though somehow making no behavioral difference.

Step 9. Deleting a far-away key doesn't change things

Suppose the system of the encrypted computation and the far-away key is conscious. Now suppose the key is destroyed. Assume this doesn't affect the system's consciousness: the encrypted computation by itself, with no key anywhere in the universe, is still conscious.

This assumption is based on locality intuition. Could my consciousness depend directly on events happening 1 light-year away, which I have no way of observing? If my consciousness depended on it in a behaviorally relevant way, then that would imply faster-than-light communication. So it can only depend on it in a behaviorally irrelevant way, but this presents similar problems as with P-zombies.

We could also consider a hypothetical where the key is destroyed, but then randomly guessed or brute-forced later. Does consciousness flicker off when the key is destroyed, then on again as it is guessed? Not in any behaviorally relevant way. We did something like "getting something from nothing" in this scenario, except that the key-guessing is real computational work. The idea that key-guessing is itself what is producing consciousness is highly dubious, due to the dis-analogy between the computation of key-guessing and the original conscious computation.

Rejecting this step: Consciousness as a non-local property, affected by far-away events, though not in a way that makes any physical difference. Global but not local natural supervenience.

Step 10. Physics does not efficiently determine encrypted mind

If a homomorphically encrypted mind (with no decryption key) is conscious, and has mental states such as belief, it seems it knows things (about its mental states, or perhaps mathematical facts) that cannot be efficiently determined from physics, using the computation of physics and polynomial time. Physical omniscience about the present state of the universe is insufficient to decrypt the computation. This is basically re-stating that homomorphic encryption works.

Imagine you learn you are in such an encrypted computation. It seems you know something that a physically omniscient agent doesn't know except with super-polynomial amounts of computation: the basic contents of your experience, which could include the decryption key, or the solution to a hard NP complete problem.

There is a slight complication, in that perhaps the mental state can be determined from the entire trajectory of the universe, as the key was generated at some point in the past, even if every trace of it has been erased. However, in this case we are imagining something like Laplace's demon looking at the whole physics history; this would imply that past states are "saved", efficiently available to Laplace's demon. (The possibility of real information, such as the demon's memory of the physical trajectory, exceeding physical information, is discussed later; "Reality exceeds physics, informationally".)

If locality of natural supervenience applies temporally, not just spatially, then the consciousness of the homomorphically encrypted computation can't depend directly on the far past, only at most the recent past. In principle, the initial state of the homomorphically encrypted computation could have been "randomly initialized", not generated from any existent original key, although of course this is unlikely.

So I assume that, given the steps up to here, the homomorphically encrypted mind really does know something (e.g. about its own experiences/beliefs, or mathematical facts) that goes beyond what can be efficiently inferred from physics, given the computing power of physics.

Rejecting this step: Temporal non-locality. Mental states depend on distinctions in the distant physical past, even though these distinctions make no physical or behavioral difference in the present or recent past. Doubt that the randomly initialized homomorphically encrypted mind really "knows anything" beyond what can be efficiently determined from physics, even reflexive properties about its own experience.

Step 11. A fork in the road

A terminological disambiguation: by P-efficiently computable, I mean computable in polynomial time with respect to the computing power of physics, which is BQP according to standard theories. By R-efficiently computable, I mean computable in polynomial time with respect to the computing power of reality, which is at least that of physics, but could in principle be higher, e.g. if our universe was simulated in a universe with beyond-quantum computation.

If assumptions so far are true, then there is no P-efficiently computable $h:P\to M$ mapping physical states to mental states, corresponding to the natural supervenience relation. This is because, in the case of homomorphically encrypted computation, h would have to run in P-super-polynomial time. This can be summarized as "mind exceeds physics, epistemically": some mind in the system knows something that cannot be P-efficiently determined from physics, such as the solution to some hard NP-complete problem.

Now we ask a key question: Is there a R-efficiently computable $g:R\to M$ mapping reality states to mental states, and if so, is there a P-efficiently computable g?

Path A: Mind exceeds reality

Suppose there is no R-efficiently computable g (from which it follows that there is no P-efficiently computable g). That is, even given omniscence about ultimate reality, and polynomial computation with respect to the computation of reality (which is at least as strong as that of physics, perhaps stronger), it is still not possible to know all about minds in the universe, and in particular, details of the experience contained in a homomorphically encrypted computation. Mind doesn't just exceed physics; mind exceeds reality.

Again, imagine you learn you are in a homomorphically encrypted computation. You look around you and it seems you see real objects. Yet these objects' appearances can't be R-efficiently determined on the basis of all that is real. Your experiences seem real, but they are more like "potentially real", similar to hard-to-compute mathematical facts. Yet you are in some sense physically embodied; cracking the decryption key would reveal your experience. And you could even have correct beliefs about the key, having the requisite mathematical knowledge for the decryption. You could even have access to and check the solution to a hard NP complete problem that no one else knows the solution to; does this knowledge not "exist in reality" even though you have access to it and can check it?

Something seems unsatisfactory about this, even if it isn't clearly wrong. If we accept step 2 (existence of mind), rejecting eliminativism, then we accept that mental facts are in some sense real. But here, they aren't directly real in the sense of being R-efficiently determined from reality. It is as if an extra computation (search or summation over homomorphic embeddings?) is happening to produce subjective experience, yet there is nowhere in reality for this extra computation to take place. The point of positing physics and/or reality is partially to explain subjective experience, yet here there is no R-efficient explanation of experience in terms of reality.

Path B: Reality exceeds physics, computationally

Suppose $g:R\to M$ is R-efficiently computable, but not P-efficiently computable. Then the real substrate computes more powerfully than physics (given polynomial time in each case). Reality exceeds physics: there really is a more powerful computing substrate than is implied by physics.

As a possibility argument, consider that a Turing-computable universe, such as Conway's Game of Life, can be simulated in this universe. Reality contains at least quantum computing, since our universe (presumably) supports it. This would allow us to, for example, decrypt the communications of Conway's Game of Life lifeforms who use RSA.

So we can't easily rule out that the real substrate has enough computation to efficiently determine the homomorphically encrypted experience, despite physics not being this powerful. This would contradict strict physicalism. It could open further questions about whether homomorphic encryption is possible in the substrate of reality, though of course in theory something analogous to P = NP could apply to the substrate.

Path C: Reality exceeds physics, informationally

Suppose instead that $g:R\to M$ is P-efficiently computable (and therefore also R-efficiently computable). Then physicalism is strictly false: R contains more accessible information than P. There is real information, exceeding the information of physics, which is sufficient to P-efficiently determine the mental state of the conscious mind in the homomorphically encrypted computation. Perhaps reality has what we might consider "high-level information" or a "multi-level map". Maybe reality has a category theoretic and/or universal algebraic structure of domains and homomorphisms between them.

According to this path, reductionism is not strictly true. Mental facts could be "reduced" to physical facts sufficient to re-construct them (by natural supervenience). However, there is no efficient re-construction; the reduction destroys P-computation-bounded information even though it destroys no computation-unbounded information. Hence, since reality P-efficiently determines subjective experiences, unlike physics, it contains information over and above physics.

HashLife is inspirational, in its informational preservation and use of high-level features, while maintaining the expected low-level dynamics of Conway's Game of Life. Though this is only a loose analogy.

Conclusion

Honestly, I don't know what to think at this point. I feel pretty confident about conscious digital computers being possible. The homomorphic encryption step (with a key nearby) seems to function as a virtualization step, so I'm willing to accept that, though it introduces complications. I am pretty sure moving the key far away, then deleting it, doesn't make a difference; denying either would open up too many non-locality paradoxes. So I do think a homomorphically encrypted computation, with no decryption key anywhere, is probably conscious, though ordinary philosophical uncertainty applies.

That leads to the fork in the road. Path A (mind exceeds reality) seems least intuitive; it implies actual minds can "know more" than reality, e.g. know mathematical facts not R-efficiently determinable from reality. It seems dogmatic to be confident in either path B or C; both paths imply substantial facts about the ultimate substrate. Path B seems to have the fewest conceptual problems: unlike path C, it doesn't require positing the informational existence of "high-level" homomorphic levels above physics. However, attributing great computational power to the real substrate would have anthropic implications: why do we seem to be in a quantum-computing universe, if the real substrate can support more advanced computations?

Path C is fun to imagine. What if some of what we would conceive of as "high-level properties" really exist in the ultimate substrate of reality, and reductionism simply assumes away this information, with invalid computational consequences? This thought inspires ontological wonder.

In any case, the disjunction of path B or C implies that strict physicalism is false, which is theoretically notable. If B or C is correct, reality exceeds physics one way or another, computationally and/or informationally. Ordinary philosophical skepticism applies, but I accept the disjunction $B\vee C$ as the mainline model. (Note that Chalmers believes natural supervenience holds but that strict physicalism is false.)

As an end note, there is a general "trivialism" objection to functionalism, in that many physical systems, such as rocks, can be interpreted as running any of a great number of computations. Chalmers has discussed causal solutions; Jeff Buenchner has discussed computational complexity solutions (in Gödel, Putnam, and Functionalism), restricting interpretations to computationally realistic ones, e.g. not interpreting a rock as solving the halting problem. Trivialism and solutions to it are of course relevant to attributing mental or computational properties to a computer running a homomorphically encrypted computation.

(thanks to @adrusi for a X discussion leading to many of these thoughts)