> And worse, the world is clearly not a computer -- as any non-trivial definition (eg., universal turing machine) cannot simulate properties the world has (non-determinism, chaos, etc.).
This is precisely wrong. With the possible exception of consciousness, in our current understanding of physics, the world is a computer. I Bieber some of your intuition fails because you have wormgly assumed this point to be obvious.
You're also mixing up the Chinese room experiment with the larger problem of an interactive AI. The Chinese room system is restricted, by definition, to only interact with the word through language. Over time, as it interacts with a person, it may decide that it likes them or not, and its responses may start changing based on this - that is allowed to be part of the algorithm that the human/machine inside the room is applying.
For a silly example, the algorithm may say that 'people who use the same word three times in a row are extremely nice', and use this state of 'extremely nice' to modify its future responses once the same word has been recorded three times in a row. This is no different, fundamentally, to me liking people whose face is symmetric, or the same gender as myself - it's not a choice I made, it's just an arbitrary rule of my (genetic) algorithm.
Quantum non-determinism isnt computable. A universal turing machine is just an implementation of a function from the Naturals to Naturals (ie., expressible as a binary -> binary).
QM systems cannot be described by such functions, hence the world isnt a computer. Eg., any quantum randomness isnt a function as it has no input state and a non-determinstic output state.
Worse, clearly at least the input and output states of a computer should be measurable -- but most systems arent. Even including classical ones.
Ie., in chaotic systems it isn't possible, due to QM, to measure x st. f in y = f(x, t) is just a set (x, y, t) -- for large t. f here cannot be a function -- as y is necessarily a distribution of states.
In abitary chaotic systems y spans the space of all possible states of abitary t, and thus we have zero ability to predict anything beyond relatively narrow time-horizons.
If you wish to claim reality is a computer: (1) state the properties of a computer; (2) show that having these properties is non-trivial (ie., it is a substantive claim that something is a computer); and (3) show that these properties are at least consistent with our best theories of physics.
An ideal Turing Machine can perfectly well predict the state of a quantum system: it can solve the Schordinger equation, and predict the time evolution of the system for each possible solution. It can then predict all possible states of the world after N steps, and even the probability of each state. N can be arbitrarily large. If you believe in the Copenhagen interpretation of QM, you then need to choose one of these states randomly according to the probability the TM predicted. If you believe in the MWI, you don't even need to do that: by this point, the TM has already done exactly what the universe does.
This same process also extends to chaotic systems: a Turing machine can take all possible measurement clause and compute the state of the system after N steps.
The mistake you are making is in the word 'predict'. Of course you can't predict the result of a non-deterministic process. But a computer doesn't predict: it computes, that is, it follows mechanical steps to transform an input value. And non-deterministic computations are nothing special, we do them every day in real computers ( rand() is a non-deterministic computation). The most famous class of algorithms in complexity theory is even called 'Nondeterministic Polynomial Time', NP for short (of P=NP? fame).
I would first note that you can keep arguing with me all you want, but it is well known and commonly practiced that a Turing machine such as a digital computer can simulate a Quantum Computer or Quantum System to arbitrary precision, given sufficiently large but finite time. There are even cloud services offering such simulation capabilities, such as Amazon Braket [0]. The simulations take exponential time to run a linear time quantum algorithm, but they will produce the same result as a QC would (after enough sampling of the QC results).
> Hilbert space is infinite-dimensional and real-valued.
Infinite-dimensional Hilbert state spaces are a useful mathematical model, but they are not physical. Physical systems have finite elements and thus finite state spaces. We could argue that space itself is infinitely divisible, so that we need infinite-dimensional state spaces to represent the infinite possible positions of a particle, but this is not a physical concern, as it is impossible to differentiate in finite time two states that only differ in an infinitesimal position change - so, we are free to choose some minimal unit of length (say, one over Graham's number of the Planck length) and get a finite-dimensional state space. Crucially, for any amount of time and for any sensitivity of instruments, we can always choose some such unit and ensure that our results will not be distinguishable with those measurement instruments within that amount of time.
> QM says that there is no `rand(seed)`, rather there is only `rand()` that is what "no hidden variables" means.
QM says no such thing, though many interpretations do. Still, that is exactly why I chose rand(), not rand(seed) in my example, so I'm not sure why you're bringing this up. rand(seed) is a deterministic computation, rand() is a non-deterministic computation.
> If you want to do `rand(s) forall s`, that "forall" requires real numbers -- and then you're out-of-luck on computability.
This is not what I was proposing. I was proposing a Turing machine + perfectly random rand() as the simulation of a random world.
Even this is not necessary if you believe in the MWI, where the whole universe actually evolves perfectly linearly and deterministically from a fixed initial state, with no randomness of any kind. This interpretation is perfectly compatible with all observations of QM, if we also add the postulate that observers can only observe one state at a time (the one they are entangled with).
Even without going there, the construction I proposed earlier, where we essentially select enough real numbers to satisfy any possible measurement device leads to such that our computable set is indistinguishable in practice from the infinite uncomputable set we started with still applies.
In general, there is no (known) way to introduce infinity into empirical science - it is a priori impossible to distinguish, in finite time, between an arbitrarily large but finite quantity, and a truly infinite quantity.
A quantum computer is just a computer. Yes, a turing machine can simulate a QC. A QC doesnt use any special quantum properties in its computation. The "quantum" part should really be read more about its storage system than its computation, ie., its mostly about how its input states are prepared.
That's neither here nor there though.
A myriad of other issues exist. The uncertainty principle precludes "in any world", infinitely precise measurement. Thus chaotic systems, at least of a large time horizon, arent deterministic.
A chaotic system is one in which "insignificant digits" in the input determine "significant digits" in the output. You need very very very low digit precession in inputs the longer the time horizon. This gets beyond the uncertainty principal.
You can run as many turing machines as you like (necessarily, uncountably many to be consistent with QM). But you cannot escape the problem.
The best theories of physics are extremely far away from "effective computation". They are phrased actively hostile to it. And you have to completely revise physics to make it even half-plausible. It's a project which is barely stated, and a long way from finished.
It seems, indeed, a little doomed to failure. Properties that physics uses day-in-day-out posses the trivial infinities of geometry; and the trivial randomness of uncountable ensembles etc. -- to claim that physics states that reality is a computer is bizarre.
A QC absolutely uses quantum properties in its probabilistic computation (the complex-valued probabilities of its qbits and their entanglement). A QC can also perfectly represent the state of a quantum system (until you want to measure its output, of course).
The laws of physics are known to be computable. Quantum Mechanics is one of the easiest to prove this for, since it is based on simple linear transforms! (Thermodynamics and GR are more complex, but still computable).
I have no idea why you believe differently.
A chaotic system requires arbitrary precision in the input measurements to be able to predict arbitrarily precisely its state after arbitrarily many steps. But, crucially, it does not require infinite precision in the input unless you want to predict its state with infinite precision (non-goal) or after infinite time (also a non-goal).
The uncertainty principle is also irrelevant. The uncertainty principle has to do with measurement of classical properties. Conversely, the wave function of a system is precise, not fuzzy. It's only in measuring properties of the wave function that we reach uncertainty or probability at all. So, a computer can precisely compute the simple linear evolution of the wave function, according to the Schrodinger equation, which is what the universe itself is doing. The result of this computation is a complex value. You can then apply the Born rule to this result to get the probability of your system being in a particular state after following the simulated evolution, just as you would for a real quantum mechanical system.
We do this kind of quantum simulation every day with great success, with a countable number of classical computers (typically 1), though it is only tractable for very very simple systems. The results don't typically differ in any perceivable way from the actual experiment.
Depends on rand() implementation. The one in C is indeed deterministic. Others use hardware entropy as the seed, and are not predictable.
Either way, extending the TM model with an idealized rand() that is purely random doesn't significantly alter the model, and it makes it capable of non-deterministic computations.
This is precisely wrong. With the possible exception of consciousness, in our current understanding of physics, the world is a computer. I Bieber some of your intuition fails because you have wormgly assumed this point to be obvious.
You're also mixing up the Chinese room experiment with the larger problem of an interactive AI. The Chinese room system is restricted, by definition, to only interact with the word through language. Over time, as it interacts with a person, it may decide that it likes them or not, and its responses may start changing based on this - that is allowed to be part of the algorithm that the human/machine inside the room is applying.
For a silly example, the algorithm may say that 'people who use the same word three times in a row are extremely nice', and use this state of 'extremely nice' to modify its future responses once the same word has been recorded three times in a row. This is no different, fundamentally, to me liking people whose face is symmetric, or the same gender as myself - it's not a choice I made, it's just an arbitrary rule of my (genetic) algorithm.