On Monday, March 16, 2020 at 9:24:17 AM UTC-5, Bruno Marchal wrote:
>
>
> On 15 Mar 2020, at 16:39, Lawrence Crowell <goldenfield...@gmail.com
> <javascript:>> wrote:
>
> This is from a post I wrote on Hossenfelder's blog:
>
> Superdeterminism has characteristics very similar to hypercomputation. A
> hyper-Turing machine is one able to circumvent the restrictions of the
> Gödel-Turing uncomputability limit. A sort of Zeno machine may illustrate
> this. If I were to flip a switch at one second, then flip again at ½
> second, then flip again at ¼ second and so forth, what is the state of the
> switch after 2 seconds? Well the rapid increase in energy required to
> oscillate the switch means that even if the switch does not fly apart it
> will face an energy limit where it is a black hole. Hence, at least from
> the outside general relativity appears to spoil the dream of circumventing
> Gödel and Turing.
>
> What if we go into the black hole? The inner event horizon is in the pure
> eternal solution continuous with I^∞. This would permit a sort of Zeno
> computation to be observed. This means one could in principle witness this
> end of switch toggling. So, if that switch is toggled or not toggled with
> each half-partitioned integral of time any possible algorithm can be
> computed and its output logged. However, this idealism is spoiled by
> quantum mechanics, for the black hole will emit Hawking radiation and the
> inner horizon is no longer continuous with I^∞. Thus, quantum mechanics
> rescues Gödel and Turing, where both QM and GR appear to respect the
> Church-Turing thesis of computation so computed outputs are from first
> order primitive recursive algorithms.
>
>
> I agree, except on some minute vocabulary use, but I will not bother you
> with this, unless necessary at some point.
> Note that a universal machine can compute/emulate all algorithms, but of
> course not decide all provability question, and less so semantical
> question, but not less than us, apparently.
>
>
>
The UTM is an emulator, and it can't determine the halting status of all
TMs. A part of the reason is that as an emulator it has to emulate itself
emulating other TMs. which leads to the Cantor diagonal slash result.
>
>
>
> As Peter Shor keeps harping on superdeterminism, with its real nesting of
> fractal loops or orbits, implies a vast or even infinite number of degrees
> of freedom. This does appear to be a serious minefield to go through,
> However, if these are nonlocal they are not real degrees of freedom that
> communicate information. They respect the no-signaling theorem of QM. If we
> could really output all of these nested loops or paths this would be a sort
> of hypercomputation. But we can’t, I illustrate this with the Matiyasevich
> theorem in the uncomputability of p-adic sets,
>
>
> I know (and teaches) Matiyasevic’s result that the diophantine polynomial
> equations are Turing Universal, and thus that Hilbert 10th problem is
> undedicadable (assuming CT). But I am not aware he worked on the p-adic
> numbers. If you have a link?
>
>
>
The Gödel incompleteness comes with the p-adic number system used to define
a field over this set or Cantor set. The set of frequencies or
periodicities means this Cantor set in a certain “limit” has an unbounded
set of primes for p-adic number systems. Then enters Martin Davis, Hilary
Putnam, Julia Robinson and in particular Yuri Matiyasevich (DPRM). Hilbert
asked if all Diophantine equations could be solved by a single method in
his 10th problem. Diophantine equations were found to be equivalent to
p-adic sets, and subsequently Yuri Matiyasevich proved these sets were not
computable by a single method. This is a Gödel incompleteness result, where
a proof for all possible p-adic is a form of Cantor diagonal slash.
Solutions to Diophantine equations, which are associated with the nested
frequencies or periodicities of orbits, can be solved locally, but there is
no global solution method. This is a form of Szangolies’ epistemic horizon.
The Cantor set here then has some undecidable properties in the p-adic
setting, where any global field of numerical operations in a p-adic setting
is incomplete.
Martin Davis has a book on computability, *Computability and Unsolvability*
and in a Dover edition, that on page 114 shows how all Diophantine
equations are equivalent to a genera p-adic set. This was solved by DPR
where M then used this in a Gödel incompleteness result. This is one of a
few cases where Gödel incompleteness impacts what might be called real
mathematics.
>
>
>
>
> while Palmer appeals to a funny idea of uncomputability of fractal or
> recursively enumerable sets by Blum, Shub, and Smale.
>
>
>
> OK. Unfortunately with a non standard notion of computability on the reals
> (or on any rings). They ahem shown that the Mandelbrot set, for example, is
> uneducable in that sense, but the question “is the (rational) Mandelbrot
> set Turing universal is still an open problem.
>
>
>
I have been arm wrestling with Hossenfelder and Palmer over this. This is a
non-standard meaning to uncomputability, which defines recursively
enumerable sets as uncomputable because a final output does not occur. It
is really the complement that is uncomputable. This is why I much prefer
the DPRM approach with p-adic sets.
>
>
> Hypercomputing is an interesting concept,
>
>
> To be honest, I have not yet find a satisfying definition of
> “hyper-computation”. It always look like computation + magic, which is not
> different from Turing computability + oracle. Note that all my results go
> through for the machine + oracle.
> Or, in some case, hypercomputaion is more than that, but then all
> functions are computable, and it is not clear if this is meant classicaly,
> (in which case it is a trivial) or intutionistically, in which case we get
> a very restricted subset of the Turing-computable.
>
>
>
A hyper-Turing machine is one able to circumvent the restrictions of the
Gödel-Turing uncomputability limit. A sort of Zeno machine may illustrate
this. If I were to flip a switch at one second, then flip again at ½
second, then flip again at ¼ second and so forth, what is the state of the
switch after 2 seconds? Well the rapid increase in energy required to
oscillate the switch means that even if the switch does not fly apart it
will face an energy limit where it is a black hole. Hence, at least from
the outside general relativity appears to spoil the dream of circumventing
Gödel and Turing.
What if we go into the black hole? The inner event horizon is in the pure
eternal solution continuous with I^∞. This would permit a sort of Zeno
computation to be observed. This means one could in principle witness this
end of switch toggling. So, if that switch is toggled or not toggled with
each half-partitioned integral of time any possible algorithm can be
computed and its output logged. However, this idealism is spoiled by
quantum mechanics, for the black hole will emit Hawking radiation and the
inner horizon is no longer continuous with I^∞. Thus, quantum mechanics
rescues Gödel and Turing, where both QM and GR appear to respect the
Church-Turing thesis of computation so computed outputs are from first
order primitive recursive algorithms.
>
> but spacetime configurations that permit it violate conditions such as
> Hawking-Penrose energy conditions or chronology protection. Quantum
> mechanics and general relativity have conditions that in some ways are
> equivalent. For instance, the no-cloning theorem of QM is equivalent to
> chronology protection, for a wormhole would permit quantum cloning of
> states.
>
>
> That is interesting. If you can explain the relation between non-cloning
> theorem (which I got from mechanism even before I heard about quantum
> mechanics), that would be interesting to get some “chronology protection”
> in arithmetic. It might help to fight the “white rabbits” which pullulated
> a priori the arithmetical reality.
>
>
>
If you have a wormhole that connects two regions of spacetime in a multiply
connected topological way, you can then accelerate one opening out to close
to the speed of light and the accelerate it back and then to rest near the
other opening. By the results of special relativity the clock on this
opening you sent on this round trip will be behind the first. So if I jump
into this opening I will exit the first opening back in time. Now suppose I
have a quantum state |ψ⟩ = a|1⟩ + b|2⟩ and I duplicate this as |ψ⟩ → |ψ⟩|ψ⟩.
However, there is an ambiguity here, for this duplication could occur in
the |1⟩ and |2⟩ basis so that |1⟩ → |1⟩|1⟩ and |2⟩ → |2⟩|2⟩. These two
duplications can easily be seen to be inequivalent. So this is not a
unitary process. The impossibility of wormholes and violations of
Hawking-Penrose energy conditions is at least consistent with there being
no unitary means for duplicating quantum states.
>
>
> For hypercomputation a spacetime with closed timelike curves would suffice
> as well. The answer to an undecidable problem could come from a time
> machine if one sends the results back in time to yourself later. Anti-de
> Sitter spacetimes permit this. Curiously AdS and de Sitter (dS) spacetimes
> have the same quantum information, for the two spacetimes share asymptotic
> infinity. The de Sitter spacetime of inflation may then have a
> correspondence with an AdS in a quantum computing system. We may think of a
> computer with a closed timelike curve CTC register plus a chronology
> respecting register. As the two are quantum states or waves they then
> constructively and destructively interfere with each other.
>
> [image: quantum computer with closed timelike curves.png]
>
> <quantum computer with closed timelike curves.png>
>
> The CTC corresponds to the chronology violating AdS and the CR is
> chronology respecting register. The self-interference of quantum waves
> imprints a quantum computing output in the R_{ctc} and quantum information
> in the R_{cr} corresponding to it has constructive interference. This is
> then how quantum computing in quantum gravitation or cosmology may work.
>
>
>
> Hmm… Once a machine is trapped on a loop, it can no more exploit its
> universality. Now, if the loop is gigantic, that does not matter in
> principle, FAPP, but it does matter in the derivation of the physical laws
> from the elementary arithmetical ontology (which is need to solve the
> mind-body problem in the Computationalist frame).
>
>
>
The closed timelike curves of the AdS and the strange hypercomputing
aspects of that are only commensurate with quantum computations in our
observable dS, or local FLRW region thereof, spacetime because of this
correlation between their asymptotic infinite information. After all Gödel's
theorem does tell us that somethings are true by self-referential
inference. We just can't causally compute them. It is like Peano
arithmetic, we can always compute the successor of a natural number, but we
can't compute the evident fact it is infinite. That is a sort of inference
by induction, which is not strictly computable. The same happens here; we
"frogish" observers in this patch on a de Sitter spacetime can't observe it
all, which in effect enforces the Church-Turing thesis.
LC
>
>
> However, event horizons protect us from observing the data stack in
> R_{ctc} and this is then not hypercomputation. In this way spacetime and in
> general classical einselected outcomes occur, but we are shielded away from
> knowing how it occurs. It is not Turing computable. If theorems of QM and
> those of GR are formally equivalent, then this carries over to
> superdeterminism. This means there is no extra information we can obtain
> from superdeterminism that beats Gödel and Turing.
>
>
>
> I agree.
>
> Bruno
>
>
>
>
> LC
>
>
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