true randomness? (finite vs infinite sequences)

[Alex Galicki]

It seems that in the current discussion on "true randomness", there is
so much confusion that it would take way too much time for anyone to
clear that up in a reasonable amount of time. So I have just one
relatively important remark and a few less so. The important one is
this:
Algorithmic Randomness is a fairly advanced branch of computability
theory that is actually dedicated to the mathematical study of various
notions of randomness. Here are two great books on the subject:
1) https://www.amazon.com/Computability-Randomness-Oxford-Logic-Guides/dp/0199652600
2) https://www.amazon.com/Algorithmic-Randomness-Complexity-Applications-Computability-ebook/dp/B00DGER3OS/

The less important are:
A) The below cited "definition" seems to imply that
uncomputable=truly/quantum random. Uncomputability, in general, is not
considered as a `sensible' randomness notion.
B) Last time I checked, nobody had any definitive idea whether
quantum-generated sequences can even be uncomputable, let alone
algorithmically random in any meaningful way.
C) One simple but important idea from Algorithmic Randomness is that
randomness is always relative, there is no "true randomness".


On Mon, 21 Mar 2022 at 16:39, José Manuel Rodríguez Caballero
<josephcmac at gmail.com> wrote:
>
>  Sam wrote
>>
>> I second Harvey here: as far as I know, quantum mechanics can provide
>>
>> finite datasets that are better than classical ?state of the art? pseudorandom
>> sources.  *However*, when talking about randomness in computability theory,
>> the definition is concerned with infinite sequences.
>
>
>> Which notion are we dealing with (in theory and practise)?
>
>
> Assuming the Physical Church Turing Thesis (Wolfram's version), each result of a quantum measurement is recorded by an observer, which is assumed to be a machine with memory (Hugh Everett's definition). Assuming an observer with arbitrarily large memory (a finite memory that increases with time), we have a potential Turing machine to record an infinite sequence of quantum measurements. Hence, we can consider whether or not the sequence recorded by the potentially infinite memory of the observer can be generated by a deterministic algorithm.
>
> To be completely rigorous, it should be proved that an observer with potentially infinite memory is possible. Indeed, we could imagine a universe that collapses after some time and no observer is possible anymore. This possibility, considered for the first time by Lord Kelvin (William Thomson)
>
> Thomson, William (1862). "On the Age of the Sun's Heat". Macmillan's Magazine. Vol. 5. pp. 388–393.
>
> is known as the heat death of the universe. Freeman Dyson
>
> Freeman J. Dyson, "Time without end: Physics and biology in an open universe," Reviews of Modern Physics, Vol. 51, Issue 3 (July 1979), pp. 447-460; doi:10.1103/RevModPhys.51.447.
> https://journals.aps.org/rmp/abstract/10.1103/RevModPhys.51.447
>
>  and Frank J. Tipler
>
> Tipler, Frank J (June 1986), "Cosmological Limits on Computation", International Journal of Theoretical Physics, 25 (6): 617–61, Bibcode:1986IJTP...25..617T, doi:10.1007/BF00670475, S2CID 59578961
> https://link.springer.com/content/pdf/10.1007/BF00670475.pdf
>
> proposed methods to construct an eternal observer in an open (but not accelerating) and a closed (contracting universe), respectively. But, the current scientific consensus is that our universe is accelerating. Hence, if it is possible to construct an eternal observer who increases its memory in time, it cannot be done following Freeman Dyson and Frank J. Tipler proposals. Therefore, the answer to your question remains open for our universe, and it is positive for the universes where either Dyson's or Tipler's constructions can be done. This problem belongs to a new approach to physics known as constructor theory, developed by David Deutsch, Chiara Marletto, and collaborators in Oxford Physics:
>
>> Constructor Theory is a new approach to formulating fundamental laws in physics. Instead of describing the world in terms of trajectories, initial conditions and dynamical laws, in constructor theory laws are about which physical transformations are possible and which are impossible, and why. This powerful switch has the potential to bring all sorts of interesting fields, currently regarded as inherently approximative, into fundamental physics. These include the theories of information, knowledge, thermodynamics, and life.
>
>
> https://www.constructortheory.org/
>
> Kind regards,
> Jose M.
>