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Let SMTM be Turing Machine, but the commands recorded in which can change to others in some random way (for example, choose with a 50/50 probability the command to move to the right or move to the left), and SM-NTM be non-deterministic Turing Machine with same property (note that NTM non-deterministic in sense of choosing next action, but not the rule, which describes this actions). A difference between a self-modifying Turing machine and TM/NTM: instead of one command for any given situation, or a set of commands in a non-deterministic Turing machine, there is a fuzzy set of commands.

In addition to the question from the title: Is there any papers which describes these sort of theoretical models of computation? In particular, I am interested in the applicability of Rice's theorem for a given computational model. Maybe, there is some connection to quantum computers, NP-hardness, etc.?

The problem that prompted me to this question:

There are neural networks that have an element of non-determinism, where at a certain stage the numbers change by adding random numbers (obtained, for example, from weather observations or observation of the decay of unstable atoms, that is, really random numbers) - do such models have restrictions superimposed on a Turing machine?

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Randomness is not the same as non-determinism -- at least, not the non-determinism that is referred to when people talk about non-deterministic Turing machines.

See Can an algorithm be truly non-deterministic?, Differences and relationships between randomized and nondeterministic algorithms?, https://cstheory.stackexchange.com/q/632/5038, Are nondeterministic algorithm and randomized algorithms algorithms on a deterministic Turing machine?.

There is plenty of study of randomized algorithms and the power of randomness. You can read about BPP, the open question whether BPP = P, etc.

There is plenty of study of non-deterministic algorithms and the power of non-determinism, too. For instance, you can read about the open question whether P = NP.

One common hypothesis/guess/speculation right now is that probably randomness doesn't add much power (maybe P = BPP) but non-determinism is very powerful (probably P != NP).

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