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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: How to write a self-referencial TM?
Date: Mon, 19 May 2025 13:41:28 +0300
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On 2025-05-18 20:35:01 +0000, wij said:

> On Sat, 2025-05-17 at 14:39 -0500, olcott wrote:
>> On 5/17/2025 2:26 PM, wij wrote:
>>> On Sat, 2025-05-17 at 15:45 +0100, Mike Terry wrote:
>>>> On 17/05/2025 04:01, wij wrote:
>>>>> On Fri, 2025-05-16 at 23:51 +0100, Mike Terry wrote:
>>>>>> On 16/05/2025 20:35, wij wrote:
>>>>>>> On Fri, 2025-05-16 at 16:33 +0100, Mike Terry wrote:
>>>>>>>> On 16/05/2025 12:40, wij wrote:
>>>>>>>>> On Fri, 2025-05-16 at 03:26 +0100, Mike Terry wrote:
>>>>>>>>>> On 16/05/2025 02:47, wij wrote:
>>>>>>>>>>> On Fri, 2025-05-16 at 01:40 +0100, Mike Terry wrote:
>>>>>>>>>>>> On 15/05/2025 19:49, wij wrote:
>>>>>>>>>>>>> On Thu, 2025-05-15 at 17:08 +0100, Mike Terry wrote:
>>>>>>>>>>>>>> On 14/05/2025 18:53, wij wrote:
>>>>>>>>>>>>>>> On Wed, 2025-05-14 at 12:24 -0500, olcott wrote:
>>>>>>>>>>>>>>>> On 5/14/2025 11:43 AM, wij wrote:
>>>>>>>>>>>>>>>>> On Wed, 2025-05-14 at 09:51 -0500, olcott wrote:
>>>>>>>>>>>>>>>>>> On 5/14/2025 12:13 AM, wij wrote:
>>>>>>>>>>>>>>>>>>> Q: Write a turing machine that performs D function (which calls
>>>>>>>>>>>>>>>>>>> itself):
>>>>>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>>>>>> void D() {
>>>>>>>>>>>>>>>>>>>           D();
>>>>>>>>>>>>>>>>>>> }
>>>>>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>>>>>> Easy?
>>>>>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>>>>> That is not a TM.
>>>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>>>> It is a C program that exists. Therefore, there must be a equivalent
>>>>>>>>>>>>>>>>> TM.
>>>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>>>>> To make a TM that references itself the closest
>>>>>>>>>>>>>>>>>> thing is a UTM that simulates its own TM source-code.
>>>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>>>> How does a UTM simulate its own TM source-code?
>>>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>>> You run a UTM that has its own source-code on its tape.
>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>>> What is exactly the source-code on its tape?
>>>>>>>>>>>>>>> 
>>>>>>>>>>>>>> 
>>>>>>>>>>>>>> Every UTM has some scheme which can be applied to a (TM & input tape) that
>>>>>>>>>>>>>> is to
>>>>>>>>>>>>>> be
>>>>>>>>>>>>>> simulated.
>>>>>>>>>>>>>> The
>>>>>>>>>>>>>> scheme says how to turn the (TM + input tape) into a string of symbols that
>>>>>>>>>>>>>> represent
>>>>>>>>>>>>>> that
>>>>>>>>>>>>>> computation.
>>>>>>>>>>>>>> 
>>>>>>>>>>>>>> So to answer your question, the "source-code on its tape" is the result of
>>>>>>>>>>>>>> applying
>>>>>>>>>>>>>> the
>>>>>>>>>>>>>> UTM's
>>>>>>>>>>>>>> particular scheme to the combination (UTM, input tape) that is to be
>>>>>>>>>>>>>> simulated.
>>>>>>>>>>>>>> 
>>>>>>>>>>>>>> If you're looking for the exact string symbols, obviously you would need to
>>>>>>>>>>>>>> specify
>>>>>>>>>>>>>> the
>>>>>>>>>>>>>> exact
>>>>>>>>>>>>>> UTM
>>>>>>>>>>>>>> being used, because every UTM will have a different answer to your question.
>>>>>>>>>>>>>> 
>>>>>>>>>>>>>> 
>>>>>>>>>>>>>> Mike.
>>>>>>>>>>>>> 
>>>>>>>>>>>>> People used to say UTM can simulate all TM. I was questing such a UTM.
>>>>>>>>>>>>> Because you said "Every UTM ...", so what is the source of such UTM?
>>>>>>>>>>>> 
>>>>>>>>>>>> Yes, a UTM can simulate any TM including itself.  (Nothing magical changes when
>>>>>>>>>>>> a
>>>>>>>>>>>> UTM
>>>>>>>>>>>> simulates
>>>>>>>>>>>> itself, as opposed to some other TM.)
>>>>>>>>>>> 
>>>>>>>>>>> Supposed UTM exists, and denoted as U(X), X denotes the tape contents of the
>>>>>>>>>>> encoding of a TM. And, U(X) should function the same like X.
>>>>>>>>>>> Given instance U(U(f)), it should function like f from the above definition.
>>>>>>>>>>> But, U(U(f)) would fall into a 'self-reference' trap.
>>>>>>>>>> 
>>>>>>>>>> There is no self-reference trap.
>>>>>>>>>> 
>>>>>>>>>> In your notation:
>>>>>>>>>> 
>>>>>>>>>> -  f represents some computation.
>>>>>>>>>> -  U(f) represents U being run with f on its tape.
>>>>>>>>>>         Note this is itself a computation, distinct from f of course
>>>>>>>>>>         but having the same behaviour.
>>>>>>>>>> -  U(U(f)) represents U simulating the previous computation.
>>>>>>>>>> 
>>>>>>>>>> There is no reason U(f) cannot be simulated by U.  U will have no 
>>>>>>>>>> knowledge that it
>>>>>>>>>> is
>>>>>>>>>> "simulating
>>>>>>>>>> itself", and will just simulate what it is given.
>>>>>>>>>> 
>>>>>>>>>> 
>>>>>>>>>> Mike.
>>>>>>>>> 
>>>>>>>>> Sorry for not being clear on the UTM issue (I wanted to mean several 
>>>>>>>>> things in one
>>>>>>>>> post).
>>>>>>>>> You are right there is no self-reference.
>>>>>>>>> I mean 'UTM' is not a complete, qualified TM because the contents of the tape
>>>>>>>>> would not be defined. Saying "UTM can simulate any TM" is misleading because
>>>>>>>>> no such TM (UTM as TM) exists.
>>>>>>>> 
>>>>>>>> What do you mean "the contents of the tape would not be defined"?  A TM 
>>>>>>>> is /equipped/
>>>>>>>> with
>>>>>>>> an
>>>>>>>> infinite tape, but the /contents/ of that tape are not a part of that 
>>>>>>>> TM's definition.
>>>>>>>> 
>>>>>>>> For example we could build a TM P that decides whether a number is 
>>>>>>>> prime.  Given a
>>>>>>>> number n,
>>>>>>>> we
>>>>>>>> convert n into the input tape representation of n, and run P with that 
>>>>>>>> tape as input.
>>>>>>>> 
>>>>>>>> It's essentially no different for UTMs.  Such a UTM certainly is a 
>>>>>>>> "complete TM",
>>>>>>>> equipped
>>>>>>>> with
>>>>>>>> its
>>>>>>>> own input tape.  Of course we don't know what's on the input tape 
>>>>>>>> because nobody has
>>>>>>>> said
>>>>>>>> yet
>>>>>>>> what
>>>>>>>> computation we are asking it to simulate!  [Similarly we don't know 
>>>>>>>> what's on P's input
>>>>>>>> tape,
>>>>>>>> until
>>>>>>>> we know what n we want it to test for primeness.]  Once you say what 
>>>>>>>> computation you
>>>>>>>> want
>>>>>>>> the
>>>>>>>> UTM to
>>>>>>>> simulate we can build a tape string to perform that particular 
>>>>>>>> simulation.  That is the
>>>>>>>> case
>>>>>>>> /whatever/ computation we come up with, so it is simply the case [not 
>>>>>>>> misleading] that
>>>>>>>> the
>>>>>>>> UTM
>>>>>>>> can
>>>>>>>> simulate any computation.
>>>>>>>> 
>>>>>>>> 
>>>>>>>> Mike.
>>>>>>> 
>>>>>>> TM has no I/O mechanism. 'Computation' always means the contents of the tape
>>>>>>> is defined (fixed before run).
>>>>>>> 
>>>>>> 
>>>>>> Correct, and correct.
>>>>>> 
>>>>>> So... What do you mean "the contents of the tape would not be defined"?
>>>>>> 
>>>>>> 
>>>>>> Mike.
>>>>> 
>>>>> In "UTM simulates itself", denoted as U(U(f)), the f would not be defined.
>>>> 
>>>> Eh?  The f was something /you/ introduced!  You said it represents some 
>>>> computation which UTM U
>>>> simulates.  How can f suddenly become undefined after you defined it?
>>>> 
>>>> Do you mean that f would not be on the input tape for (outer)U?  That's 
>>>> not the case at all.  In
>>>> U(f), the input tape for U contains a representation of f.  When 
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