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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: comp.theory
Subject: Re: How to write a self-referencial TM?
Date: Sun, 18 May 2025 11:20:27 +0300
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References: <1e4f1a15826e67e7faf7a3c2104d09e9dadc6f06.camel@gmail.com> <1002akp$2i4bk$2@dont-email.me> <479eebef3bd93e82c8fe363908b254b11d15a799.camel@gmail.com> <1002jkk$2k00a$3@dont-email.me> <05e306f20fcb7c88c497e353aaecd36b30fc752a.camel@gmail.com> <10053hb$3759k$1@dont-email.me> <879b3c552bad9da9885e41a298b570c92bef1aaf.camel@gmail.com> <10061h6$3de5f$1@dont-email.me> <4bce5af2b2b8cd198af611e5d8d56598cab15b0a.camel@gmail.com> <10067ok$3ib39$1@dont-email.me> <e63d3083ddf6b9ab172cc24c07155410d81ce5b4.camel@gmail.com> <1007lrp$3r388$1@dont-email.me> <0cbe88d46c63af596e4d2ad6a846e61b7efb14bb.camel@gmail.com> <1008fhf$53u$1@dont-email.me> <cd31647abcc33f0978415df34ec2c8d41d886591.camel@gmail.com> <100a7e4$efgi$1@dont-email.me> <f94f006b40c3ca204d41be9b4507280a3a4fc17b.camel@gmail.com> <100aolc$hq2u$1@dont-email.me>
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On 2025-05-17 19:39:24 +0000, olcott said:

> 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 
>>> (outer)U simulates (inner)U
>>> simulating f, (outer)U's tape contains a representation of computation 
>>> U(f), which internally
>>> contains the original representation of f.  The f is still there and 
>>> equally well defined in
>>> U(U(f)).
>>> 
>>> I think you would benefit from being more explicit and generally more 
>>> careful in your notation!
>>> 
>>> Using notation <P,I> to mean U's input tape representation of "TM P, 
>>> running with input I":
>>> 
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