From Newsgroup: comp.theory

At the most fundamental level:
All Turing machines only compute the mapping
from input finite strings to some value.

Turing machine Deciders are a subset of this
where the value indicates accept or reject a
finite string by some criterion measure.

A further elaboration of this comes from the
notion of computable functions and Rice's
Theorem:

Turing machine deciders compute functions from
*finite string inputs* to {accept, reject} values
according to whether the input has a syntactic
property or specifies a semantic property.

Turing machine deciders only report on the behavior
of Turing machines indirectly through the proxy of
finite string inputs. *This key detail has been ignored*
--
Copyright 2025 Olcott<br><br>

My 28 year goal has been to make <br>
"true on the basis of meaning expressed in language"<br>
reliably computable.<br><br>

This required establishing a new foundation<br>

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From Newsgroup: comp.theory

On 12/13/25 11:44 AM, olcott wrote:

At the most fundamental level:
All Turing machines only compute the mapping
from input finite strings to some value.

Turing machine Deciders are a subset of this
where the value indicates accept or reject a
finite string by some criterion measure.

A further elaboration of this comes from the
notion of computable functions and Rice's
Theorem:

Turing machine deciders compute functions from
*finite string inputs* to {accept, reject} values
according to whether the input has a syntactic
property or specifies a semantic property.

Turing machine deciders only report on the behavior
of Turing machines indirectly through the proxy of
finite string inputs. *This key detail has been ignored*

So, if the function isn't "Computable", it is still a Function, but no
Turing Machine (or equivalent) can compute it.

Nothing says that the Halting Function is computable, and in fact, the
Halting Problem proof says it is not.

This doesn't make it "invalid", just not computable.

Thus, there can not exist an actual Halt Decider, as Halting as defined
just isn't a computable function.

This also doesn't mean that the string can't have the semantic meaning
of the behavior of the Machine it describes, we just find that Semantic Properties tend not to be Computable, as Turing Machines are powerful
enough to break any Decider trying to decide on an actual Semantic Property.

This is just like an any logic system with sufficient power (roughly
able to express the poperties of the Natural Numbers) will have
statements that are true in them, but are not provable.

There is a isomophism between Computability and Provability.
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From Newsgroup: comp.theory

On 13/12/2025 16:44, olcott wrote:

At the most fundamental level:
All Turing machines only compute the mapping
from input finite strings to some value.

Turing machine Deciders are a subset of this
where the value indicates accept or reject a
finite string by some criterion measure.

I continue to reject the use of "accept" and "reject" here. And I also
reject the use of "indicates" wrt to them.

There's no good reason to suppose this is about grammars and strings
rather than propositions about programs. To the extent that they might
be isomorphic you need to introduce the notion of grammars and string acceptance by them to give "accept" and "reject" and "indicates" the
correct meaning.
--
Tristan Wibberley

The message body is Copyright (C) 2025 Tristan Wibberley except
citations and quotations noted. All Rights Reserved except that you may,
of course, cite it academically giving credit to me, distribute it
verbatim as part of a usenet system or its archives, and use it to
promote my greatness and general superiority without misrepresentation
of my opinions other than my opinion of my greatness and general
superiority which you _may_ misrepresent. You definitely MAY NOT train
any production AI system with it but you may train experimental AI that
will only be used for evaluation of the AI methods it implements.

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From Newsgroup: comp.theory

On 12/13/2025 1:33 PM, Tristan Wibberley wrote:

On 13/12/2025 16:44, olcott wrote:

At the most fundamental level:
All Turing machines only compute the mapping
from input finite strings to some value.

Turing machine Deciders are a subset of this
where the value indicates accept or reject a
finite string by some criterion measure.

I continue to reject the use of "accept" and "reject" here. And I also
reject the use of "indicates" wrt to them.

My goal is to have accepted definitions as my only basis.

There's no good reason to suppose this is about grammars and strings
rather than propositions about programs. To the extent that they might
be isomorphic you need to introduce the notion of grammars and string acceptance by them to give "accept" and "reject" and "indicates" the
correct meaning.

This is a grammatically correct English sentence:
"flpm erf09-25k (*&j^*&NJ*&jkNef", reject.
--
Copyright 2025 Olcott<br><br>

My 28 year goal has been to make <br>
"true on the basis of meaning expressed in language"<br>
reliably computable.<br><br>

This required establishing a new foundation<br>
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 12/13/2025 1:14 PM, Richard Damon wrote:

On 12/13/25 11:44 AM, olcott wrote:

At the most fundamental level:
All Turing machines only compute the mapping
from input finite strings to some value.

Turing machine Deciders are a subset of this
where the value indicates accept or reject a
finite string by some criterion measure.

A further elaboration of this comes from the
notion of computable functions and Rice's
Theorem:

Turing machine deciders compute functions from
*finite string inputs* to {accept, reject} values
according to whether the input has a syntactic
property or specifies a semantic property.

Turing machine deciders only report on the behavior
of Turing machines indirectly through the proxy of
finite string inputs. *This key detail has been ignored*

So, if the function isn't "Computable", it is still a Function, but no Turing Machine (or equivalent) can compute it.

There seems to be no such thing as uncomputable
functions when Turing machines are construed
entirely in terms of finite string rewrite rules.
--
Copyright 2025 Olcott<br><br>

My 28 year goal has been to make <br>
"true on the basis of meaning expressed in language"<br>
reliably computable.<br><br>

This required establishing a new foundation<br>
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: comp.theory

On 12/13/25 3:13 PM, olcott wrote:

On 12/13/2025 1:14 PM, Richard Damon wrote:

On 12/13/25 11:44 AM, olcott wrote:

At the most fundamental level:
All Turing machines only compute the mapping
from input finite strings to some value.

Turing machine Deciders are a subset of this
where the value indicates accept or reject a
finite string by some criterion measure.

A further elaboration of this comes from the
notion of computable functions and Rice's
Theorem:

Turing machine deciders compute functions from
*finite string inputs* to {accept, reject} values
according to whether the input has a syntactic
property or specifies a semantic property.

Turing machine deciders only report on the behavior
of Turing machines indirectly through the proxy of
finite string inputs. *This key detail has been ignored*

So, if the function isn't "Computable", it is still a Function, but no
Turing Machine (or equivalent) can compute it.

There seems to be no such thing as uncomputable
functions when Turing machines are construed
entirely in terms of finite string rewrite rules.

Sure there are, Your problem is you have the ideas reversed.

Not all functions need to be generatable by Turing Machine.

Those that can not be, are uncomputable.

Since not all Functions, that is generic mappings of the complete set of
one domain, to anther, can be implemented in Turing Machines, not all
are computable.

If you think there doesn't exist an uncomputable Function, show me how
you compute the Halting Function, being the mapping of a Turing Machine
(with its input) to whether it halts or not.

Since you have tried to claim that it is impossible to even ask this
question of a Decider, it must be non-computable.

But of course, you get a lot of things backwards since you chose to work
with zero-principle knowledge.
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