On 9/19/2021 2:23 PM, André G. Isaak wrote:
> On 2021-09-19 12:33, olcott wrote:
>> On 9/19/2021 1:24 PM, André G. Isaak wrote:
>>> On 2021-09-19 11:54, olcott wrote:
>>>> On 9/19/2021 12:25 PM, André G. Isaak wrote:
>>>>> On 2021-09-19 11:07, olcott wrote:
>>>>>> On 9/19/2021 11:56 AM, André G. Isaak wrote:
>>>>>>> On 2021-09-19 10:39, olcott wrote:
>>>>>>>> On 9/19/2021 11:11 AM, André G. Isaak wrote:
>>>>>>>>> On 2021-09-19 09:46, olcott wrote:
>>>>>>>>>> On 9/19/2021 10:40 AM, André G. Isaak wrote:
>>>>>>>>>>> On 2021-09-19 08:34, olcott wrote:
>>>>>>>>>>>> On 9/18/2021 11:19 PM, André G. Isaak wrote:
>>>>>>>>>>>>> On 2021-09-18 22:10, André G. Isaak wrote:
>>>>>>>>>>>>>
>>>>>>>>>>>>>> And note that Rice is talking about properties of Turing
>>>>>>>>>>>>>> Machines (or, more properly, of the language accepted by a
>>>>>>>>>>>>>> TM), not computations.
>>>>>>>>>>>>>
>>>>>>>>>>>>> I realized immediately after hitting 'send' that the above
>>>>>>>>>>>>> will no doubt confuse you since people have been telling
>>>>>>>>>>>>> you that Turing Machines can only express computations
>>>>>>>>>>>>> whereas C/x86 aren't thus constrained.
>>>>>>>>>>>>>
>>>>>>>>>>>>> A computation is a Turing Machine description PLUS an input
>>>>>>>>>>>>> string.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Rice's theorem is concerned with the Turing Machine's
>>>>>>>>>>>>> themselves.
>>>>>>>>>>>>>
>>>>>>>>>>>>> The Linz proof shows that you cannot construct a halting
>>>>>>>>>>>>> decider, i.e. a decider which correctly determines for any
>>>>>>>>>>>>> given TM + input string pair, whether that pair represents
>>>>>>>>>>>>> a halting computation.
>>>>>>>>>>>>>
>>>>>>>>>>>>> Rice's theorem, on the other hand, would rule out the
>>>>>>>>>>>>> construction of something like a Decider Decider, i.e. a TM
>>>>>>>>>>>>> which takes as its input a TM description and determines
>>>>>>>>>>>>> whether that TM qualifies as a decider, i.e. is guaranteed
>>>>>>>>>>>>> to halt on *any* possible input.
>>>>>>>>>>>>>
>>>>>>>>>>>>> André
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> Here is where I referred to my code defining a
>>>>>>>>>>>> a decidability decider nine days before you did:
>>>>>>>>>>>
>>>>>>>>>>> Where did I define or even mention a 'decidability decider'?
>>>>>>>>>>> Above I discussed (but did not define) a DECIDER decider,
>>>>>>>>>>> i.e. a TM which determines whether another TM qualifies as a
>>>>>>>>>>> decider.
>>>>>>>>>>>
>>>>>>>>>>>> On 9/9/2021 10:25 AM, olcott wrote:
>>>>>>>>>>>>  > It is the case that H(P,P)==0 is correct
>>>>>>>>>>>>  > It is the case that H1((P,P)==1 is correct
>>>>>>>>>>>>  > It is the case the this is inconsistent.
>>>>>>>>>>>>  > It is the case that this inconsistency
>>>>>>>>>>>>
>>>>>>>>>>>>  > defines a decidability decider that correctly
>>>>>>>>>>>>  > defines a decidability decider that correctly
>>>>>>>>>>>>  > defines a decidability decider that correctly
>>>>>>>>>>>>
>>>>>>>>>>>>  > rejects P on the basis that P has the
>>>>>>>>>>>>  > pathological self-reference(Olcott 2004) error.
>>>>>>>>>>>
>>>>>>>>>>> So what exactly is it that the above is deciding? And how
>>>>>>>>>>> does this relate to Rice? If you want to claim this is
>>>>>>>>>>> somehow related to rice, you need to identify some semantic
>>>>>>>>>>> property that you claim to be able to decide.
>>>>>>>>>>>
>>>>>>>>>>> André
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> It is deciding that either H/P or H1/P form the pathological
>>>>>>>>>> self-reference error(Olcott 2004). As I said in 2004 this is
>>>>>>>>>> the same error as the Liar Paradox. This means that either H/P
>>>>>>>>>> or H1/P are not correctly decidable.
>>>>>>>>>
>>>>>>>>> The post in which I mentioned a 'decider decider' which you
>>>>>>>>> somehow misread as 'decidability decider' was intended to
>>>>>>>>> clarify a single sentence from an earlier post which you
>>>>>>>>> entirely ignored. Why don't you go back and actually read that
>>>>>>>>> earlier post and address the points made therein.
>>>>>>>>>
>>>>>>>>> Your ill-defined notion of a 'pathological self-reference
>>>>>>>>> error' doesn't appear to be a property of a Turing Machine,
>>>>>>>>> which is what Rice's theorem is concerned with. To the extent
>>>>>>>>> that it is a property at all, it appears to be a property of
>>>>>>>>> specific computations, so this has absolutely no relevance to
>>>>>>>>> Rice.
>>>>>>>>>
>>>>>>>>> André
>>>>>>>>>
>>>>>>>>
>>>>>>>> It is the property of H(P,P).
>>>>>>>
>>>>>>> Right, which means this is entirely irrelevant to Rice's theorem.
>>>>>>>
>>>>>>
>>>>>> Many historical posts in comp.theory say otherwise.
>>>>>> If a function can be provided that correctly decides that a
>>>>>> specific TM cannot correctly decide a specific input pair
>>>>>> (according to many historical posts on comp.theory) this does
>>>>>> refute Rice's theorem.
>>>>>
>>>>> So what is the semantic property *of P* that you claim your
>>>>> PSR_Olcott_2004 is capable of identifying?
>>>>>
>>>>> You can't claim that there is anything 'erroneous' about P. There
>>>>> isn't.
>>>>>
>>>>> The fact that you run into a problem when you pass a description of
>>>>> P to some *other* TM is not an indicator that there is something
>>>>> wrong with P. That's like saying that the expression 40 + 50 is
>>>>> 'erroneous' because I can't pass it as an argument to tangent().
>>>>>
>>>>>>>> u32 PSR_Olcott_2004(u32 P)
>>>>>>>> {
>>>>>>>>    return H1(P,P) != H(P,P);
>>>>>>>> }
>>>>>>>>
>>>>>>>> Decides that H(P,P) cannot correctly decide the halt status of
>>>>>>>> its input,
>>>>>>>
>>>>>>>
>>>>>>> No, it does not. It decides that the results of H1(P, P) doesn't
>>>>>>> match the results of H(P, P). It 'decides' that one of these is
>>>>>>> correct and the other is not but it cannot tell you which one,
>>>>>>> which means it can't decide whether H(P, P) can correctly decide
>>>>>>> the halt status of its input.
>>>>>>>
>>>>>>
>>>>>> I simply have not gotten to that part yet.
>>>>>
>>>>> And yet you claimed to be able to 'decide' that H(P, P) cannot
>>>>> correctly decide the halt status of its input. Until you actually
>>>>> 'get to that part' you have no business making such a claim.
>>>>>
>>>>
>>>> Odd man out:
>>>> When H1(P,P) != H(P,P) we test
>>>>    HX(P,P) != H1(P,P)
>>>>    HX(P,P) != H(P,P)
>>>>
>>>> The one that HX(P,P) agrees with is the correct one.
>>>
>>> There are a potentially infinite number of Hns. Your 'decider' only
>>> qualifies as a decider if it works for *every* one of them. Your 'odd
>>> man out strategy' only works if the case you are testing happens to
>>> be among the three which are included in your test function.
>>>
>>
>> The halting theorem is based on an artificially contrived example that
>> would never actually occur in actual practice. In actual practice we
>> would only actually need H.
>
> There is no formal distinction between a Turing Machine and an
> 'artificially contrived' Turing Machine.
>

A set of three Turing machines can defeat the previously undecidable
inputs. When these machines are fully elaborated such that they can
handle every combination of conditional branch instructions then we have
a universal halt decider whenever at least two machines agree.

> A universal halting decider must be able to correctly decide *all
> possible* Turing Machine/Input pairs. That includes both useful and
> useless Turing Machines, contrived machines and machines that no one has
> ever even thought about. If H_Hat is a possible Turing Machine, then it
> exists, has always existed, and will always exist regardless of whether
> anyone has ever even thought about it. If it is part of the set of
> possible TMs, a universal halt decider must be able to deal with it.
>
>> The odd-man-out system does correctly decide an input that was
>> artificially contrived to specifically have the pathological
>> self-reference error (Olcott 2004) for either H or H1. All of the rest
>> of the cases are simply correctly decided by either H or H1.
>
> But this still fails to solve the halting problem.
>
> Using your 'odd man out' approach, you are claiming that neither H, nor
> H1, nor HX are the actual halt decider, but rather that halting is
> decided by some larger program that runs each of these and then compares
> their results. Call that decider UH.
>
> One can, using Linz's method, construct a UH_Hat from UH which cannot be
> decided by UH.
>
>>>>>>>> thus a semantic property of H relative to P.
>>>>>>>
>>>>>>> That's not a property (semantic or otherwise) of *either* P or H.
>>>>>>>
>>>>>>>> The Liar Paradox works this same way.
>>>>>>>
>>>>>>> The liar's paradox is completely irrelevant to Linz.
>>>>>>>
>>>>>>> André
>>>>>>>
>>>>>>
>>>>>> The pathological self-reference error (Olcott 2004) is
>>>>>> specifically relevant to:
>>>>>> (a) The Halting Theorem
>>>>>> (b) The 1931 Incompleteness Theorem
>>>>>> (c) The Tarski undefinability theorem
>>>>>> (d) The Liar Paradox (upon which (c) is based).
>>>>>
>>>>> So why don't you:
>>>>>
>>>>> (a) provide a proper definition of "pathological self-reference error"
>>>>>      That means a definition which will allow one to unambiguosly
>>>>> determine whether an arbitrary expression/entity has this error.
>>>>
>>>> I just provided a reference to function that correctly decide this.
>>>
>>> No. You provided an ad hoc function which decides it for a *single*
>>> case. That's not the same thing as a function which correctly decides
>>> it.
>>>
>>
>> This will come later when my "pure function of its inputs" has been
>> totally validated.
>
> There is absolutely no difference between a 'pure function' and a 'pure
> function of its inputs'. Either your H is a pure function or it isn't.
>
> André
>
>

--
Copyright 2021 Pete Olcott

"Great spirits have always encountered violent opposition from mediocre
minds." Einstein