Turing Computations <are> finite string transformations of inputs

On 4/25/2025 5:28 PM, André G. Isaak wrote:

On 2025-04-25 10:31, olcott wrote:
 

Once we understand that Turing computable functions are only
allowed to derived their outputs by applying finite string
operations to their inputs then my claim about the behavior
of DD that HHH must report on is completely proven.

 You're very confused here.
 Computable functions are *functions*. That is, they are mappings from a domain to a codomain, neither of which are required to be strings. Functions don't involve finite string operations at all.
 

All Turing Machine based computation applies the
finite string transformations specified by the TM
language to the input finite string.

What distinguishes a computable function from other functions is that it is possible to implement an algorithm which computes that function. That algorithm might involve string operations, but the algorithm is not the function anymore than the function is the algorithm.
 The halting *function* is a mapping from the set of computations (not strings) to the set of boolean values (also not strings). A given computation maps to true if and only if it is finite.
 

Ultimately every semantic meaning that can be expressed in
language can be encoded as some type of relation between
finite strings.
 From a set of basic facts all knowledge that can be expressed
in language can be derived. It is derived through semantic
logical entailment as the ONLY rule of inference.

A halt *decider* (were such a thing possible) would be an algorithm which computes the halting function. Such an algorithm (assuming we're talking in terms of Turing Machines) would have to encode the elements of the halting function's domain as strings, but the mapping it computes

would still be from the computations which those strings represent to their associated boolean values.
 André
 

It is an easily verified fact that when HHH applies
the finite string transformations specified by the
x86 language to its input DD that for each HHH/DD
pair no emulated DD can possibly reach its final
halt state.
  _DD()
[00002133] 55         push ebp      ; housekeeping
[00002134] 8bec       mov ebp,esp   ; housekeeping
[00002136] 51         push ecx      ; make space for local
[00002137] 6833210000 push 00002133 ; push DD
[0000213c] e882f4ffff call 000015c3 ; call HHH(DD)
[00002141] 83c404     add esp,+04
[00002144] 8945fc     mov [ebp-04],eax
[00002147] 837dfc00   cmp dword [ebp-04],+00
[0000214b] 7402       jz 0000214f
[0000214d] ebfe       jmp 0000214d
[0000214f] 8b45fc     mov eax,[ebp-04]
[00002152] 8be5       mov esp,ebp
[00002154] 5d         pop ebp
[00002155] c3         ret
Size in bytes:(0035) [00002155]
--
Copyright 2025 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer