On 2025-03-24 16:43, olcott wrote:
Computable functions don't have inputs. They have domains. Turing
machines have inputs.p
Maybe when pure math objects. In every model of
computation they seem to always have inputs.
https://en.wikipedia.org/wiki/Computable_function
Computable functions *are* pure math objects. You seem to want to
conflate them with C functions, but that is not the case.
The crucial point is that the domains of computable functions are *not* restricted to strings, even if the inputs to Turing Machines are.
While the inputs to TMs are restricted to strings, there is no such
such restriction on computable functions.
The vast majority of computable functions of interest do *not* have
strings as their domains, yet they remain computable functions (a
simple example would be the parity function which maps NATURAL
NUMBERS (not strings) to yes/no values.)
Since there is a bijection between natural numbers
and strings of decimal digits your qualification
seems vacuous.
There is not a bijection between natural numbers and strings. There is a one-to-many mapping from natural numbers to strings, just as there is a one-to-many mapping from computations (i.e. turing machine/input string pairs, i.e. actual Turing machines directly running on their inputs) to strings.
André
On 3/24/2025 5:49 PM, André G. Isaak wrote:
On 2025-03-24 16:43, olcott wrote:
Computable functions don't have inputs. They have domains. Turing
machines have inputs.p
Maybe when pure math objects. In every model of
computation they seem to always have inputs.
https://en.wikipedia.org/wiki/Computable_function
Computable functions *are* pure math objects. You seem to want to
conflate them with C functions, but that is not the case.
The crucial point is that the domains of computable functions are
*not* restricted to strings, even if the inputs to Turing Machines are.
While the inputs to TMs are restricted to strings, there is no such
such restriction on computable functions.
The vast majority of computable functions of interest do *not* have
strings as their domains, yet they remain computable functions (a
simple example would be the parity function which maps NATURAL
NUMBERS (not strings) to yes/no values.)
Since there is a bijection between natural numbers
and strings of decimal digits your qualification
seems vacuous.
There is not a bijection between natural numbers and strings. There is
a one-to-many mapping from natural numbers to strings, just as there
is a one-to-many mapping from computations (i.e. turing machine/input
string pairs, i.e. actual Turing machines directly running on their
inputs) to strings.
André
_III()
[00002172] 55 push ebp ; housekeeping
[00002173] 8bec mov ebp,esp ; housekeeping
[00002175] 6872210000 push 00002172 ; push III
[0000217a] e853f4ffff call 000015d2 ; call EEE(III)
[0000217f] 83c404 add esp,+04
[00002182] 5d pop ebp
[00002183] c3 ret
Size in bytes:(0018) [00002183]
When III is emulated by pure emulator EEE for any finite
number of steps of emulation according to the semantics
of the x86 language it never reaches its own "ret"
instruction final halt state
THUS DOES NOT HALT.
Op 25.mrt.2025 om 00:04 schreef olcott:
On 3/24/2025 5:49 PM, André G. Isaak wrote:
On 2025-03-24 16:43, olcott wrote:
Computable functions don't have inputs. They have domains. Turing
machines have inputs.p
Maybe when pure math objects. In every model of
computation they seem to always have inputs.
https://en.wikipedia.org/wiki/Computable_function
Computable functions *are* pure math objects. You seem to want to
conflate them with C functions, but that is not the case.
The crucial point is that the domains of computable functions are
*not* restricted to strings, even if the inputs to Turing Machines are.
While the inputs to TMs are restricted to strings, there is no such >>>>> such restriction on computable functions.
The vast majority of computable functions of interest do *not* have >>>>> strings as their domains, yet they remain computable functions (a
simple example would be the parity function which maps NATURAL
NUMBERS (not strings) to yes/no values.)
Since there is a bijection between natural numbers
and strings of decimal digits your qualification
seems vacuous.
There is not a bijection between natural numbers and strings. There
is a one-to-many mapping from natural numbers to strings, just as
there is a one-to-many mapping from computations (i.e. turing
machine/input string pairs, i.e. actual Turing machines directly
running on their inputs) to strings.
André
_III()
[00002172] 55 push ebp ; housekeeping
[00002173] 8bec mov ebp,esp ; housekeeping
[00002175] 6872210000 push 00002172 ; push III
[0000217a] e853f4ffff call 000015d2 ; call EEE(III)
[0000217f] 83c404 add esp,+04
[00002182] 5d pop ebp
[00002183] c3 ret
Size in bytes:(0018) [00002183]
When III is emulated by pure emulator EEE for any finite
number of steps of emulation according to the semantics
of the x86 language it never reaches its own "ret"
instruction final halt state
If fails to complete the simulation of a program that halts
On 3/25/2025 4:19 AM, Fred. Zwarts wrote:
Op 25.mrt.2025 om 00:04 schreef olcott:
On 3/24/2025 5:49 PM, André G. Isaak wrote:
On 2025-03-24 16:43, olcott wrote:
Computable functions don't have inputs. They have domains. Turing >>>>>> machines have inputs.p
Maybe when pure math objects. In every model of
computation they seem to always have inputs.
https://en.wikipedia.org/wiki/Computable_function
Computable functions *are* pure math objects. You seem to want to
conflate them with C functions, but that is not the case.
The crucial point is that the domains of computable functions are
*not* restricted to strings, even if the inputs to Turing Machines are. >>>>
While the inputs to TMs are restricted to strings, there is no
such such restriction on computable functions.
The vast majority of computable functions of interest do *not*
have strings as their domains, yet they remain computable
functions (a simple example would be the parity function which
maps NATURAL NUMBERS (not strings) to yes/no values.)
Since there is a bijection between natural numbers
and strings of decimal digits your qualification
seems vacuous.
There is not a bijection between natural numbers and strings. There
is a one-to-many mapping from natural numbers to strings, just as
there is a one-to-many mapping from computations (i.e. turing
machine/input string pairs, i.e. actual Turing machines directly
running on their inputs) to strings.
André
_III()
[00002172] 55 push ebp ; housekeeping
[00002173] 8bec mov ebp,esp ; housekeeping
[00002175] 6872210000 push 00002172 ; push III
[0000217a] e853f4ffff call 000015d2 ; call EEE(III)
[0000217f] 83c404 add esp,+04
[00002182] 5d pop ebp
[00002183] c3 ret
Size in bytes:(0018) [00002183]
When III is emulated by pure emulator EEE for any finite
number of steps of emulation according to the semantics
of the x86 language it never reaches its own "ret"
instruction final halt state
If fails to complete the simulation of a program that halts
It is stupid to ignore the pathological relationship
that III defines with EEE.
| Sysop: | DaiTengu |
|---|---|
| Location: | Appleton, WI |
| Users: | 1,029 |
| Nodes: | 10 (1 / 9) |
| Uptime: | 182:28:16 |
| Calls: | 13,337 |
| Calls today: | 4 |
| Files: | 186,574 |
| D/L today: |
5,519 files (1,528M bytes) |
| Messages: | 3,356,613 |