On 7/14/2024 5:39 AM, John Clark wrote:
On Sat, Jul 13, 2024 at 10:34 PM Brent Meeker <[email protected]>
wrote:
/> The machine is universal. You don't need a different machine
with different internal states./
*
*
*First of all, the very definition of "a different Turing Machine" is
a machine with a differentset of internal states*. And there is not
just one Turing machine, there are an /infinite/number of them. *There
are 64 one state two symbol(zero and one) Turing Machines,20,736 two
state, **16,777,216three state, **25,600,000,000four state, and
**634,033,809,653,824five state two symbol T**uring Machines. **
*
A Turing Machine with different sets of* internal states**will
exhibit*/**different behavior even if given identical input states. /I
think What confuses you is that it is possible to have a machine in
which the tape not only provides the program the machine should work
on but also the set of internal states that the machine has. In a way
you could think of the tape as providing not only the program but also
the wiring diagram of the computer. A universal Turing Machine is in
an undefined state until the input tape, /or something else/, puts it
in one specific state.
*
*
*Consider the Busy beaver function, if you feed in a tape with all
zeros on it into all 4 state Turing Machines and ask "which of those
25,600,000,000 machines will print the most ones before stopping"
(it's important that the machine eventually stops), you will find this
is not an easy question. All the machines are operating on identical
input tapes (all zeros) but they behave differently, some stop almost
immediately, others just keep printing 1 forever, but for others the
behavior is vastly more complicated. It turns out that the winner is a
set of states that prints out 13 ones after making 107 moves.
*
*You're one confused, John. Every Turing machine can compute whatever
is computable...which means stopping with the answer. That's the
definition of "Turing machine". That Turing machines can be constructed
with different numbers of internal states and using different operations
is old news.
Brent
*
*
*
*A five state Turing Machine behaves differently, we just found out
that a particular set of internal states prints 2098 ones after making
47,176,870 moves. I wouldn't be surprised if the sixth Busy Beaver
number is not computable, we know for a fact that any busy beaver
number for a 745 State Turing machine or larger is not computable.
Right now all we know about BB(6) is that it's larger than
10^10^10^10^10^10^10^10^10^10^10^10^10^10^10.*
*
*
*The point of all this is thatTuring Machines with different sets of
internal states behave very differently. *
*
*
John K Clark See what's on my new list at Extropolis
<https://groups.google.com/g/extropolis>*
*
mth
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