Am 04.01.2011 23:48, schrieb Darren New:
> clipka wrote:
>> And that's no problem, because CA are allowed to have that /by
>> definition/.
>
> That right there is what I'm asking about. I've never seen a definition
> that allows for a CA to have an infinite initialization other than to a
> default state. Do you have a citation to a definition that describes
> this? I am having trouble coming up with a trivial way to define what
> would be the allowable initialization and what would be the disallowed
> initialization without making reference to something outside the realm
> of CAs. Perhaps if you can describe its state based on a RE function of
> its index or something?

"An initial state (time t=0) is selected by assigning a state for each 
cell."

(Wikipedia)

Isn't that plain enough? You assign a state for /each/ cell. CA's don't 
seem to make any limitations on this.

Just like you "create" a TM [tape] to be initially empty except for a 
limited section, you "create" a CA with an explicit initial state for 
/each/ cell. How you pick that state per cell is purely a CA design 
decision, and it doesn't matter whether you fetch that state out of thin 
air already providing (part of) your solution. Just name /any/ rule for 
that pattern - I guess the only limitation is that if you define the 
initial state at time t=0 as a function of any states at time t>0 you 
have a CA with oracle - and in fact may find yourself running straight 
into a paradoxon.

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On 22/12/2010 11:07 PM, Darren New wrote:

> Initializing an infinite tape for a
> turing machine at least makes it more powerful than one with an
> uninitialized tape.

OK.

> And rule 110 needs an infinitely initialized tape.

I'm still not convinced that this is actually necessary.

In other words, I suspect that rule 110 *is* Turing complete, it's just 
that the proof provided is a bit weak.

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clipka wrote:
> "An initial state (time t=0) is selected by assigning a state for each 
> cell."
> 
> (Wikipedia)
> 
> Isn't that plain enough? You assign a state for /each/ cell. CA's don't 
> seem to make any limitations on this.

Well, from that same page:

"""
It is usually assumed that every cell in the universe starts in the same 
state, except for a finite number of cells in other states, often called a 
configuration. More generally, it is sometimes assumed that the universe 
starts out covered with a periodic pattern, and only a finite number of 
cells violate that pattern. The latter assumption is common in 
one-dimensional cellular automata.
"""

So you "generally" don't start with an infinite aperiodic pattern.

What I was really interested in was a proof that starting with a periodic 
pattern (except a finite number of cells) didn't give more computing power 
to the CA than starting with a single state (except a finite number of 
cells). Because Wolfram is arguing about the computational power of Rule 110.

> Just name /any/ rule for that pattern 

I don't think that works, because then it lets the CA calculate stuff a TM 
cannot, and I think that would lead to a trivial disproof of the 
Church-Turing thesis.

Now, a pattern that differs from a regular repeating pattern in only a 
finite number of places? Sure, that's likely good.

But if Wikipedia says a repeating pattern for linear CAs is common, I'll 
assume that's sufficient to quell my disquiet with the Rule 110 proof in 
that regard. I'd still like to see a proof that having the repeating pattern 
doesn't add computational power, which is what the Crawley paper was 
apparently discussing. I'll have to take time to look at that closer.

-- 
Darren New, San Diego CA, USA (PST)
   Serving Suggestion:
     "Don't serve this any more. It's awful."

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Invisible wrote:
> In other words, I suspect that rule 110 *is* Turing complete, it's just 
> that the proof provided is a bit weak.

I suspect you can build a linear CA that's turing complete. I'm not 
convinced that 110 without the infinite initialization can do that.

-- 
Darren New, San Diego CA, USA (PST)
   Serving Suggestion:
     "Don't serve this any more. It's awful."

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