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From: gerryg@il.us.swissbank.com (Gerald Gleason)
Subject: Re: Algorithmic Information: Computable?
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Date: Mon, 12 Dec 1994 18:53:22 GMT
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Hal writes
> [ . . . ]
> I guess the crux of my confusion is how the halting problem can be
> solvable on any finite domain without this rendering it solvable in
> practice for all programs.  Thanks -
> 
> Hal Finney
> hfinney@shell.portal.com

I think there is some confusion in this thread about what is meant by a  
'finite domain'.  It is not sufficient to limit the "input strings" to any  
particular finite length, you have to limit the possible state space of  
the running turing machine.  If you can show that the machine never uses  
an infinite amount of tape, then your Turing machine reduces to an finite  
state machine, and therefore, no halting problem.

It seems to me that any consideration of algorithmic information content  
must take into account both time and space, and so with a suitable  
definition, there should be no problem making this kind of measure  
computable.

The problem is that any such definition cannot help but be artificial.   
You might be able to rank information content relative to a specific  
definition, but other equally reasonable measures will produce completely  
different rankings.  Turing designed his machines to help with proofs  
about computability, not complexity of algorithms.  As such, it has  
features that are necessary and sufficient to compute all computable  
functions, stripped of anything that might complicate proofs.  There is no  
consideration of things like expressive power, or compact representations  
of programs.  In short, Turing machines are the wrong tools to investigate  
this property, and it is unclear what would count as a good tool for such  
a task much less the best tool or how we would know we had such a tool if  
it was right in front of us.

Gerry Gleason
