On 25 Dec 2008, at 08:05, Abram Demski wrote:

> Bruno,
> I agree with Gunther about the two types of machine. The broader
> machine is any system that can be logically described-- a system that
> is governed by rules and has a definite description.

Then Church thesis entails it is not broader, unless you mean that the  
rules are not effective.

> Such machines are
> of course not necessarily computable; oracle machines and so on can be
> logically described (depending of course on the definition of the word
> "logical", since they cannot be described using 1st-order logic with
> its standard semantics).

UDA still works with very big weakening of comp, which I don't mention  
usually for pedagogical purpose. The fact that the first person cannot  
be aware of delays, together with the fact that the UD generates the  
reals extend the comp consequences to machine with all kind of oracles.
The AUDA is even less demanding, and works for highly non effective  
notion of "belief". Instead of using the Gödel provability predicate  
we can use non effective notion like "truth in all model of ZF", or  
even "truth in all transitive models of ZF". In that last case G and  
G* can be effectively extended.

To my knowledge the only scientist being explicitly non mechanist is  
Penrose. Even Searle who pretends to be a non mechanist appears to  
refer to machine, for the brain, which are Turing emulable. Then  
Searles make error in its conception of "mind implementation" and  
"simulation"  like Hofstadter and Dennett have already very well  
criticized. The comp reasoning begins to be in trouble with machines  
using discrete set of actual infinities. Analog machine based on  
notion of interval are mostly Turing emulable. You have to diagonalize  
or use other logical tools in some sophisticate way to build  
analytical machine which are non turing emulable. Nothing in physics  
or in nature points on the existence of those "mathematical  
weirdness", with the notable "collapse of the wave packet" (exploited  
by Penrose, but also by many dualists).

> The narrower type of machine is restricted to be computable.

It is logically narrower. But no weakening of comp based on nature is  
known to escape the replicability. Even the non cloning theorem in QM  
cannot be used to escape the UDA conclusion. You have to introduce  
explicit use of actual infinities. This is very akin to a  
"substantialisation" of soul. I respect that move, but I have to  
criticize unconvincing motivations for it.
Comp entails the existence of uncomputable observable phenomena. It is  
normal to be attracted to the idea that non computability could play a  
role in the mind. But this consists to build a machine based on the  
many sharable computations going through the turing state of the  
machine and this gives "quantum machine", which are turing emulable  
although not in real time, but then they play their role in the  
Universal Deployment.

Of course you can just say "NO" to the doctor. But by invoking a non  
turing emulable "machine", you take the risk of being asked which one.  
Up to now, as far as I know, this exists in mathematics, but there are  
no evidence it exists in nature, except those using the kind of  
indeterminacy which can be explain with the comp hypothesis.

>> All known physical causal system are Turing emulable.
> I am no physicist, but I've been trying to look up stuff on that
> issue... Schmidhuber asserts in multiple places that the fact that
> differential equations are used to describe physics does not
> contradict its computability, but he does not explain.

The SWE is linear. It makes the quantum object directly turing  
emulable (mostly by dovetailing if you are using a sequential  
processor). The solution are linear combination of complex  
exponential. Obviously e, PI and i are computable reals.
It is far more difficult, and perhaps false, to say that Newtonian  
Physics is Turing emulable. Newton himself was aware of action at a  
distance for its gravitational law. But anything so weird has been  
usually considered as an evidence that Newtonian Physics could not be  
taken literaly.
To reintroduce such bizarre feature in nature just to contradict the  
comp hyp is a bit ironical. It is like Bohmian reformulation of  
Quantum Mechanics: to make a theory more complex to avoid  
interpretation judged as unpleasant.

This subject is made difficult because there are no standard notion of  
computability with the real numbers (despite many attempts to find one).
If someone know better ... Non comp theories have to be rather exotic.  
Of course this is not an argument for the truth of comp.

> I know that,
> for example, Wolfram is attempting a computable foundation for
> physics, but I don't know about any real progress... so any info would
> be appreciated.

Wolfram like Schmidhuber believes there could be a computable  
universe. The "whole" could be computable. But in that case the UDA  
shows that the universe is a mathematical one, and indeed can be  
described by any universal dovetailer. But the *physical* universe  
emerging from inside will have some uncomputable feature. This comes  
from the fact that we are necessarily ignorant about which  
computations support us, and that, below our substitution level, we  
have to take into account of a a non enumerable set of computations.  
(More in the UDA).

An interesting book on the computability with the reals containing   
some of the construction mentioned above, is:

POUR-EL M. B., RICHARD J. I., 1989, Computability in Analysis and  
Physics, Springer-Verlag, Berlin.

If I remember well you can find in this book a computable function  
having a non computable derivative!

See also this link, and related links for the analog/digital problem,  
or search on "universal analog machine" or "turing analog machine".



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