Re: Symbol Grounding Problem (attn: Vend)

forbisgaryg@xxxxxxx wrote:
On May 7, 9:28 pm, HMSBeagle <jsb...@xxxxxxxxxxxxx> wrote:
On Sat, 05 May 2007 00:12:56 -0700, Stephen Harris
I can now state the physical version of the Church-Turing
principle: "Every finitely realizable physical system can be perfectly
simulated by a universal model computing machine operating by finite
means." This formulation is both better defined and more physical than
Turing's own way of expressing it. (Deutsch 1985: 99.)

Okay then Computationalism is on the high ground scientifically. We
have evidence that physical processes can be modelled on computers.
If this is what "Computationalism" meant all along I simply do not see
how anyone can reasonably argue an alternative. They would inevitably
be arguing for a "soul" or otherwise some sort of homonculous or
"oracle" in the brain. There is simply no evidence for any of these
things at this time. Because of the preponderance of evidence, I
will have to follow the doctrine of Computationalism.

Stephen can correct me if I have this wrong.


Actually, it seems right to me! :-) I will quote Piccinini and
you can compare your ideas to his. I will also attempt to defend
his view if you disagree with it, even if I mount a feeble defense.

Are analog devices finitely realizable? I believe they can only be
approximated to some reducingly small but always present error.

I think so because some real numbers are infinitely long, so they are
truncated which creates roundoff errors. I'm going to look it up in Piccinini since I was just reading about this yesterday and know where to find it. Also I think he writes very well about approximations which I will just quote since you ask how good does an imperfect simulation have to be, which relates to approximations. Piccinini_ETD_2003.pdf :

"7.3.1 Modest Physical CT
A first version of Physical CT pertains to what functions can be computed, in a generalized sense that need not rely on an effective procedure, by a mechanism or machine. In order to formulate it, we need to say what we mean by “computation in a generalized sense” and by “mechanism.”
A mechanism in the present sense is something that has proper functions, such as the cooling function of a refrigerator or the pumping function of the heart. A computing mechanism in the present sense is a mechanism whose proper function is to obtain certain output strings of tokens (or symbols) from certain input strings of tokens (and perhaps internal states), according to a general rule that applies to all inputs and outputs.18 Turing Machines, digital computers, and certain kinds of neural networks are computing mechanisms in this sense.

Strictly speaking, ordinary analog computers are not computing mechanisms in this sense, because their inputs and outputs are real-valued quantities and not finite strings. In the general case, real-valued quantities cannot even be represented as finite strings. But using an analog computer to perform computations requires preparing the input and measuring the output. Any input and output of an analog computer can only be prepared or measured with finite precision, and values measured with finite precision can be (and normally are) represented by strings of symbols. Hence, by taking only values measured with finite precision as the inputs and outputs of analog computers, which needs to be done in order to use analog computers to perform computations, we can subsume analog computers under the present notion of computing mechanism."

SH: To me, that agrees with you.

One doesn't have to rely upon some "soul" or "homonculous" or
"oracle". Zeno's paradox strikes. The out is that not all physical
systems are finitely realizable and therefore cannot be perfectly
simulated by a UTM (assuming that all "universal model computing
machine operating by finite means" are UTMs and I'm not sure that's
so.)

Piccinini_ETD_2003.pdf

"7.3.3 Bold Physical CT
A second version of Physical CT pertains to the computational properties of all physical systems, whether or not they are computing mechanisms. As a first approximation, it can be formulated as follows:
(Bold Physical CT) Any function whose values are generated by a physical system is Turing-computable.

Evaluating Bold Physical CT requires that we make it more precise, by saying how the functions whose values are generated by physical systems are to be identified. Once one moves away from mechanisms operating on strings into the realm of physical processes in general, it is quite difficult to formulate Bold Physical CT in a clear and meaningful way.
Most physical systems are usually mathematically represented by systems of differential equations, which give rise to a state space and a set of trajectories through that space. For any initial condition, the equations pick out the state space trajectory that the system goes through. TMs are dynamical systems in the same sense. They have a fixed starting state, but their different inputs may be used to define different initial conditions, and they have a dynamical evolution that can be represented as a trajectory through a state space.
Given this, a natural explication of Bold Physical CT is that for any physical system, there is a TM whose state transitions map onto the state transitions of that system under its ordinary dynamical description.24 This proposal faces two related problems. First, state-variables of ordinary physical systems are generally characterized by real numbers. Hence, there are uncountably many initial conditions of a physical system. But TM tapes contain discrete strings of tokens, of which there are only countably many. Therefore, there are too few initial conditions of TMs for them to map onto initial conditions of ordinary physical systems.25 The second problem is a direct consequence of the first. Since every initial condition gives rise to a different state space trajectory, ordinary physical systems have uncountably many state space trajectories, whereas TMs have only countably many state space trajectories. Therefore, there are too few state space trajectories of TMs for them to map onto state space trajectories of ordinary physical systems."

How good must an imperfect simulation be to be good enough?


Piccinini_ETD_2003.pdf

7.3.4.2 Computational Approximation

"The popularity and usefulness of computational approximations of physical systems, not only in physics but in many other sciences, may have been a motivating factor behind Bold Physical CT. Some authors state forms of CT according to which every physical system can be “simulated,” by which they presumably mean computationally approximated in the present sense, by TMs.31 But the question of whether every system can be computationally approximated is only superficially similar to Bold Physical CT.
The importance of computational approximation is not that it embodies some thesis about physical systems, but that it is the most flexible and powerful tool ever created for scientific modeling. An approximation may be closer or farther away from what it approximates. The way in which and the degree to which an approximation should mimic the system it models is largely a pragmatic factor, which depends on the goals of the investigators who are building the model.
If one allows computational approximations to be arbitrarily distant from the dynamical evolution of the system being approximated, then the thesis that every physical system can be computationally approximated becomes trivially true. If one is stricter about what approximations are acceptable, then that same thesis becomes nontrivial but much harder to evaluate. Formulating stricter criteria for acceptable approximations and evaluating what systems can be approximated to what degree of precision is a difficult question, which would be worthy of systematic investigation. Here, I can only make a few obvious points.
First, strictly speaking, unpredictable (e.g., non-deterministic) systems cannot be computationally approximated. A computational approximation can only indicate the possible dynamical evolutions of such systems, without indicating which path will be followed by any given system.
Second, if there are any (deterministic or non-deterministic) physical systems whose state transitions are not Turing-computable, e.g. if genuine hypercomputers are possible, then there is a strict sense in which those systems cannot be computationally approximated."

Sensitive Dependence on Initial Condition. There can be basins of
attraction in Chaotic systems. Many years ago I was trying to write
a program to investigate the Mandelbrot Set. It turned out many of
the problems I was having was due to the number of iterations I was
trying to use and the math functions I was using.

SH: Continuing the quote from above, this portion below seems most relevant to this (immediately above portion of your post).

"Finally, as soon as the state-variables of a system are more than two and they interact nonlinearly in a sufficiently complex way, the system may exhibit chaos (in the mathematical sense). As is well known, chaotic systems are so sensitive to initial conditions that their dynamical evolution can only be computationally approximated for a relatively short time before diverging exponentially from the observed behavior of the system.

In conclusion, the extent to which physical systems can be computationally approximated depends both on the properties of physical systems and their mathematical descriptions, and on the criteria that are adopted for adequate approximation. The same computational model may count as producing adequate approximations for some modeling purposes but not for others. At any rate, on any nontrivial criteria for adequate approximation, it is far from true that every physical system can be computationally approximated." [SH: various papers can be found at
http://www.umsl.edu/~piccininig/my%20research.html ]


Little errors add
up and the behaviors were an artifact of the rounding errors rather than of the Set itself.

I think the best trajectory to arrive at a target has to be recalculated
along the way in long interplanetary journey because of rounding errors.



It seems like your summary was quite good.

Regards,
Stephen
.

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