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AIList Digest Volume 4 Issue 249
AIList Digest Wednesday, 5 Nov 1986 Volume 4 : Issue 249
Today's Topics:
Philosophy - The Analog/Digital Distinction
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Date: 31 Oct 86 02:45:56 GMT
From: husc6!Diamond!aweinste@think.com (Anders Weinstein)
Subject: Re: The Analog/Digital Distinction
In article <20@mind.UUCP> harnad@mind.UUCP (Stevan Harnad) writes:
> I suspect that some people will find Goodman's
>considerations a little "dense," not to say hirsute, ...
Well you asked for a "precise" definition! Although Goodman's rigor may seem
daunting, there are really only two main concepts to grasp: "density", which
is familiar to many from mathematics, and "differentiation".
>> A scheme is syntactically dense if it provides for infinitely many
>> characters so ordered that between each two there is a third.
>
>I'm no mathematician, but it seems to me that this is not strong
>enough for the continuity of the real number line. The rational
>numbers are "syntactically dense" according to this definition. But
>maybe you don't want real continuity...?
Quite right. Goodman mentions that the difference between continuity and
density is immaterial for his purposes, since density is always sufficient to
destroy differentiation (and hence "notationality" and "digitality" as
well).
"Differentiation" pertains to our ability to make the necessary distinctions
between elements. There are two sides to the requirement: "syntactic
differentiation" requires that tokens belonging to distinct characters be at
least theoretically discriminable; "semantic differentiation" requires that
objects denoted by non-coextensive characters be theoretically discriminable
as well.
Objects fail to be even theoretically discriminable if they can be
arbitrarily similar and still count as different. For example, consider a
language consisting of straight marks such that marks differing in length by
even the smallest fraction of an inch are stipulated to belong to different
characters. This language is not finitely differentiated in Goodman's sense.
If, however, we decree that all marks between 1 and 2 inches long belong to
one character, all marks between 3 and 4 inches long belong to another, all
marks between 5 and 6 inches long belong to another, and so on, then the
language WILL qualify as differentiated.
The upshot of Goodman's requirement is that if a symbol system is to count as
"digital" (or as "notational"), there must be some finite sized "gaps",
however minute, between the distinct elements that need to be distinguished.
Some examples:
A score in musical notation can, if certain conventions are adopted, be
regarded as a digital representation, with the score denoting any performance
that complies with it. Note that although musical pitches, say, may take on
a continuous range of values, once we adopt some conventions about how much
variation in pitch is to be tolerated among the compliants of each note, the
set of note extensions can become finitely differentiated.
A scale drawing of a building, on the other hand, usually functions as an
analog representation: any difference in a line's length, however fine, is
regarded as denoting a corresponding difference in the building's size. If we
decide to interpret the drawing in some "quantized" way, however, then it can
be a digital representation.
To quote Goodman:
Consider an ordinary watch without a second hand. The hour-hand is
normally used to pick out one of twelve divisions of the half-day.
It speaks notationally [and digitally -- AW]. So does the minute hand
if used only to pick out one of sixty divisions of the hour; but if
the absolute distance of the minute hand beyond the preceding mark is
taken as indicating the absolute time elapsed since that mark was passed,
the symbol system is non-notational. Of course, if we set some limit --
whether of a half minute or one second or less -- upon the fineness of
judgment so to be made, the scheme here too may become notational.
I'm still thinking about your question of how Goodman's distinction relates
to the intuitive notion as employed by engineers or cognitivists and will
reply later.
Anders Weinstein <aweinste@DIAMOND.BBN.COM>
------------------------------
Date: 26 Oct 86 16:37:53 GMT
From: rutgers!princeton!rocksvax!oswego!dl@spam.ISTC.SRI.COM (Doug
Lea)
Subject: Re: The Analog/Digital Distinction: Soliciting Definitions
re: The analog/digital distinction
First, I propose a simple ground-rule. Let's assume that the "world"
somehow really is "discrete", that is, time, energy, mass, etc., all
come in little quanta. Given this, the differences between analog and
digital processes seem forced to lie in the nature of representations,
algorithms to manipulate them, and the relations of both to actual
quantities "out there".
I offer a very simple example to illustrate some possibilities. It is
intended to be somewhat removed from the sorts of interesting problems
encountered in distinguishing analog from digital mental processes.
Consider different approaches to determining population growth, given
this grossly simplistic model: an initial population, P, a time period in
question, T, (expressed in time quanta), and a "growth rate", R, the
number of quanta between the times that each member of this (asexual)
population gives birth to a new member (supposing that no more than
one birth per quantum is possible and no deaths).
Approach 1: (digital)
Simulate this process with an O(PT) algorithm, repeating T times a
scan across each member of the population, determining whether it gave
birth, and if so, adding a new member. If the population actually does
grow in this fashion, then the result is surely correct, as one might
verify by mapping the representation of the P individuals to real
individuals at time 0, and again at time T. Several efficiency
improvements to this algorithm are, of course, possible.
Approach 2: (analog)
An alternative method may be constructed by first noting that both
population size and time have very simple properties with respect
to this process. For purposes of the problem at hand, the difference
between the state of having a population of size N and one of size N+1
lies only in the difference between N and N+1. Similarly with time. To
develop an algorithm capitalizing on this, construct a nearly
equivalent problem in which population states differ only according to
the difference between N and N+epsilon, for any epsilon. Now, we know
that if epsilon is infinitessimally small, we can exploit the
differential and integral calculus to derive an exponential function
describing this process, and compute the population value at time T
with one swift calculation. Of course, the answer isn't right: we
solved a different problem! But it is close, and methods exist to
determine just how close this approximation will be in specific
instances. We may even be able to apply particular corrections.
Approach 3: (digital)
Use techniques developed for difference equations and recurrence
relations to come up with an exact answer requiring nearly as little
calculation as in the analog approach.
Approach 4: (digital?)
Place P cents in a bank account with compound interest rate
corresponding to R, and then see how much money you have at time T.
Approach 5: (analog)
Build a RLC integrating circuit with the right parameters. Apply
input voltage P and measure the output voltage at time T.
Approach 6: (analog?)
Observe some process with an exponential probability distribution
of events. Apply lots of transformations to get an answer.
There are probably many other interesting approaches, but I'll
leave it there.
Morals:
1. The notion of "analogy" or simulation does not do much to
distinguish analog from digital processing. Perhaps due to the nature
of our physical world, often there do seem to be more and better
analog analogies than digital analogies for many problems.
2. Speed of calculation also seems secondary. For example, the
calculus allows manipulation of representations involving infinite
numbers of states with a single calculation. But some digital methods
are fast too. Similarly with the fact that analog methods sometimes
allow compact representations (with single numbers and simple well
behaved functions representing entire problems). But one could
probably match, one-for-one, problems in which analog and digital
approaches were superior with respect to these attributes. This all
just amounts to acknowledging that the choice between ANY two
algorithms ought to take computational efficiency into account. And,
of course, the notion of "symbolic" vs. "non-symbolic" processing
plays no role here. All of the above approaches were symbolic in one
way or another.
3. The notion of approximation seems to be the most helpful one.
Again, for example, processing that involves implicit or explicit use
of the calculus can ONLY (given the above ground-rule) provide
approximations. Most such processing should probably be considered
analog. However, the usual conceptualization of approximation in
current use doesn't seem good enough. There are many digital
"heuristic" algorithms that are labelled as "approximations". (Worse,
discrete computational techniques for numerically solving "analytic"
problems like integration are also labelled "approximations" in nearly
a reverse sense.) For example, the nearest-neighbor heuristic is
considered as an approximation algorithm for the travelling
salesperson problem. But this seems to be a different sort of
approximation than using exponential equations to solve population
problems.
I'm not at all sure how to go about dealing with such
distinctions. Considerations of the robustness and the arbitrary level
of precision for approximations in the first sense might be useful,
but aren't the whole story: For example, several clearly digital
heuristics also have these properties (see, e.g., Karp's travelling
saleperson heuristic), but in somewhat different (e.g., probabalistic)
contexts. See J. Pearl's "Heuristics" book for related discussions.
Doug Lea
Computer Science
SUNY Oswego
Oswego, NY 13126
seismo!rochester!rocksvax!oswego!dl
------------------------------
Date: 4 Nov 86 01:55:22 GMT
From: rutgers!husc6!Diamond!aweinste@SPAM.ISTC.SRI.COM (Anders
Weinstein)
Subject: Re: Analog/Digital Distinction: 8 more replies
>Stevan Harnad:
>
>> Goodman mentions that the difference between continuity and density
>> is immaterial for his purposes, since density is always sufficient to
>> destroy differentiation (and hence "notationality" and "digitality" as
>> well).
>
>There seems to be some difference of opinion on this matter from the
>continuity enthusiasts, although they all advocate precision and rigor...
I don't believe there's any major difference here. The respondants who
require "continuity" are thinking only in terms of physics, where you don't
encounter magnitudes with dense but non-continuous ranges. Goodman deals with
other, artificially constructed symbol systems as well. In these we can, by
fiat, obtain a scheme that is dense but non-continuous. I think that
representation in such a scheme would fit most people's intuitive sense of
"analog-icity" if they thought about it.
>> Objects fail to be even theoretically discriminable if they can be
>> arbitrarily similar and still count as different.
>
>Do you mean cases like 2 vs. 1.9999999..., or cases like 2 vs. 2 minus epsilon?
>They both seem as if they could be either "theoretically
>discriminable" or "theoretically indiscriminable," depending on the
>theory.
I'm not sure what you mean here. I don't see how a length of 2 inches would
count as "theoretically discriminable" from a length of 1.999... inches; nor
is a length of 2 inches theoretically discriminable from a length of 2 minus
epision inches if epsilon is allowed to be arbitrarily small. On the other
hand, a length of 2 inches IS theoretically discriminable from a length of
1.9 inches.
In his examples, Goodman rules out cases where no measurement of any finite
degree of precision would be sufficient to make the requisite distinctions.
>> "Consider an ordinary watch without a second hand. The hour-hand is
>> normally used to pick out one of twelve divisions of the half-day.
>> It speaks notationally [and digitally -- AW]. So does the minute hand
>> if used only to pick out one of sixty divisions of the hour; but if
>> the absolute distance of the minute hand beyond the preceding mark is
>> taken as indicating the absolute time elapsed since that mark was
>> passed, the symbol system is non-notational. Of course, if we set
>> some limit -- whether of a half minute or one second or less -- upon
>> the fineness of judgment so to be made, the scheme here too may
>> become notational."
>
>So apparently it does not matter whether the watch is in fact an
>"analog" or "digital" watch (according to someone else's definition);
>according to Goodman's the critical factor is how it's used.
Right. Remember, Goodman is not talking about whether this is what an
engineer would class as an analog or digital WATCH (ie. in its internal
workings); he's ONLY talking about the symbol system used to represent the
time to the viewer. And he's totally relativistic here -- whether the
representation is analog or digital depends entirely on how it is to be
read.
------------------------------
Date: Tue, 4 Nov 86 17:10:39 pst
From: ladkin@kestrel.ARPA (Peter Ladkin)
Subject: analog/digital distinction
Here's a quick shot at an A/D distinction.
The problem with the rationals was that the ordering and the operations
are easily translatable into computations on the natural numbers.
So, the proposal is:
DIGITAL: computations on a structure S that is recursively
isomorphic to a definable fragment of Peano Arithmetic.
ANALOG: computations on a dense structure that
is not recursively isomorphic to a definable fragment of Peano
Arithmetic.
Note there can be computations which are neither analog nor
digital according to this definition.
The rationale for this choice depends on two considerations.
(1) One must not be able to transform one kind of computation
into the other, which can be done only if there is a machine
(aka recursive function) that can do it.
(2) The distinction must not collapse in the face of the
possibility that physics will tell us the world is
fundamentally discrete (or fundamentally continuous), since
if Gerald Holton is to be believed, physical science has
been wavering between one and the other for thousands of years.
So the discrete/continuous nature of nature can be regarded
as a metaphysical issue, and we want to finesse this in our
definition to make it physically realistic.
I chose Peano Arithmetic as the base structure because it is
intuitively discrete, and all the digital structures that have
been proposed fit the criterion that they can be recursively
mapped into simple discrete arithmetic.
The density-of-values criterion for analog computation seems
intuitively plausible, and if one wants to make the distinction
between analog and digital into a feature of the world, not merely
of the representation chosen, one needs to assure consideration
(1) above.
If quantum physics ultimately tells us that the world is discrete,
there is no reason to assume that the discreteness in the world
will provide us with recursive functions mapping that discreteness
into the natural numbers, so analog computations will survive that
discovery.
Peter Ladkin
ladkin@kestrel.arpa
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End of AIList Digest
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