harnad@mind.UUCP (Stevan Harnad) (10/21/86)
I'd like to test whether there is a coherent formulation of the analog/digital distinction out there. I suspect that the results will be surprising. Engineers and computer scientists seem to feel that they have a suitable working definition of the distinction, whereas philosophers have argued that the distinction may not be tenable at all. Cognitive scientists are especially interested because they are concerned with analog vs. nonanalog representations. And neuroscientists are interested in analog and nonanalog processes in the nervous system. I have some ideas, but I'll save them until I sample some of what the Net nets. The ground-rules are these: Try to propose a clear and objective definition of the analog/digital distinction that is not arbitrary, relative, a matter of degree, or loses in the limit the intuitive distinction it was intended to capture. One prima facie non-starter: "continuous" vs. "discrete" physical processes. Stevan Harnad (princeton!mind!harnad)
dl@oswego.UUCP (Doug Lea) (10/26/86)
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
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!dlaweinste@Diamond.BBN.COM (Anders Weinstein) (10/28/86)
Philosopher Nelson Goodman has distinguishes analog from digital symbol systems in his book _Languages_of_Art_. The context is a technical investigation into the peculiar features of _notational_ systems in the arts; that is, systems like musical notation which are used to DEFINE a work of art by dividing the instances from the non-instances. The following excerpts contain the relevant definitions: (Warning--I've left out a lot of explanatory text and examples for brevity) The second requirement upon a notational scheme, then, is that the characters be _finitely_differentiated_, or _articulate_. It runs: For every two characters K and K' and every mark m that does not belong to both, determination that m does not belong to K or that m does not belong to K' is theoretically possible. ... A scheme is syntactically dense if it provides for infinitely many characters so ordered that between each two there is a third. ... When no insertion of other characters will thus destroy density, a scheme has no gaps and may be called _dense_throughout_. In what follows, "throughout" is often dropped as understood... [in footnote:] I shall call a scheme that contains no dense subscheme "completely discontinuous" or "discontinuous throughout". ... The final requirement [including others not quoted here] for a notational system is semantic finite differentiation; that is for every two characters K and K' such that their compliance classes are not identical and every object h that does not comply with both, determination that h does not comply with K or that h does not comply with K' must be theoretically possible. [defines 'semantically dense throughout' and 'semantically discontinuous' to parallel the syntactic definitions]. And his analog/digital distinction: A symbol _scheme_ is analog if syntactically dense; a _system_ is analog if syntactically and semantically dense. ... A digital scheme, in contrast, is discontinuous throughout; and in a digital system the characters of such a scheme are one-one correlated with compliance-classes of a similarly discontinous set. But discontinuity, though implied by, does not imply differentiation...To be digital, a system must be not merely discontinuous but _differentiated_ throughout, syntactically and semantically... If only thoroughly dense systems are analog, and only thoroughly differentiated ones are digital, many systems are of neither type. To summarize: when a dense language is used to represent a dense domain, the system is analog; when a discrete (Goodman's "discontinuous") and articulate language maps a discrete and articulate domain, the system is digital. Note that not all discrete languages are "articulate" in Goodman's sense: Consider a language with only two characters, one of which contains all straight marks not longer than one inch and the other of which contains all longer marks. This is discrete but not articulate, since no matter how precise our tests become, there will always be a mark (infinitely many, in fact) that cannot be judged to belong to one or the other character. For more explanation, consult the source directly (and not me). Anders Weinstein <aweinste@DIAMOND.BBN.COM>