MacLennan: Grounding Analogue Computers

From: Hudson Joe (
Date: Sat Mar 17 2001 - 01:59:40 GMT

'Grounding Analogue Computers'

The paper by Bruce MacLennan, is the subject of this skywriting.

The issues central to MacLennan's paper are:

1/ Whether there is a computationally relevant difference between what
is called analogue and digital/discrete computation, i.e. are there
some computations that can be done with one but not the other.

2/Is any difference between the two sorts of computation (if indeed
they are distinct) relevant to cognition and symbol grounding?

Addressing 1/:

>In traditional terminology, analog computers represent variables by
>continuously-varying quantities, whereas digital computers represent
>them by discretely-varying quantities (typically, voltages, currents,
>charges, etc. in both cases). Thus the difference between analog and
>digital computation lies in a distinction between the continuous and
>the discrete, but it is not the precise mathematical distinction. What
>matters is the behavior of the system at the relevant level of
>analysis. For example, in an analog computer we treat charge as though
>it varies continuously, although we know it's quantized (electron
>charges). Conversely, in a digital computer we imagine we have
>two-state devices, although we know that the state must vary
>continuously from one extreme state to the other (voltage cannot change
>discontinuously). The mathematical distinction between discrete and
>continuous is absolute, but irrelevant to most physical systems.

So the implementations of analogue and digital computers are both
subject to the same behavioural characteristics of the matter they are
made from. I think because we are unable to make EXACT measurements of
objects in our physical reality we cannot know things like whether a
quantity of charge is discrete or in constant flux (at whatever scale).
Sure your could say that charge is defined by the number of electrons,
but then how do you know exactly which electrons to count? Also
electrons are themselves made from smaller elements (quarks?) and it
would seem unlikely that these remain perfectly static within the
electron over its life, and if they don't then what effect (however
small and unmeasurable) does this have on the electrons electrical
properties? And so on.

But then does this have any baring on the computations being done? In
the case of the 'digital' computer although voltages don't change
instantly and don't settle at exactly the ideal value (e.g. 5V) the
behaviour of the computer is such that the voltages might as well
change instantly to exact values. So the fact that the digital computer
at a fine scale seems more analogue than discrete is made irrelevant by
the design or structure of the machine. There is nothing analogue about
the computational operations a digital computer performs.

Is there an 'analogue' parallel to this? To answer this we need to be
clear on what is ment by analogue computation. Lets say that it is
something which doesn't rely on Boolean logic for a start. Because if
it did it wouldn't be doing anything a digital computer couldn't, in
which case we couldn't say 'There is nothing digital about the
computational operations an analogue computer performs.' in the same
way we said for digital computers. It is also important to note that
whatever analogue computation is a digital computer could approximate
the physical behaviour of the implementation to the nth degree (in the
same way a sine wave can be digitally approximated, or PSPICE software
models analogue circuits). If this always resulted in essentially the
same results/outputs then analogue computation would be in no way
distinct in its computational scope from digital computation. But then
if a digital computer could approximate to the nth degree the physical
behaviour of our imaginary analogue computer then how could its
approximated outputs be anything but similar to the nth degree also?
So it seems that we can't segregate analogue from digital computation.

But still there seems to be something about the 'analogue' idea that
'digital' can't capture. Perhaps this has to do with a type of analogue
computation where ANY approximation could not be tolerated. Then a
digital approximation however close would not be good enough. Of course
we couldn't even appreciate the significance of this fine difference so
we certainly couldn't verify its purely analogue status let alone build
build an implementation. These intractable qualities seem similar to
those of cognition. If cognition (see definition below) is a form of
this type of analogue computation (answers 2/) then we should find
something slightly less impossible to puzzle over. If it isn't then its
all to play for.

>Many complex systems are discrete at some levels of analysis and
>continuous at others. The key questions are: (1) What level of analysis
>is relevant to the problem at hand? (2) Is the system approximately
>discrete or approximately continuous (or neither) at that level? One
>conclusion we can draw is that it can't matter whether an analog
>computer system (such as a neural net) is "really" being simulated by a
>digital computer, or for that matter whether a digital computer is
>"really" being simulated by an analog computer. It doesn't matter
>what's going on below the level of relevant analysis. So also in the
>question of whether cognition is more discrete or more continuous,
>which I take to be the main issue in the symbolic/connectionist debate.
>This is a significant empirical question, and the importance of
>connectionism is that it has tipped the scales in favor of the

What does MacLennan mean by cognition? The Chambers dictionary
definition is: "the act or process of knowing in the widest sense,
including sensation, perception, etc., ...". If this is what MacLennan
means then we have a similar problem of verification of our any design
as we would have with purely analogue computation (something where ANY
digital approximation would invalidate the resultant output). But if
MacLennan means something which is able to perform the mechanical
elements of thinking like distilling information and formulating plans,
then I agree, it would be useful to know if these capacities are best
modeled with continuous or digital methods. From this point I will use
'analogue computation/computer' to mean something which is most
effectively viewed as continuous, even though it may be implementable
using discrete methods. An example would be an audio DSP (digital
signal processing) chip that filtered and boosted certain frequency
ranges. Inside the chip its all 1's and 0's but looking at the input
and output where there is an effectively smooth change in amplitude
across the frequency bandwidth it appears to be analogue.

MacLennan associates a symbolic view of cognition with a
discrete one and a connectionist view with an analogue one.
Why must/should symbols be static or discrete? The archetypal
object referenced by a symbol might be fixed from an omniscient point
of view but from our stand-point it may change. For instance if at an
early age an artificial intelligence (with a body), Tom is introduced
to a selection of cakes and Tom...

1/ likes every one he tries.

2/ knows they were all baked in an oven.

3/ also likes pies and knows that they are baked in an oven too.

Tom might surmise that cakes are those nice sweet things which are
baked in an oven. But later Tom tastes a few over cooked cakes which
aren't that sweet and he don't like that much. What happens now?

If Tom's mind is viewed as a symbol system implemented by a digital
computer, this experience could cause the addition of a few exceptions
to the cake symbol definition. But a more intuitively sensible response
would be to adjust the strength of the associations linked to the cake
symbol. So now 'cake' would be nice and sweet in most cases instead of
in all cases and so would be less strongly linked to the group of
things which are nice all the time (e.g. chocolate). Also if Tom knew
the cakes he didn't like spent considerably longer in the oven than the
other cakes he might make a new associate with the cooking time to
the goodness of a cakes taste. Because he knows pies are also baked in
an oven this association might flow over to pies. As more good and bad
cakes are encountered the strength of these associations are varied
accordingly. Thus the interpretable sense of the cake symbol changes.

All this seems like a connectionist description but we started with a
cake symbol and we still have a cake symbol. We have just manipulated
it in a fluent way which is more akin to connectionism. Of course these
symbols and associations might all be stored in a digital memory chip
as 64bit floating point numbers, but it is still more conceptually
instructive to consider them as fluent continuous variables.

It could be argued that the 'cake' symbol is redundant and just
something that has been put there artificially(excuse the pun). But if
there wasn't something concrete in the place of the symbol what would
the associations attach themselves to? And how would these grouped
associations be referenced collectively without using a symbol? Maybe
you could get away with not explicitly instantiating a symbol place
holder by placing the associations in a special area of memory but
effectively this is still symbolizing 'cake' by using a special area of

So this symbolic system is best viewed as a CONTINUOUS system,
not a discrete one.

I don't know how a connectionist view of Tom would work without the
neural-net (or whatever mechanism is used) being able to capture and
maintain something that could be interpreted as a symbol.

>In general, a {em computational system is characterized by: (1) a {em
>formal part, comprising a state space and processes of transformation;
>and (2) an {em interpretation, which (a) assigns meaning to the states
>(thus making them {em representations), (b) assigns meaning to the
>processes, and (c) is {em systematic. For continuous computational
>systems the state spaces and transformation processes are continuous,
>just as they are discrete for discrete computational systems.
>Systematicity requires that meaning assignments be continuous for
>continuous computational systems, and compositional for discrete
>computational systems (which is just continuity under the appropriate

Just a minute, "assigns meaning...". How does that work? Meaning is
experienced not 'assigned'. Symbols, tags, references and values are
assigned. Maybe a computational system could assign syntactically
defined values to states and symbols but meaning itself? I think not.

>Despite our differences, I agree with Harnad's requirement that
>meaningful symbols be grounded. Furthermore, representational states
>(whether discrete or continuous) have sensorimotor grounding, that is,
>they are grounded through the system's interaction with its world. This
>makes transduction a central issue in symbol grounding, as Harnad has

First off how can a state be continuous? State implies something which
is bounded, so perhaps a sine wave at frequency F1 could be used to
represent state S1. But then only the representation of the state (the
sine wave) would be continuous, the state S1 itself would be static and
discrete. Perhaps MacLennan ment 'the representation of states' rather
than, "representational states".

Secondly why is transduction a "central issue in symbol grounding"? So
long as the method of energy conversion from sound pressure, light,
mechanical resistance, etc. to electrical signals provides enough
information for the computational bits to function properly why worry
about it? Isn't the symbol grounding problem more, 'how do we use the
transduced electrical signals to terminate the hierarchical symbolic
definition chain?' (But then even if symbols were optimally grounded
I've no idea how this would conjure up meaning in the system.) Did
Harnad really say that transduction was the central issue?

>Harnad seems to be most interested in continuous-to-discrete
>transduction, if we interpret his "analog world" to mean the world of
>physics, which is dominated by continuous variables, and we assume the
>output of the transducers are discrete symbols. The key point is that
>the specific material basis (e.g. light energy) for the information
>"out there" is converted to the unspecified material basis of formal
>computation inside the computer. Notice, however, that this is not pure
>transduction, since in addition to changing the substance of the
>information it also changes its form; in particular it must classify
>the continuous image in order to assign it to one of the discrete
>symbols, and so we have computation as well as transduction. (We can
>also have the case of an "impure" discrete-to-continuous transduction;
>an example would be an effector that interpolates between discretely
>specified states. Impure continuous/continuous and discrete/discrete
>transducers also occur; an analog filter is an example of the former.)

The fact that input is quantized and represented as 0's and 1's is
irrelevant for reasons discussed in earlier paragraphs. Isn't any
realizable form of transduction impure in that it loses some
information or adds some noise of its own through the transduction

To sum up, I wasn't able to extract a coherent massage from MacLennan's
paper. I thought the notion of 'analogue' although partially delt with
earlier was later confused (e.g. 'continuous states' ) as were the
notions of discrete with symbolic, and state with meaning.
Personally I don't think the essence of cognition has much do do
with computation. Certanly computation of some kind plays an important
supporting role but that is all.

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