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Hi All,

I hope that with the rest of this term's readings everything will

begin to fall in place (with Penny it clearly seems to have done).

You will find lecture notes from last year's course at:

http://www.cogsci.soton.ac.uk/~harnad/CM302/

I's appending them to this list directly, in case you want to Q/C

them.

(The readings this year are different, so don't feel obliged to read the

unfamiliar ones mentioned below unless you are especially interested.

Next term we will be reading looking at recent papers on hybrid

systems.)

Chrs, S

WHAT IS COMPUTATION?

First, What was there before computation? Differential Equations and

Dynamical Systems.

The Turing Machine is a hypothetical mechanical device that reads and writes

symbols on a tape according to internal states that causally dictate what it

does. An example would be:

Machine is in state I, which is such that it makes it read its

input tape position now.

(Read input.)

(Input is 0.)

State I is such that if the input is 1, the machine goes into

state J, which is Halt; if the input is 0, the machine goes into

state K, which is, advance the tape, write 1 and Halt.

For a more formal definition of a Turing Machine, see:

http://i12www.ira.uka.de/~pape/papers/puzzle/node4.html

or

http://obiwan.uvi.edu/computing/turing/ture.htm

or

http://aleph0.clarku.edu/~anil/math105/machine.html

These simple mechanical operations are what computation consists of. They

are things any mindless device can do. They are based on making all

operations explicit and automatic. No "thinking" is required. (That is the

point!)

The Turing Machine is only a hypothetical device. (A digital computer is a

finite physical approximation to it, differing in that its tape is not

infinitely long.)

Implementation-Independence:The physical device that actually implements the

Turing Machine is irrelevant (except of course that it has to be implemented

by a physical device). This fact is critical. It is simple, but often

misunderstood or forgotten, yet, as you will see, it is essential to the

definition of computation. It is also the basis of the hardware/software

distinction.

Let's call the marks on the Turing machine's tape "symbols" (actually

"symbol tokens," because "symbol" really refers to a symbol-type, a kind of

generic pattern, like "A," whereas a symbol token is an actual instance of

A; but we will use "symbol" for both symbol types and symbol tokens except

where the difference matters).

The implementation-independence or hardware-independence of computation is

related to the notation-independence of a formal system: Arithmetic is

arithmetic regardless of what symbol or notational system I use, as long as

the system has the right formal properties (something corresponding

systematically to "0," "+" "=" etc.). This is exactly the same as the

implementation independence of computation: A Turing Machine is performing a

particular computation if it implements the right formal properties. The

physical details matter no more than the details of the shapes of the

symbols in a notational system.

The counterpart of the hardware-independence of computation is the

shape-independence and arbitrariness of symbols and symbol systems: It does

not matter whether I designate "add" by "add," "+" "&" or "PLUS" -- as long

as I use the it consistently and systematically to designate adding.

A symbol cannot really be defined in isolation. Or rather, a single symbol,

unrelated to any symbol system, is trivial. (There is a joke about a

wonder-rabbi at his death-bed, with all his disciples gathered together to

hear his last words. The wonder-rabbi murmurs "Life.... is like.... a

bagel." All his disciples are abuzz with the message: "Pass it on: The

wonder-rabbi says life is like a bagel!" The word is passed on till it

reaches the synagogue-sweeper, the lowliest of the flock. He asks" "Life is

like a bagel? How is life like a bagel?" The buzz starts again as the

question propagates back the the deathbed of the wonder-rabbi:

"Wonder-Rabbi, How is life like a bagel?"

The wonder-rabbi pauses for a moment and then says "Okay, so life's not

like a bagel."

The point is that you can read anything and everything into a point-symbol:

It only becomes nontrivial if the symbol is part of a symbol system, with

formal relations between the symbols. And, most important, the symbol system

must be semantically interpretable. That is, it must mean something; it must

make sense -- systematic sense.

For example, it doesn't matter what symbol you use for "addition" in

arithmetic, but then whenever you refer to addition, you must use that

(arbitrary) symbol, and the strings of symbols in which it occurs must be

systematically interpretable as denoting addition. In particular, "1 + 1"

must equal "2" no matter what notation you use for "1" "+" "=" and "2".

Now we are ready to define computation (the Turing Machine was more an

example than a definition): Computation is symbol manipulation. Nontrivial

computations are systems of symbols with formal rules for manipulating them.

The shape of the symbols is arbitrary. That is just part of a notational

system. But the symbols and the manipulations must be semantically

interpretable: It must be possible to interpret them (systematically, not

point-wise, like the bagel) as meaning something.

For example, arithmetic is a formal symbol system, consisting of its

primitive symbols (o, +, =, etc.) and strings of symbols (axioms) and

manipulation rules (rules for forming well-formed formulas, for making

logical inferences, and for making arithmetic calculations).

See:

http://www.csc.liv.ac.uk/~frans/dGKBIS/peano.html

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Peano.html

Now there is one and only one way to interpret all the axioms, theorems and

calculations of arithmetic: as referring to numbers and their properties.

(There are also so-called "nonstandard" interpretations, but these only

apply to things very much like numbers and isomorphic to them in critical

respects.) Unlike "Life is like a bagel," the Peano's arithmetic symbol

system has, to all intents and purposes, only one coherent interpretation.

It doesn't make sense if interpreted as a military manual, a planetary map,

or a Shakespeare play (and vice versa). This systematic mappability into

meanings and vice versa is the central property of symbol systems.

Symbol systems are also compositional: They consist of elementary symbols

that are combined and recombined according to the symbol manipulation rules.

Yet all the (well-formed) combinations are semantically interpretable too,

and all the interpretations cohere. This is not a trivial property. It is

easily to invent an arbitrary code, consisting of symbols and symbol

manipulation rules. It is much harder to invent one in which the symbols and

symbol combinations all make sense.

And, conversely, it is hard to take an undeciphered symbol system that does

have a unique, nontrivial interpretation, and decipher it so as to find that

interpretation.

All formal or artificial symbol systems (including all of mathematics and

logic, computer programmes, and artificial "languages") are subsets of

natural language: We don't change languages when we begin to talk

"geometry," "boolean algebra," or "C," we simply use a specialized subset of

the vocabulary of English.

So all formulas are really sentences in English (or any other natural

language). This means that natural language is the "mother of all symbol

systems."

THE CHURCH/TURING THESIS

The Turing machine is one attempt to formalise the mathematician's working

notion of "computation." The mathematician has an idea of what it is that

he is doing when he is doing mathematics. There are differences of opinion

among mathematicians about what mathematics is about. Four schools of

thought (at least) about the foundations of mathematics exist:

(1) The Realists (also called "Platonists") hold that mathematics is about

the eternal truths of the universe: that numbers, for example, are

properties of the world too, except that they are even more real than

objects, because what is true of objects only happens to be true in the

actual world, whereas what is true of numbers is true in every possible

world, on pain of contradiction.

(2) The Formalists do not think numbers (or other mathematical entities)

really exist, like objects. They think that their properties are just the

formal consequences of the formal rules we choose to adopt (i.e., which

symbol strings we take as axioms, which symbol-manipulation rules we adopt

for making deductions from the axioms, etc.). ("Formal" means based on

arbitrary shape conventions or notations, as we discussed earlier in

connection with symbol systems.)

(3) The Logicists are really a special kind of formalist, in that they too

think that mathematical objects are just formal, rather than real, but they

think all of mathematics can be reduced to and derived from logic, which is

to say a particular formal system, whereas the formalists just say it's all

formal, with no commitment to reducibility to logic. One thing that all

formalists, including logicists, have in common is that they emphasise

"apagogic" proof: proving that things are true or that things "exist" on

pain of contradiction (i.e., because otherwise it would lead to a

contradiction). You may remember the method of proof called "reductio ad

absurdum." For the logicists, and for all formalists, this is the model for

what a proof really is.

(4) The Intuitionists (also called "Constructivists") think that

mathematical objects such as numbers are neither real, as the realists do,

nor merely formal, as the formalists/logicists do. They think they are ideas

constructed by human minds. They differ from the formalists (and perhaps the

realists) in that they don't believe that a theorem is true, or that a

mathematical object exists, merely because it can be shown that if it did

not exists, that would lead to a contradiction. They believe only in things

that have been proved by a construction, that is, an algorithm that will

actually find or generate the object in question. [The famous example of the

function that exists for everyone else but not for intuitionists is the one

that takes the value "1" if, somewhere in the decimal expansion of pi there

occurs a string of 7 consecutive 7's (i.e., pi =

2.1416.........7777777...), and the value "0" if it does not. No one knows

whether there is such a string of 7's in pi so no one knows whether or not

the search for it in the decimal expansion would ever come to a halt. For

the intuitionists, this is not a well-defined mathematical function, even

though we know, on pain of contradiction, that it would have to have either

value 0 or 1.]

Now I mention these 4 foundational views because one of them, formalism,

looks so much like what we have said about symbol systems, and that is no

coincidence, because symbol systems are formal systems (and symbol shapes

are just the shapes used in a formal notational system).

All mathematicians are formalists at least to this extent:

(a) They all agree that a statement is false if it leads to a contradiction.

(b) They all agree that mathematics involves "computation,"

but until the 20th century that was just an intuitive idea.

Computation was thought of as an "effective procedure," a way to get a

result mechanically and unambiguously.

The formalists (who, by the way, are the vast majority) had the further

belief, expressed most prominently by David Hilbert, that is will eventually

be possible to compute all the theorems that follow from axioms: That

mathematical proof is computational, and that all mathematical truths are

computable.

Goedel, as you probably know, went on to show famously that Hilbert's

formalist programme for mathematics was doomed to fail because even in

arithmetic it was provable that there were always truths that were

unprovable within any particular formalisation (axiomatisation) of

arithmetic. We will not be talking about that, although some thinkers

(including Lucas

<http://cogprints.soton.ac.uk/abs/phil/199807022>) and Penrose) have thought

Goedel's proof showed that all of intelligence could not be captured by

computation.

But another thing Goedel did was to produce another formalisation of

mathematicians' intuitive notion of "computation," and his formalisation

turned out to be equivalent to Turing's. The logician Alonzo Church (and

another called Post) also had a go, and although their formalisations all

looked different, they too turned out to be just notational variants of the

Turing Machine.

Given that all these independent attempts to capture what mathematicians

have in mind by "computation" all turned out to be equivalent, it was

natural to propose the thesis (and note that it is a "thesis" or conjecture,

rather than a theorem that has been or can be proved) that the Turing

Machine and its various equivalent variants all capture what is meant by

computation, and hence that anything a mathematician can "compute" can be

computed by a Turing Machine. This is has come to be known as the

Church/Turing Thesis.

TURING EQUIVALENCE

So according to the C/T Thesis, computation is "universal." What ever can be

computed at all, can be computed by a Turing Machine (i.e., a symbol system

with the right symbols and rules). This also led to the idea of the

"Universal Turing Machine" and "Turing Equivalence," for if everything that

is computable is Turing-computable, then for every computation, in whatever

symbol system, there should be an equivalent Turing Machine version of the

computation (and the Universal Turing Machine was simply one that could

perform any computation).

Computation, however, is just formal, and a Turing Machine is just an

abstraction. To make computation happen in the real world, you need a

physical device to do it, something very like a Turing Machine, but with the

physical details needed to make it work. The modern digital computer, with

the stored-programme architecture due to von Neumann, is a physical

approximation to such a machine. It is a programmable symbol manipulator, a

Universal Turing Machine that can be transformed, via software, into any

particular Turing Machine (and particular symbol system).

So far this is standard stuff, known (if only vaguely) to every computer

scientist. Here is something new: What is the relation between Universal

Turing Machines and other things, both other machines (such as bridges,

cars and planes), and other physical systems in the world (such a galaxies,

gases, molecules, and avalanches)? If, according to the C/T Thesis,

computation captures mathematics, does it capture physics and engineering

too? Are machines and other physical systems also Turing-Equivalent to some

symbol system?

The answer is yes: Apart from (1) true continuity (which calls for the

differential equations of Newton rather than discrete difference equations),

plus (2) turbulence and (3) quantum effects, digital computers can simulate

anything. This means symbol systems and algorithms can formally encode

anything and everything (and even 1-3, to an approximation). You might call

this the Physical version of the C/T Thesis: A computer can not only do any

computation, hence "imitate" any other computer, it can also imitate any

other physical system.

But to be able to imitate anything and everything is not to be able to be

anything and everything. It is good to remind yourself now and then that a

computer can simulate flight, but it cannot fly; it can simulate evaporation

and liquefaction, but it cannot evaporate or liquify; it can simulate a

fire, but not by burning.

"Turing Equivalence" means computational or formal equivalence. That means

equivalence up to a systematic interpretation. A symbol system, if it is the

right symbol system (i.e., if it really is a set of symbols and rules

systematically interpretable, squiggle for squiggle, as corresponding to

some other system in the physical world), can be designed to be equivalent,

for example, to the solar system. So it will encode the positions,

movements, and other properties of the sun and planets to as close an

approximation as we like (and as we have the algorithms for). We can then

make it work as a calculating oracle, cranking out the positions of the

planets at any time of the year or the millennium. Or, if we want to

translate the symbols not just into a verbal interpretation in terms of

plantetary position, but into something that looks to our senses like the

planets (viewed through a powerful telescope, for example), the semantic

interpretability of the symbols can be cashed in as a Virtual Reality

simulation. That is all part of Turing Equivalence -- the fact that,

property for property, a symbol system will formally match what it is

equivalent to.

Turing Equivalence leads naturally to the Turing Test.

THE TURING TEST

Each of you should read Turing's Paper on this topic. Even though he did

not put it in words that were immune to misinterpretation, the paper is has

become a classic.

Turing was interested in whether a machine could be "intelligent." Normally,

we only use this word to describe people and some animals, and normally we

think of intelligence as almost synonymous with "having a [conscious] mind."

One way to settle the matter of whether machines can be intelligent is

simply to drop these two features we normally ascribe to intelligence (that

only people/animals with minds have it) and simply say that intelligence is

merely what it takes to be able to do what has hitherto required a person or

an animal with a mind to do (until today, when we can build very capable

machines).

This solution -- to simply define "having intelligence" as "being able to do

the kinds of things that could until now only be done by organisms with

minds" -- is appealing to some people, but it does trivialise the matter.

For, according to this definition, a thermostat is intelligent (it can turn

on a furnace when it's cold, and off when it's warm, something that only

people could do until now), and so is a desk calculator, etc.

Is there another approach, one that does not trivialise intelligence? We can

always pick a more difficult task, and say "a machine is only intelligent if

it can do this," but that sounds arbitrary: What do we know, in advance,

about "intelligence," whatever that is, that allows us to pick and choose

tasks in deciding whether or not the executor is intelligent?

Turing did not propose to define intelligence. (A good idea not to do so, in

advance of having any real idea of what it is.) In fact, one interpretation

of Turing's paper is that he suggested forgetting about what intelligence

was altogether, and simply pushing on with getting machines to do all the

things we can do. That's not bad advice, and in the end it's probably the

methodological moral of his paper, but I think he can be interpreted as

saying something more substantial than that.

He tries to influence our intuitions about intelligence having us imagine a

party game in which a man and woman leave the room and interact with us only

through messages on paper. The game is to figure out which is the woman and

which is the man. (They try to fool us.) The intuition comes here: He

suggested that if we continued to exchange messages with both players --

indeed, if we played many games, with many pairs of candidates like this --

we would sometimes be right about which was the man and which the woman, and

sometimes right. But suppose, unbeknownst to us, sometimes one of the

candidates was neither a man nor a woman, but a machine -- but one capable

of interacting with us in the same way.

Turing suggested that if the machine never did anything to make us suspect

it was a machine -- if we kept guessing that it was a man, or a woman, as

the case may be, but never was that it was neither, but a machine -- then,

when the game was over, if we were told that it had been a machine, we would

really have no non-arbitrary reason for revising our judgments about it. Our

judgments might be wrong about whether it was a man or a woman, just as they

might be wrong about a real man or woman, but what basis do we have for

saying that they are wrong about the fact that it was a person, with a mind,

with intelligence? In discovering it was a machine, what have we really

learnt?

Turing might have added a bit more to clarify this intuition, and strengthen

the case for his conclusion that intelligence is as intelligence does, and

that this is no less true of us than it is of machines. For a "machine" is

merely a man-made physical system, obeying the cause/effect laws of the

universe like everything else in the universe. Surely the property being

"man-made" or not "man-made" has nothing to do with being intelligent. If I

had been cobbled together in a lab, would that make me any the more or less

intelligent?

So Turing might have been more explicit about the fact that he was pumping

our intuitions about our ignorance about what machines really are, hence

what is or is not one; also about our ignorance about what does and does not

have a mind. Each of us knows in his own private case that he has a mind,

but how do we know about anyone else, other than on the basis of what they

do? This has been called the "other-minds" problem by philosophers, and it

turns out that the way we solve it from day to day is by Turing Testing.

Does anyone know another way? Does anyone have a periscope for peering into

a candidate to see whether he has a mind? (Do brain scans do that? But how

do we know that?)

Turing might also have been more careful to point out that although he

introduced it as a party games, he was not talking about games or trickery,

and that far from being a one-time test, Turing Testing is the game of life:

It is not enough to fool a few people briefly at a party. The candidate

would have to be indistinguishable ("Turing-Indistinguishable") from any of

the rest of us for a lifetime.

And even the out-of-sight, message-passing version of the Turing Test in

this paper is (arbitrarily) restricted, for if the candidate is to be

Turing-Indistinguishable from us in what it can do, then there is a great

deal that we can do besides just sending and receiving messages: Messages

are, after all, just strings of symbols. That is what we get from a

life-long pen-pal, but people can also see and hear one another, and, more

important than what people look like (for other-minds testing is based more

on what we can do -- including what we can say and write -- than on what we

look like) people can also interact with the world of objects that their

strings of symbols are systematically interpretable as being about.

In other words, for full generality, the Turing Test should be thought of in

its robotic version, not merely its disembodied pen-pal version, for

something more substantial than intuitions rides on that difference: For the

pen-pal version of the Turing Test (let's call it T2) could in principle be

passed by a just a computer implementing a symbol system, symbols in and

symbols out, whereas the robotic version (T3) necessarily also calls for the

capacity to interact causally with the world that those symbols are

interpretable as being about, and it could not be passed by just a computer

receiving, manipulating and sending symbols. The difference is critical, and

Searle's Chinese Room Argument is based on this critical difference (though

Searle does not quite seem to realise or admit it).

SYMBOLISM VS DYNAMISM

Last term you had Bob's course on neural nets and hybrid modeling, so you

should have a practical idea of what the difference between the

logical/symbolic and neural-net approaches to AI and cognitive modeling

amounts to in practise. We will now turn to a closer analysis of what it

amounts to in theory. First, have a look at the readings under " SYMBOLIC AI

CRITIQUES OF CONNECTIONISM" in

<http://www.cogsci.soton.ac.uk/~harnad/topics.html> There you will find

theorists disagreeing about the scope and limits of the two approaches. What

underlies this disagreement?

Let's look at the distinction in terms of what we have learned already:

First we have to ask what is and is not computation, and then we have to ask

what neural nets are (we already know that logical/symbolic computation is

computation!)

Computation is implementation-independent, systematically interpretable,

symbol manipulation. Symbols are just arbitrary objects. They are

manipulated according to rules or "algorithms" that are applied on the basis

of the shape of the symbols, not on the basis of what the symbols mean

(i.e., not on the basis of their interpretation, yet they are nevertheless

systematically interpretable as meaning something). The shapes of the

symbols are arbitrary: they neither resemble nor are physically connected in

any way with what it is that they mean; they are merely a notational system.

And all symbol systems, no matter what their shape, are equivalent if they

follow the same symbol-manipulation rules. This is the sense in which symbol

systems are independent of their physical implementations: Their shapes are

arbitrary, and systems with completely different shapes may still be exactly

the same symbol system, if the rules are the same.

The intuitive example you should keep in your mind whenever you think of a

symbol system is binary digits, 0's and 1's (because all other symbol

systems can be translated into 0's and 1's, among other possibilities) and

the simplest kind of manipulation rule might be: replace all 0's by 1's.

That is a (trivial) symbol system. And physical implementation of it -- that

it, any system that will actually perform according to that rule -- would be

able to take, as input, any string of 0's and 1's, and produce, as output an

equally long string, consisting of all 1's.

The idea is that the symbol manipulation rule is mechanical; that means that

any mindless machine could do it, without needing to "understand" what it

was doing, or why. The remarkable property of such mindless, mechanical,

symbol-manipulation systems is that, nevertheless, the symbols and

manipulations (if they are not trivial or arbitrary ones) can be

systematically interpreted. The system does not understand, yet what it is

doing is understandable (and useful) to someone who is capable of

understanding, and has some use for what it is doing.

Now it is obvious that logical/symbolic programmes are examples of symbolic

computation. What is not an example of symbolic computation? Well one

general set of examples is the very things that computations are

interpretable as being about: in general, unless the computation is itself

meant to be about symbols (i.e., about arbitrarily shaped objects that are

interpretable as standing for still further objects), objects themselves are

not computational.

Apples are not symbols or computations. Symbol systems can be systematically

interpretable as being about apples; they may be systematically

interpretable as being like apples in every respect. (If there is any apple

property missing, you can fix the algorithm so it includes that too.) But,

unlike apples, which are not just systematically interpretable as apples,

but they really are apples, symbol systems are just systematically

interpretable as apples. They are not apples. And apples are not symbol

systems. Nor is just about any other object under the sun that you might

think of (except, say, a computer, that happens to be implementing a symbol

system that is systematically interpretable as, say, an apple!).

So most objects (e.g., apples) are not symbol systems. Any object can be

used as a symbol, of course, so, for example, an apple could be part of a

symbol system. It could stand for, for example, a banana, just as the symbol

"0" could stand for a banana. But it should be obvious that then its shape

would be irrelevant to what it was being used for. S even an apple that is

being used as one of the arbitrary objects in a symbol system is not itself

a computational object, as an apple, because the physical details of its

shape are irrelevant. Any other shape would have done the same job, just as

well.

Now, assuming that it is clear that not every object is a symbol system

(though symbol systems can probably be devised that are systematically

interpretable as just about any object), it follows that if we are using an

object as a model for something or other, if it is the object's physical

properties that are doing the work (i.e., the object is not just being used

as an arbitrary symbol in a symbol system), then the model is not a

computational model. In this sense (and this is important), so-called

"analog" computation is not really computation (or at least it is not

symbolic computation, and symbolic computation is what we mean by

computation in this course; it is also what Goedel, Church, Turing, von

Neumann, etc. meant by "computation" too).

So a sun-dial, for example, is not a computer, even though it "computes" all

the times of day for us. Analog computers manage to deliver the results they

deliver because of their physical "shapes." This means that they are not

shape-independent, or implementation-independent, and hence do not "compute"

in the formal, classical, Turing-Machine sense. In the same sense, a system

whose performance is explained by a set of differential equations is not a

computer either. In fact, it's a fairly good rule of thumb is that if a

system's performance is best explained as conforming to a set of

differential equations rather than as implementing a computer programme,

then the system is not a computer.

In general, physics is the science of dynamical systems -- physical systems

that change in time. Newtonian mechanics explains the dynamics of things

like billiard ball interactions and celestial mechanics. Quantum mechanics

explains the interactions on the subatomic scale. None of the systems in

question -- billiard balls, stars, electrons -- is a symbol system, and the

explanations are not computational. (It is almost always true that one can

do a discrete computational simulation of any dynamical system, but that is

only an approximation, and, like the simulation of an apple, it is not the

dynamical system.)

Computers are of course dynamical systems too: every physical system is a

dynamical system. But their dynamics is irrelevant to what they are

computing, because computation is implementation-independent [ =

dynamics-independent]. A computation must be physically implemented somehow,

to be sure -- even if it is only by a person doing the symbol manipulations

on paper -- but the dynamical details of the implementation can vary wildly:

the difference can be as big as the difference between a nuts and bolts PC

pushing flip-flops and a flesh and blood person pushing a pencil on paper.

So there is a true difference between dynamical and symbolic models. Now we

at last come to the question of neural nets: If something is explained using

a neural net, is that a symbolic/computational or a dynamic/noncomputational

model?

First let's set aside two tricky trivial cases:

(1) It is known that a neural net architecture can be used to implement a

symbol system. This is irrelevant, because computation is

implementation-independent. If the only use to which you are putting your

neural net in your "hybrid" symbolic/nonsymbolic model is that you are using

it as the hardware to perform the computations of the symbolic component,

then you don't have a hybrid model at all, but a symbolic one (and you have

added to it, for some unknown reasons, some details of your particular

implementation of the symbol system, details that we know are irrelevant,

because we could implement the same symbol system in wildly different ways).

(2) Most neural nets are not "real" neural nets: They are not really

parallel, distributed, systems of nodes, with interconnections of

continuously variable and mutable strength. In reality, they are

computational simulations of parallel, distributed nets. In other words,

they are symbol systems that are interpretable as neural nets. That's fine,

but then the question arises: For what the models are meant to be able to do

-- for the functional capacity that they are meant to deliver -- is there

anything about a real parallel, continuous, distributed, modifiably

interconnected hardware that is essential? Or could a discrete, serial

system (like the net simulation itself) have done the same job? For if so,

then the "neural net" is really just a symbolic algorithm, and the "hybrid"

model is really just a combination of different symbolic algorithms.

How could parallelness or distributedness be essential to a system, rather

than merely one arbitrary way of implementing something that was serial,

symbolic, and very fast? One case is suggested by the actual nervous system.

There is some evidence that global oscillations (EEG) might play a role in

regulating brain activity. This means that brain waves going on

simultaneously in different regions interact to produce a global pattern.

This would not be possible if the areas were only active serially.

But don't strain your brains too hard to find cases where some dynamical

property is essential to a system, because it almost never is. That's one of

the implications of he Church/Turing Thesis: Computation can approximate any

other system as closely as you like. The most famous case is in which people

tried to show that a system had to be analog was mental rotation: Roger

Shepard used a computer to generate unfamiliar new 3-D geometric shapes. He

showed pairs of them to human subjects at various spatial orientations and

asked them to judge whether they were the same or different. When he showed

two that were the same, but one was rotated into a different orientation

from the other, the amount of time it took subjects to report that they were

the same was directly proportional to the degree that they were rotated. It

was natural to conclude that the way they solved the problem was to

"mentally" rotate one to see whether it would match the other.

Now a mental image rotation is an analog process, and that was what Shepard

concluded that the brain must be doing (and he was right, as was later

confirmed by more complicated studies, including brain scans during the

task). But that did not stop the computationalists from saying "Not

necessarily: It could be a fast, serial, digital approximation, operating

on, say, the numerical coordinates of the vertices with an algorithm, rather

than by any analog shape rotation."

And the computationalists were right. It could have all been done by a

symbolic algorithm (although then the correlation between the degree of

rotation and the reaction time would have required a more strained, ad hoc

explanation, because there is no reason the algorithm would be more complex

or time-consuming, the greater the degree of rotation!). But the fact is

(almost certainly) that it is not done by a serial, symbolic algorithm in

the brain, for the simple reason that the analog implementation would be so

much more economical and efficient than the serious one in this case.

And so it might be with neural nets: For some problems, real parallelism and

distributedness might be more economical and efficient than any digital

approximation. And in such cases you would be better off with a hybrid model

than a purely symbolic one.

But can we do better than that? Are there cases where it is essential that a

system be analog (e.g., parallel/distributed) rather than simply more

economical and efficient?

One big domain in which such cases can be found is in sensorimotor

systems: systems whose function is to transduce optical or acoustic or

mechanical energy, or to generate movement output. A retina, for examples,

transduces photons, not unlike the way a synthetic photoreceptor does. There

are no computational options for this: If you simulate the light and you

simulate the retina, then you can simulate the transduction too. But if the

system has to deal with real photons, then it has to have real transducers.

Exactly the same is true for motor output systems: In a virtual world,

movement can be simulated symbolically. But if the system has to move around

in the real world, there is no computational substitute for real motor

effectors.

Remember the symbol grounding problem? the problem that squiggles and

squoggles don't have any meaning? that their meanings are just projected

onto them by the mind of their interpreter/users? The proposed solution to

the symbol grounding problem (which had the added advantage of being immune

to Searle's Chinese Room Argument) turned out to be sensorimotor

grounding: The meanings of symbols need to be grounded in robotic

capacities: in robotic interactions with the objects, events and states that

the symbols are interpretable as standing for.

So perhaps it is not coincidental that sensorimotor transduction/actuation

is the paradigmatic case of nonsymbolic function. What might neural nets

have to do with this?

Let's consider the most elementary relation between a symbol and an

object: The symbol stands for the object: "Apples" stands for apples.

(Symbols are objects too, but their physical details as objects are

irrelevant to this; the objects are just used for an arbitrary notational

system.) How is the meaning of "apples" grounded in the objects it stands

for, namely, apples?

Let us consider a system (we won't say yet whether it is symbolic, dynamic,

or hybrid): We want that symbols in that system to be "grounded." We don't

want it to be subject to the criticism: "Its symbols only seem to mean what

they mean because an outside interpreter interprets them as such: It is

nice, and nontrivial, that they can indeed be systematically interpreted as

such, but that is not enough, because without the mediation of the

interpreter, there is no connection whatsoever between the symbols and the

objects they refer to." So how could we design a system that had symbols

like "apples," that were systematically interpretable as referring to

apples, but a system in which the connection between "apples" and apples was

autonomous and direct, rather than being dependent on the mediation of an

outside interpreter.

Who knows? But here is one candidate: Suppose the system was a robot, one

that could go about in the world, and it could learn (just as we do), what

objects are called "apples," and what are the sensorimotor interactions we

can have with them (pick them, eat them, call them "apples"). You could step

aside from such a system and say: "So you think its symbol "apples," just

means apple because I can interpret it that way? Well then let me step

aside, and get out of the loop, so to speak, and you just watch it interact

with the real world of apples, and see whether the connection between its

symbol "apples" and apples is just in my head, or its in its head too!"

First, notice that we now have something much more than just a bunch of

symbols that can be given a systematic interpretation. We have that too, but

not just that. We also have the objects the symbols stand for, and we have

interactions between the system and those objects. And those interactions

are direct and autonomous: It doesn't pick, eat, and name apples only

because I'm interpreting it that way: It really picks, eats and names

apples.

Now a robot that could do that (and only that) would only be an impressive

toy. Even on the purely symbolic side of its capacities, "apples" would not

really be systematically interpretable as meaning apples, because all it

could do was pick, eat and name apples. That is trivial, and could be done

by countless different systems. And if you re-labeled everything, it could

be interpreted as referring bananas, and if you relabeled it yet again, it

could be referring to prime numbers. Which is to say that it really wasn't

even systematically interpretable as referring to anything at all.

This is where the Turing hierarchy comes in: For whereas that criticism

(that it's just a trivial toy) is valid enough for the apple-talk model, it

loses more and more of its force as we scale up toward T2 (on the inside)

and T3 (on the outside, in its robotic capacities). For the symbols inside a

T3 robot would be as grounded as the symbols in the head of its interpreter

(or you, or me).

Where do neural nets come it? They are a natural candidate (though not the

only one) for making the "connection" between (1) the shadow that the

outside object casts on the robot's sensorimotor transducer surfaces and (2)

its internal symbols. If there is one thing that neural nets do well, it is

pattern learning: They can learn the mapping between a pattern and its name

(or, more complexly, a sensorimotor interaction with it, such as taking it,

eating it, beating it up, fleeing from it, mating with it, etc.). For the

connection between object and symbol is not the connection between one

unique input and one unique output. It is the connection between a kind of

thing, and its name.

The pattern falling on a robot's sensorimotor surfaces -- say, its optical

transducer surface -- is usually the 2-dimensional projection of a 3-D

object of some kind. A neural net would have to learn to reliably get from

that kind of optical pattern (for typically, the same kind of object will

cast many potential shadow patterns) to the arbitrary symbol that is the

object's name, and to the many nonarbitrary motor patterns corresponding to

all the kind of things the robot might need to learn to do with the object

from which the sensorimotor projection originates. It is not for nothing

that neural net modeling is also called "connectionism," and the connections

are not just between internal units, but also between analog input

configurations and analog output configurations -- with the possibility of

further internal configurations in between, both analog and symbolic.

Now, in principle, because of the Church/Turing Thesis, the very next stage

after the sensory input projection could be digital and symbolic, all the

way through to the very last stage before the motor output projection, but

the need for even as simple a capability as mental rotation [and T3 entails

a lot more than that] already calls for internal analog projections and

transformations too. So it begins to become clear why a grounded T3 system

would need to be hybrid through and through.

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