Analogy is far more powerful than transitive reasoning; nonetheless, according to the present

analysis it is nothing more than a subtler way of manipulating the pattern distance. I will

introduce three forms of analogical reasoning — structural analogy, modeling, and contextual

analogy — and propose a unified structure for analogical reasoning which encompasses all of

them. It is perhaps not obvious that all conceivable cases of analogical reasoning are included in

this formulation — but I have been unable to find a counterexample. What I am attempting here is

vaguely similar to Aristotle’s list of the seventeen forms of syllogistic logic. He did not prove his

list exhaustive, but it nonetheless served well for a long time.

STRUCTURAL ANALOGY

Consider the following approach to recognizing patterns in an entity x.

1. dS(x%,x) is small.

2. (y,z) is, approximately, a pattern in x%.

3. Thus something near (y,z) is, approximately, a pattern in x.

The justification of the process is obvious: dS is an approximation of d#, and d# is a measure of

difference in pattern. In general, if d#(A,B) is small, then most of the patterns in A are also

patterns in B, and vice versa. And as this distance gets smaller, the reasoning gets more certain.

This process I will call structural analogy; it is the simplest of the three forms. Quite simply,

X is structurally analogous to x% if x and x% have similar structure. Roughly speaking,

structural analogy is the assumption that if x and x% have some similar structures, then they

have other similar structures.

For a less formal example, consider the following experiment, described by Ornstein (1986,

p.159):

A man comes in to give a lecture to students. Afterward, the students are told that he is

either a student or a professor and are asked to rate the lecture. Not surprisingly, they rate the

lecture as being better if a professor has given it, but what is surprising is that they rate the

lectureras taller if he was identified as a professor.

It would appear that what is happening here is an unconscious, erroneous structural analogy. The

students observe that the lecturer has something in common with the loose collection of patterns

that might be called "high status" or "superiority". Therefore they assume, unconsciously, that he

is likely to have other patterns in common with this same collection. This assumption is not

overpoweringly strong, but it is enough to bias their perceptions.

MODELING

Let us begin with a rough example. Suppose one seeks a method of making housing less

expensive (i.e. one seeks a certain pattern in the domain of knowledge about housing, which

domain we may call x). Then one might decide to reason by analogy to the automobile industry.

Automobiles bear some similarity to houses — both are designed to contain humans, both are

manufactured in great number throughout the developed world, both represent fairly major

investments for the average person, etc. There are obviously significant differences also, but the

association is by no means arbitrary. One might reasonably ask: how does one make inexpensive

cars?

In reasoning thus, one would be beginning (Step 1 in the process of structural analogy) with

the observation that the entity "housing" is in some measure similar to the entity "cars". In the

notation given above, if the domain of knowledge about housing is x, then the domain of

knowledge about cars is x%.

Cars, one might then observe, are much less expensive when mass-produced than when

custom-built and custom-designed. And cars are being made cheaper and cheaper by more and

more efficient methods of mass production. This is Step 2 in the process of structural analogy:

one is recognizing a pattern in x%. Finally, one might map this reasoning back to the domain

of houses, searching in the vicinity of "cars are made less expensive through mass production"

and finding "houses are made less expensive through mass production" (Step 3). The validity of

this hypothesis could then be tested by exploring whether it is in fact possible to make the

production of housing less expensive through mass production — by exploring the feasibility of

trucking or air-lifting a pre-assembled house to a given location, et cetera.

This may be interpreted as an illustration of structural analogy: the structure of the housing

industry is somewhat similar to the structure of the auto industry, so further similarities are

assumed to exist. But it also suggests a slightly different form of analogical reasoning, which I

shall call modeling. For in this example the relation between x% and x was exactly the same as

the relation between (y,z) and the pattern finally found in x. That is, if we define a function f by

f(g(cars))= g(housing), then f(inexpensive cars) = inexpensive housing, and also f(mass-

produced cars) = mass-produced housing. This suggests thefollowing general form of reasoning:

1. dS(f(x),x) is small.

2. (y,z) is, approximately, a pattern in f(x).

3. Thus something near (f-1(y),f-1(z)) is, approximately, a pattern in x.

In general, f-1 may be multivalued; it may be a relation rather than a function. This poses a

moderate computational difficulty (Grandy, 1985), which may be dealt with by maximum-

entropy methods like those discussed in Chapter 9.

BRAINSTORMING

Why would modeling analogy work? If f(w) is always structurally similar to w, for any w,

then it is just a special form of structural analogy, and it is justified whenever structural analogy

is justified. Otherwise, however, it is a question of whether the function f "preserves patterns".

There might be some functions f for which Steps 2 and 3 taken exclusively of Step 1 would be a

plausible form of reasoning. In such cases it would follow from "(y,z) is a pattern in x" that

"something near (f(y),f(z)) is probably a pattern in f(x)." But, on the other hand, there are

certainly many functions which do not preserve patterns in this sense.

In some cases, the choice of a function f which does not generally tend to preserve patterns

may be highly valuable. This is one way of understanding the process of "brainstorming": one

transforms a problem x into a context f(x) which is fairly unlikely to have much to do with the

original context, seeks an answer in this context, and then tries to map the answer back. For

instance, in thinking about international combat, one might say to oneself "now, pretend each

country is a little boy, and the various parties of the war are just little boys squabbling." In this

case we would be selecting f so that f(g(nations))=g(boys). Let us assume that one has mapped

the particularities of a war into the particularities of a squabble, and then thought of a way to

resolve the squabble. Then in order to get anything useful one would have to map one’s answer

back into the domain of nations, i.e. one would have to take f-1 of one’s answer. In this case it

would seem that f is only moderately pattern-preserving, and such an experiment would have a

fair but not excellent chance of yielding a relevant suggestion.

Mathematically this form of reasoning may be viewed as the attempt to find a function f,

mapping a given domain D into a given range R, that is approximately "topologically

semiconjugate" to the structure operator S(x). That is, f maps D to R and, ideally,

S(f(x))=f(S(x)). Often R may be understood as a model of D, in the sense that each element x of

D may be associated with an element of R in such a way that the relation of x to other elements y

of D is similar to the relation of f(x) to the elements f(y) of R.

CONTEXTUAL ANALOGY

Only one form of analogy remains. What if x and x% have no inherent structural

commonality, but are related to other patterns in similar ways? One might still reason

analogically that x and x% are similar. Or one might reason analogically from the observation

that x and x% are related to different patterns in similar ways. Let us call this contextual

analogy. Such reasoning is not encompassed under structural analogy or modeling — a new

formula is required. To say that x and x% are related to other patterns in similar ways, is to say

that d#(Em(x%,w%),Em(x,w)) is small. Thus contextual analogy rests on those aspects of x

which do not manifest themselves in the internal structure of x, but which nonetheless emerge

when x is conjoined with other entities.

For instance, suppose w=w% is a codebook which contains several different codes, including

Code A and Code B. Suppose x is a message in Code A, and x% is a message in Code B — and

suppose x and x% both convey the same message. Then x and x% may not appear similar; they

may have no virtually no patterns in common.

To see this, observe that the meaning conveyed by a coded message, say x, is not generally a

pattern in that coded message. If the meaning of the message x is called Mx, and the function

(contained in the codebook) which translates x into Mx is called F, then we may write F(x)=Mx.

But (F,Mx) is a pattern in X only if a%F% + b%Mx% + cC(F,Mx) < %x%, and this is not at all

inevitable. Perhaps %Mx% is less than %x%, but if the codebook is large or difficult to use, then

%F% or C(F,Mx) may be sizeable. And the same could be said for x%.

So in this example, d#(x,x%) may be small, but what about d#[Em(x%,w), Em(x,w)]? Clearly,

Mx is an element of Em(x%,w), and Mx% is an element of Em(x%,w), so that if Mx and Mx% are

equal, this distance is small. Contextual analogy would occur, for instance, if one reasoned that,

because the messages x and x% have similar meanings, they may have come from the same

person.

This example is not at all artificial; in fact it may turn out to be central to mental function. The

mind works in terms of symbol systems, which may be considered as highly sophisticated

codebooks. Street signs are coded messages, as are sentences. Suppose one reasons that sentence

A has a similar meaning to sentence B, another, and therefore that the person uttering sentence A

may have a similar attitude to the person uttering sentence B. This is contextual analogy, because

the meaning of a sentence is not actually a pattern in that sentence, but rather a pattern emergent

between that sentence and the "codebook" of language.

A GENERAL ANALOGY ALGORITHM

One may incorporate all three forms of analogy in one general process:

1. dS(St(f(x%)%w%)-St(w%),St(x%w)-St(w)) is small.

2. (y,z) is, approximately, an element of St(f(x%)%v%)-St(v%).

3. Thus something near (f-1(y),f-1(z)) is, approximately, an element of

St(x%v)-St(v), where v=f-1(v%) is perhaps a good guess.

In the case of structural analogy, f is the identity mapping, and w%, v and v% are the empty

set, but x% is different from x.

In the case of modeling, x=x% and w%, v and v% are the empty set, but f is not the identity

mapping.

In the case of contextual analogy, f is the identity mapping but w% is not the empty pattern; v

and v% may or may not be the empty set. If v and v% are the empty set, then one has the form of

reasoning "x and x% have similar relations to y and y% respectively; therefore x and x% may be

structurally similar." And if neither v nor v% is the empty set, one has the form of reasoning "x

and x% have similar relations to y and y%; therefore they may also have similar relations to v

and v%."

The reason for expressions such as St(x%v)-St(v) is to specify that, for instance, either

patterns in x or patterns in Em(x,v) are desirable, but not patterns in v alone.

The general algorithm — which I will call, simply, analogy — hence includes the three forms

of analogy discussed above. It also encompasses hybrid forms. For instance, if x=x%, but f is not

the identity and neither v, v% nor w% are the empty set, then one is mapping x to a model range

via f and executing contextual rather than structural analogy in this range.

This general idea could be implemented in many different ways. For instance, consider the

following hybridization of structural and modeling analogy:

1. dS(x%,x) is small.

2. Select (y,z) which is, approximately, a pattern in f(x).

3. Check if something near (f-1(y),f-1(z)) is, approximately, a pattern in x%.

If this doesn’t turn out to be the case, and f is reasonably near the identity, then check if

something near (y,z) is, approximately, a pattern in x%.

In this implementation, modeling is considered partly as a tool for searching the space near (y,z).

Each step of the process of analogy involves a significant effort: first, to pick x% or f; second,

to recognize a pattern in f(x%); third, searching in a certain vicinity for a pattern in x, and fourth,

inverting f. It would be possible to execute the latter three tasks by some general optimization

algorithm, and to pick x% or f at random. But in order to reap the full power of

analogicalreasoning, it is necessary to have access to an intelligently structured database. In the

next section, we shall consider the hypothesis that memory is structured specifically so as to

efficiently supply the process of analogy with the data it requires.

ANALOGIES AMONG DIGRAPHS

It should be noted that an entity which characteristically seeks certain types of patterns will

therefore characteristically make certain types of analogies. In other words, different analogies

may occur to different entities, depending on the idiosyncracies of the pattern recognition

algorithms involved. This point of view may be helpful in relating the present approach to other

analyses of analogy.

For instance, Poetzsche’s (1988) theory of analogy deals only with structural analogy and

applies to precisely those entities which:

1) recognize patterns only in digraphs.

2) compute structural similarity using only patterns (x,y) of the form y={v,w},

where v = "construct x by connecting node n1 of z to node m1 of w, node n2 of z to node m2 of

w,…" and * is defined accordingly.

In other words, according to Poetszche’s definitions, two digraphs are structurally similar if they

have a common subgraph.

This scheme deals well with processes that can be easily decomposed into subprocesses. In

particular, Poetszche is concerned with robot motion: his goal is to construct a robot which can

figure out how to complete a task based on knowledge of how to complete similar tasks. Toward

this end, he has designed a robot which internally expresses each task as a digraph of

"elementary tasks". Given a new task, the robot searches its memory for a task with a structurally

similar digraph and then (by structural analogy) uses this as the starting point for figuring out

how to do the task at hand.

Kaynak: A New Mathematical Model of Mind

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