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C00067 00013	@SubSection(Different Meanings of Analogy)
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C00089 00018	@Section(Analogy Applications)
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C00121 00022	@Subsection(Dimensions of Analogy PROBLEM)
C00125 00023	@Section(Conclusion)
C00128 00024	@Appendix(Properties of Analogy)
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@Begin[Center]
@F1[What's in an Analogy?]

Russell Greiner
@End[Center]

@Section(Introduction)

A goal of this overall research project is a computer program 
capable of understanding and using an analogy.
One obvious prerequisite for this task
is a solid understanding of just what an analogy is.
Despite the reams written on this topic, there seems no adequate
(and certainly no "universally accepted")
definition, nor even a "complete" set of characteristics.
This available literature shows, instead, that each person uses his own
(usually unstated) definition for this term.

This report is another stab into this morass.
Following the lead established by @Cite[NaivePhysics],
(as well as @Cite[Isaac] and @Cite[Envisage],)
we describe a "naive view" of analogy.
This semi-formal specification 
includes
many of the"commonly understood" meaning and senses of analogy,
motivated by various examples of each.
Any analogizing program which aspires to achieve a human-like performance
(which is, we conjecture, a necessary prerequisite for being useful,)
should exhibit these characteristics.@Foot{
Two quick (meta-)notes:@*
(1) This report will, hopefully, evolve into a chapter of a much larger
document -- the author's (eventual) thesis.
This may explain both the particular perspective presented,
as well as the numerous "unfulfilled" pointers which appear throughout this paper.@*
(2) While this report attempts to provide a general description of the 
phenomenon of analogy,
my personal biases will undoubtedly appear,
possibly compromising its generality.
Comments -- particularly those which plug obvious gaps noticed --
are actively solicited.}

This "scruffy" (see @Cite(Scruffy)) first pass analysis is in four parts
(plus appendices).
First, Section @Ref(Examples) lists various examples of analogies.
Their breadth and diversity indicates how pervasive, (and hence important,)
this process is to perhaps all of our cognitive processes.
Section @Ref(Categories) then begins to characterize the space of analogies,
based on (generalizations from) these examples.
This section discusses the different senses of the term "analogy",
and what types of tasks are considered "analogizing".
It abstracts from these the primitive operations needed to answer an analogy
question, and verifies their coverage by listing
a (complete?) set of analogy questions.

Next Section @Ref(Applications) considers the purposes of an analogy,
answering the question:
why anyone would use an analogy.
Each of these sections suggested some of the relevant "dimensions"
in the space of analogies,
which could be used as a basis with which to compare
the different types of analogy.
Section @Ref(Dimensions) first succinctly reiterates these axes,
and then lists a few other dimensions as well.
Here many of the examples given in Section @Ref(Examples) are
shown to represent widely-divergent points on this analogizing space.

Each of the appendices elaborates some point(s) which the core text
glossed over.
Appendix @Ref(Properties) presents some
properties which can be used to characterize analogies.
Appendix @Ref(Reform) demonstrates why (we feel) it is necessary to include
a reformulation step when attempting to understand a given analogy.
Appendix @Ref(Analogy-Vocab) gives some standard English phrases which
help delimit an analogy.

@Section(Smattering of Examples)
@Tag(Examples)

Each of the examples shown below is, by common definition, an instance of
an analogy/metaphor/similarity.
(This list includes various linguistic tropes, such as metaphor and simile,
along with other forms of analogy.  The reflects our view that
all of these uses are applications of the same general analogizingin process.)
The particular clusters are is designed to demonstrate some general application,
which is spelled out immediately before that set of examples.
(The following Section @Ref(Categories)
presents other, more significant classifications.)

As mentioned above, 
we feel any reasonable model of analogy 
must be able to cover (that is, be able to explain) this full range of examples.
The definition of analogy proposed later in this thesis@Foot{
But NOT in this NaivePaper report.}
is indeed capable of "solving" all of these cases.

@BEGIN(Enumerate)
@BEGIN(Multiple)	@TAG(Comparison)
@B(Used for Comparisons)@*
@i{Implicit question:} "Is A like B?" or "How is A like B?"@*
The analogy establishes a connection between two models.
The underlying purpose is to explain some fact(s) about B.@*
@i{Variables:} The explicitness of these comparisons,
the "size" of the compared models, ...@*
Notes: Once the analogy is understood, it can be thrown away.

@BEGIN(ENUM1)
@i<similarity metaphor>@*
The description omits not only the basis of the comparison,
but also the fact that this is, in fact, only a comparison.
@BEGIN(ITEM1)
People are birds.

She's a packrat.

He was dynamite!
@END(ITEM1)

@i<simile>@*
Here the sentences include a word like "like", 
to indicate that this is a comparison.
@BEGIN(ITEM1)
John is like a bird.

Computers behave like people.

"West Side Story" is like "Romeo and Juliet".

"Love's Labor's Lost" is like "King Lear".
@END(ITEM1)

@i(Only one feature mapped over)
@BEGIN(ITEM1)

"Learning at CalTech is like trying to sip water from a fire hydrant."@*
i.e. a lot is forced into the student, under great pressure.

Consider almost any instance of a "that reminds me of" situation
-- where connection may be obscure to everyone
(possibly) excluding the speaker.  In general
the topic of the current conversation and the digression share
some single common feature.
@Comment{On hearing that B, B and N went to BBN -- they found it convenient
that the (existant) company already had the right name,
I remembered the famous case of the Shell sort:
"Shell" does not refer to some nuance of the operation,
it is, instead, the name of the fellow who first suggested this approach.}
@END(ITEM1)

@i<idiom>@*
Some stereotypic fact, associated with the idiom, is transfered to the referent.
(Like the case above, only that one fact is mapped over.)
@BEGIN(ITEM1)
"... his Achilles tendon ..."@*
i.e. that is his single penetrable part.

"... hanging like a sword of Damacles ..."@*
i.e. temporarily safe, but in a very tense situation.

"... like a fiddler on the roof..."@*
i.e. this considers day to day existance as a delicate operation,
in precarious situation.
@END(ITEM1)

@i<equation>@*
In this degenerate case of analogy, 
all the information needed for the comparison is made explicit.
@BEGIN(ITEM1)
John ate as many sun-flower seeds on June 24 as Polly parrot ate that day.

The nucleus is in the center of an atom, just as the sun is at the center of
the solar system.
@END(ITEM1)

@END(ENUM1)
@END(Multiple)

@BEGIN(Multiple)	@TAG(Predication)
@B<Used for Predication>@*
@i{Implicit question:} Given that A is like B, and P(A), is P'(B) true?@*
The analogy is used to understand properties of one object, based on facts
about another.@*
@i{Variables:} the "completeness" of the model -- from quite thorough through just
the few facts needed to be carried over,
the size of the models (varying from a single situation or example,
to a complete known, general model.)@*
@i{Notes:} In these situations it is often useful to keep the analogy,
as it may be a source of new conjectures/facts about B;
in general the new assertions about B will be merely plausible conjectures,
rather than guaranteed true facts;
in some (rare) cases the analogy may work both ways -- one might notice
new facts about A based on facts about B.
(@Cite(Interfield) makes this point.)

@BEGIN(ENUM1)
@i(Large and Comprehensive)@*
Here the models involved are large 
-- such analogies are used to define an entire scholarly pursuit.
@BEGIN(ITEM1)
The atom is like a miniature solar system.@*
I.e. electrons revolve about a heavy nucleus, like the planets about the sun.

Electricity is like water flow.@*
I.e. batteries can be likened to dammed resouviors.

Quantum mechanical particles behave as elements in an abstract group.@*
I.e. ?? predication of new particles ??

The genetics concept of a "gene" shares many properties with the biochemical concept
of chromosomes.@*
<take something from Lindley's Interfield Connections paper>>

The `Solar metaphor' theory used to explain the creation of myths.@Foot{
This theory held that people would use anecdotes to describe the movement
of the sun in the heavens --
phrasing its movements in terms of the
life story of a particular god or hero.
This view was popular @i(ca.) 1890-1910.  See @Cite(SolarMetaphor).}@*
I.e. Hercules would peak at the "noon" of his life, then descend.

Physiologically, people are like electronic curcuits.@*
I.e. the many loops in our endocrin system would behave much like
feedback curcuit.
(In particular, the same types of problems would arise when the loop becomes open, ...)

Cognitively, people are like computers.@*
I.e. we have interrupts, etc. (see the list of borrowed terms, shown below.)
@END(ITEM1)

@i(Moderate in Size - Interfield)@*
These comparisons help to refine (but not define) a domain.
Here the "other" analogue is outside the current domain.
@BEGIN(ITEM1)

"How is a Knowledge Representation language like a piano?"@*
(both need to be tuned and refined)

"Metaphor is like a Solar Eclipse"@*
(see @Cite(Ortony1), p169)

Music is like poetry.@*
Both are art forms, designed for esthetics, and based on symmetry and ...

Music (as sequences) are like math sequences.@*
There is a regularity in both -- leading to a predictive behaviour.

Doing reseach is lke climbing to the top of a mountain.@*
(Take any path which leads to the top, attending only to things on the
"critical path."
Hence everything to the side, or behind you, is ignored.)

Doing reseach is like constructing a building.@*
(Each and every layer from the foundation up should be solid.
This means it may be worth the time now
to fix known flaws on any underlayer, as they will only be harder to
reach and correct later.)

Consider three models for a text editor:@Foot{
This example is taken directly from @Cite(R&N).}
@BEGIN(ITEM1)
Secretary@*
Pro: either will follow your instructions in
performing the tasks you prescribe.
(This includes "taking dictation", etc.).@*
Con: The "still in append bug" -- an editor can't distinguish commands from text.

card file@*
(That is, the text file being editted is like a file of cards.)@*
Pro: Permits insertion of text into arbitrary places; multiple files; ...@*
Con: There can be other sub-units (not just cards);
nothing like a global Substitute, ...

tape recorder@*
(That is, the file is like the tape, on the recorder.)@*
Pro: Handles overwriting.  Insertions are like splicing, multiple tapes, ...@*
Con: Still no global finds/substitute commands, ...
@END(ITEM1)

HPP Executive Council is like a council of barons.@*
Each has absolute say within his own project, except in cases when
such decisions influence the rest of the empire.

@COMMENT{Broken glass metaphor" for initial state of EMYCIN --
delicate, precarious, reaching thru...

Diamonds from coal -- can be extracted, once the rest has been
cleared away.  Even then few and far inbetween; nonetheless worth
extracting.}

@END(ITEM1)

@i(Moderate in Size - Intrafield)@*
Here we compare two objects from the same domain.
@BEGIN(ITEM1)
A circle is like a sphere.

Recursion is like iteration.

Abstraction is like simplification/analogy...
@END(ITEM1)

@i<Instance to Instance>@*
Here the "models" are at the level of a single instance.@Foot{
This may be consider the standard case of analogy.
In many cases the novel object is a specific situation 
(usually the current focus).
When the vehicle (@{i.e.} the already known analogue) is also specific,
this is considered "learning from example".}
@BEGIN(ITEM1)
Patient#75 today is similar to Patient#75 a week ago, but ...

LegalCase#92 is like LegalCase#53, except ...

This "group" proof is just like that "ring" one. 
(@i(c.f.) @Cite(Kling).)

@END(ITEM1)
@END(ENUM1)
@END(Multiple)

@BEGIN(Multiple)	@TAG(ProblemXForm)
@B<Problem Transformation>@*
@i{Implicit question:} Given a problem A, 
find a similar, (but easier to solve,) problem B.
(Then solve B, and map the result back to A.)@*
@i{Variables:} Constraints on the space from which to find a suitable B,...@*
@i{Notes:} This is a super-task of the above category -- in that we have to first
find the analogue, and then use it in the manner shown above.@*
We may consider solving a problem as travelling from a starting position, A,
to a destination, B.
Problem transformation suggests taking the circuitous route
from A up to @G(t), over to @G(w), 
(i.e. solve the problem in that space,) and then map down to the "answer" B.
Picturally,
@BEGIN(Example)
        @G(t)     @G(w)
        @Z(s) -@Z(g)- @Z(Y)
        |      |
        .      .
        A      B
@END(Example)

@BEGIN(ITEM1)
To generate a program to perform task X, find one which performs a similar
task, X', and modify that program appropriately.
See @Cite(Evolution) or @Cite(M&Ua).
@Cite(RBrown) addresses the same problem, but does the mapping from the plan
for a program, rather than the code itself.

Consider the following problem from @Cite(Polya1):
Given two points lying in one of the half planes defined by a line.
Find the minimal path joining these point, subject to the constraint
that that path must go touch the line.@*
Solution: Reflect one of the points about the line -- then draw a STRAIGHT
line to the original point.

@BEGIN(Multiple)
This are certain standard problem transformations, which use certain mappings.
They are all alike in that they can be described in terms of the diagram
above.
(NOTE: we are NOT claiming, here, that A and @G(t) are analogous.  
In the below cases they are totally different problems.)

Consider A to @G(t) mappings of
@BEGIN(ITEM1)
hopping to the Meta-level

simulating another system@*
[this occurs often in MTC and logic proofs --
eg showing NP-completeness, Turing equivalence, ...]

finding mapping between pairs of theories going through some other model
@END(ITEM1)
@END(Multiple)

@END(ITEM1)
@END(Multiple)

@BEGIN(Multiple)	@TAG(ProblemRestate)
@B<Problem Restatement>@*
@i{Implicit question:} Find an alternative representation of problem A 
(which renders the problem easier to solve).@Foot{
It has been claimed that many of the significant advances science has made
have been via restructuring of known problems into tractable forms
(? Polya, Kuhn ?).}@*
@i{Variables:}
This new language may be totally different from the original descriptive
language, or may just involve small embellishments.
Also, the transformation may be information preserving, or may abstract
out certain facts...@*
@i{Notes:}
This is the critical step in problem reformulation.
@i(N.B.) we are still solving the same problem, out in the real world --
but now we have a better, more apt, handle on that situation.
(This is NOT true in the previous problem transformation case 
-- there, a different problem was solved.)@*
In the "correct" representation,
it is easy to see the differences and similarities connecting the two analogues.
Ideally, the differences can be seen as simply substituting one value of
a parameter for another.  (Many of these points are explicated in
Appendix @Ref(Reform).)

@BEGIN(ENUM1)
@i(Fixed Algorithmic Transformations)
@BEGIN(ITEM1)
Transposing a musical piece into a different key.

Switching from a rectangular to a spherical coordinate system.

"Visualizing" a problem -- using lines rather than equations,
or charts, ... @Foot{
This was discussed in @Cite(Simon-Repn) -- where it was claimed that experts
used apt representations, which let the answer be "read off" the diagram.}
(Note we present a 2D reformulation of problem transformation below.)

Our visual system does a lot of work transforming the initial retinal image
into higher level, useful information.
E.g., we readily recognize a friend,
independent of his current position and orientation relative to our eyes 
-- the received retinal images is automatically
"normalized" (or canonicalized).@Foot{
The work of @Cite(HubelWeisel) shows that much of this is learned --
as opposed to hardwired from birth.}
@END(ITEM1)

@i<Heuristic Methods>
@BEGIN(ITEM1)

Playing a musical piece with a different instrument.@*
(Or playing it in a different style -- baroque rather than classical.)

@Cite(Amarel)'s Missionary & Cannibal reformulations

Adding a new point or line in a geometric proof. 
(E.g. dropping a perpendicular, forming an object which was NOT stated in
the problem itself.
This issue was addressed by @Cite(RBrown), which considered
this ability as a critical needed step -- i.e. it was often the
factor which limited how well a program could perform.)

Defining a new slot, or relation, in terms of existing ones --
and then mapping over facts from those.

Consider the standard representation of a fraction as a piece of a pie.
This is quite useful when teaching how to add and subtract fractions.
(This is mentioned in @Cite(R&N).)

This is closely tied with the representation issues discussed in @Cite(ArchMRS).
Here the task is to find a (possibly new,) representation for a given problem,
in which a certain set of queries can be solved efficiently.
(This is analogous to much of the Analysis of Algorithms work -- see
@Cite(AHU).)

@END(ITEM1)
@END(ENUM1)
@END(Multiple)

@BEGIN(Multiple)	@TAG(Literal)
@B<Literary uses, to conjure images and to exploit 
(often cute, almost serendipity) coincidences.>@*
@i{Implicit question:} The goal is to describe some feature, P(x), of a certain
situation, B.
It is achieved by finding a commonly known object, A, which has a pronounced
salient feature corresponding to this P.@*
@i{Variables:} <varied -- except the salient features of the
vehicle must be fairly obvious>@*
@i{Notes:}
this can be used for similarity as well as proportional metaphors --
but is done with considerably less frequency.
Basically, the transference should be obvious -- i.e some particular
property should be mapped over.  This feature should be easy to understand,
else the cleverness of this case will be lost -- very counterproductive.
(This point is also discussed in @Cite(Gentner), p.45-47.)

@BEGIN(ITEM1)
"Now is the winter of our discontent;@*
Made bright by the sun of Bolingbrook" - or something like that
@Cite(Shakespeare)

"Oh, what a rogue and peasant slave am I"

"It is the east, and Juliet is the sun"
@END(ITEM1)
@END(Multiple)

@BEGIN(Multiple)	@TAG(Proportion)
@B<Proportional Metaphors>@*
@i{Implicit question:} Find @i{y} which is to @i{B} as @i{x} is to @i{A}.@*
@i{Variables:} How constrained the possible values of @i{y} are,
(in simple, artificial cases, there are only a few known values allowed for @i{y}.
In general, there will be no additional constraints imposed on that new object.)
How explicit the @i{A} term is; whether the speaker realizes he is using a metaphor,
...@*
@i{Notes:}
This can be used either to specify an object, @i{y}, or to communicate some
additional properties of this @i{y}.
There can be a full collection of terms borrowed over 
-- see especially the @Ref(Lak) case below.

@BEGIN(ENUM1)
@BEGIN(Multiple)
@i<explicit question>
@BEGIN(ITEM1)
"a ewe's calf"@*
I.e. Cow : Calf :: Ewe : ?@*
(note that cow was implicit)@*
? = Lamb

"the phonemes of music"@*
Language : Phoneme :: Music : ?@*
(note that Language was implicit)@*
? = Interval, Timbre, or ...

Who is the first lady of England?@*
(note that US was implicit)@*
Find @i{?} such that  US : First-lady :: England : @i{?}.

<Any of the geometric puzzles @Cite(Evans) examined.>@*
Here everything is explicit, and the range of accepted answers is limited.

@END(ITEM1)
@END(Multiple)

@i<Extending a term for its "native" domain to a foreign one.>@*
@Tag(Borrow)
Here a familiar term is used in an inappropriate context; and
its extended "metaphoric" meaning is extended (loosened)
by figuring how some facts about that known object 
could be transfered into this other field.
@BEGIN(ITEM1)
Consider "@i(tree)" --
Originally a botanical term, it is now used to refer to almost any
instance of a hierarchy.
Consider family trees (Geneology),
language trees (Linguistics), 
the data structure "tree" (Computer Science)
[Notice it often brings along the terms @i(root, leaves, branchiness),
but in all cases, neither its subspecies, nor bark.]

Terms like @i(Anatomy, Physiology, Diagnosis)
once applied only to the study of people, but now have been applied to
arbitrary physical systems (such as computers) as well.

@i(Channels, bugs, swapped out, processing) went the other way -- from
a describing only a particular device (a computer) to apply to people.

Notice that almost any use of scare quotes means the embedded term is really
metaphoric.  Consider `@i("native")' above.
@END(ITEM1)

@i<antequation>@*
@TAG(Anteq)
Many terms are no longer even regarded as a metaphor 
-- i.e. these meanings of the terms have been totally assimilated into the
language.
@BEGIN(ITEM1)
the @i{leg} of a chair

the @i{?} of ?
@END(ITEM1)

@BEGIN(Multiple)
@TAG(Lak)
@i<Sub-conscious use of metaphors>@*
In the above "standard" above case of borrowed terms,
the user is assumed aware that this usage is metaphoric.
Like the @Ref(Anteq) case, though,
the speaker does not even realize that this is only a metaphor.
Other than that, these cases are quite similar to the @Ref(Borrow) case.
In particular,
such systems of metaphors are largely consistent, systematic, and coherent.@Foot{
This observation was taken from @Cite[Lakoff].
which listed many cases where an entire collection of terms were
taken from one application and applied to another.
It is necesary for both the topic and vehicle
(using the nomenclature defined in @Cite(Paivio))
to satisfy a common theory (called "ground".}@*
One might conjecture that we people have some
special purpose internal hardware which has been "evolutionally"
tuned to handle certain common situations (eg Up vs Down),
which is here being applied
to cases which require the same type of efficient reasoning -- for any
linear ordering, such as degree of happiness.
Such efficient routines are faster than the alternative approach --
of reasoning from the underlying theory.  This permits efficient algorithms,
developed for one model, to be applied to a different model -- where
that model satisfies the same theory.
(This theme is further exploited in @Cite(ArchMRS).)}

@BEGIN(ITEM1)
"he was feeling down"

"time is money"

"argument is war"

"ideas are playthings"
@END(ITEM1)
@END(Multiple)

@BEGIN(Multiple)
@i(Mathematics)@*
Mathematicians often extend familiar functions to apply to new domains:
The @i{+} operator was originally meaningful only over reals,
but now this same symbol,
and much of its associated semantics 
(given below,)
have been borrowed by other (mathetical sub)fields 
-- such as fields, matrices, transfinite ordinals, sets, @i{et cetera}.
The times operator, @i{x}, has been similarly extended.
There seems an underlying unity to all uses of a given symbol:
for example
@i{+} traditionally has the following properties:
its arity is 2,
it is commutative and associative,
there is a zero element (another metaphor),
terms invariably have a unique inverse,
it distributes over the multiplication operation (when multiplication is defined),)
and it is over used to define that multiplication operation.
If there is a binary non-commutative operator, it is usually @i{x} 
-- consider non-abelian groups, matrices, or cross product.

Are there other examples of non-linguistic applications?
@END(Multiple)

@END(ENUM1)
@END(Multiple)

@BEGIN(Multiple)	@Tag(Nary)
@B(Familiar resemblence)@*
@i{Implicit Question}: Given {A@-<i>}, find how they are all similar.
(Note there may be no single pairwise commonality -- see @Cite(Wittgenstein).)@*
@i{Variables:} Presence of "negative examples" to curtail size of commonality,
existence of single shared common feature, ...@*
@i{Notes:}  This is clearly related to the question of membership --
given this class {A@-<i>}, should a new object, B, be considered a member of
this?  (e.g. was this work really composed by Bach, or written by Shakespeare,
or sculpted by Rodin, ...)

@BEGIN(ITEM1)
<Any example from @Cite(Baumgard).>

All works by Bach.

All works by Shakespeare.

All recordings of a given musical piece -- by various conductors with
different orchestra, different versions of different instrument sets,
with different styles...

Different senses of "game".

Different senses of "analogy".  (See @Ref(Senses).)

Retinal images of the same object.@*
Here we regard all (reasonable, non-degenerate) retinal images
corresponding to the same 3D object as analogous to one another.
(Of course we are not conscious of this process, having
rather powerful, parallel hardware to do much of the work.)@Foot{
The moral here is that in the correct representation,
analogous things will be be "seen" as being identical. 
(See Appendix @Ref(Reform).)
Indeed neurophysiologists have (seriously) talked about a "Fred" neuron,
which is triggered by the sight of your friend Fred.}
@END(ITEM1)
@END(Multiple)

@END(Enumerate)

@Section(Observations -- Types of Analogies)
@Tag(Categories)

There are several observations we can make from the examples.
We can approximate the general analogy question as 
@BEGIN[Quotation]
What does it mean to state that 
"A is like B (possibly, within the constraint @G(a))"?
@END[Quotation]
In general this mapping is non-decompositional -- that is,
one need not explicitly specify which properties A and B share.
In this respect it is similar to the general process of reasoning about
one thing based on a familiar model of some other thing.
Appendix @Ref(Analogy-Vocab) lists some of the standard descriptions
used for further specifying the type of analogy sought.

This section presents additional specifications of analogy,
by quickly overviewing some of the dimensions which can be used to
specify an analogy.

@SubSection(Different Meanings of Analogy)
@TAG(Senses)
First, there are three, clearly distinct senses of the term "analogy".
(While we tried to leave the listing and our choice of apt headings
relatively unbiased,
this distinction was impossible to avoid.)
These case are:
@BEGIN(Itemize)
@i{Similarity case}@*
@i{Given} objects A, B, and formula P(@G(u)),@*
@w(         )where A satisfies P(@G(u));@*
@i{Find} a predicate P'(@G(u)) which is like P(@G(u)),
and which is satisfied by B.

@i<Proportional@Foot{
The term "analogy" comes from the Greek word ?@G(analogy)?,
which means "proportion".}
case>@*
@i{Given} objects A, B, and x,@*
@w(         )where x bears some relation (call it R(@G(u),@G(m)),)
to A -- that is, R(A,x));@*
@i{Find} an object y which bears a similar relation to B.@*
This requires finding a R'(@G(u),@G(m)) which is similar to R(@G(u),@G(m)),
of course.@Foot{
@Tag(Unify)
We could "unify" the proportional case into the similarity case by
simply regarding the
R(@G(u),x) as the one place predicate of @G(u), P(@G(u)),
which is mapped onto another unary predicate, P'(@G(u)),
which corresponds to R'(@G(u),y).
These x and y serve to define the relations R and R' respectively.
Two final notes:@*
(1) @Cite(Miller) uses a similar description.@*
(2) Appendix @Ref(Reform) suggests why these two senses of analogy seem
so similar, by proposing one way of blurring the distinction.}

@i<Familiar Resemblence@Foot{
This is a form of "rough classification".
See also @Cite[Wittgenstein]'s discussion of "family resemblences."}>@*
@i{Given} objects {A@-<i>},@*
@i{Find} how they are all similar (possibly within some constraint).@*
[This may be regarded as finding a set of unary predicates,
{P@-<i>}, such that P@-<i>(A@-<i>), where these P@-<i>s are all similar.]
@END(Itemize)

@Subsection(Type of Analogizing Task)
@TAG(TaskTypes)

Another obvious dimension (orthogonal with the senses mentioned above)
is the type of task.
There are several types of task which require generating an analogy.
@BEGIN(ITEMIZE)
Find the "analogy"

Find an "analogue"

Judge/Select analogue(s)
@END(ITEMIZE)
(In all three cases there may be some @i(a priori) specification which
restricts the allowed answers.  We will say more about this later.)

We might (naively) think of these tasks as @i(generating) the analogy.
There are two other types of tasks which use the derived analogy --
taking it as input, along with the pair of analogues.

The known analogy can be used to
@BEGIN(ITEMIZE)
derive (i.e. conclusively deduce)

suggest (i.e. plausibly conjecture)
@END(ITEMIZE)
a new fact (or conjecture) about B, based on some fact(s) known about A.
@SubSection(Analogy Primitives)
@TAG(Primitives)

There seem too basic analogy-related operations:
analogy generation and analogy use.@Foot{
There is nothing terribly deep about this generation/comprehension
dichotomy -- it is identical to the split found in natural language programs.}
We will show in the next Subsection @Ref(AnalQuests) that
(combinations of) these two primitives
are sufficient to answer any of the analogy questions.
This subsection will attempt to describe these two actions.

The "analogy generation process"
takes as input the two (proposed) analogues,
A and B, possibly augmented with some contraint, @G(a), and generates
a reason, @G(b)@-(AB), why A and B should be considered analogous.
(In general this reason may be consider a mapping of the various parts
of A onto parts of B.)
The "analogy understanding process", uses these reasons or mapping.
It takes the "vehicle" analogue, A, and the reason @G(b)@-(AB),
and uses them infer (or simply conjecture) some new (i.e. previously unrealized) 
property of "topic analogue", B.
(Actually, it need not be given the other analogue, B.  
Indeed, in one case its task is actually determining this B.)

A few notes:
@BEGIN(ITEM1)
The intended use of the analogy 
(i.e. if it is to be used to find a proportional term,)
can help guide the generation of the analogy.
Such information is communicated via the constraints passed to the generator.

Constraints and reasons seem very similar
-- perhaps even interchangable.
For this reason they should probably be expressed in the same language.

The triple <A, B, @G(b)@-(AB)> map fairly nicely in the <vehicle, topic, ground>
triplet @Cite(Paivio) discusses.  
(Of course many other researchers discuss these "objects" as well.)

We may have to extend the analogy generating primitive to take an N-tuple of
analogues, rather than just two, to handle the familiar case.
(I.e. I'm not sure we can simulate this N-ary performance using only this
binary operator.)

@BEGIN(Multiple)
It was misleading to word the first list of tasks
mentioned in SubSection @Ref(TaskTypes) above as "analogy generating",
and the other two as "analogy using".
Among other problems, this seems to lead to a funny asymmetry --
why are the only analogy understanding tasks involved with similarity analogies,
and not with the two other senses mentioned in the previous Subsection
@Ref(Senses)?

While the division is correct, the tasks have been mis-assigned.
We will see in the next subsection that each of the latter two types of 
"generating" tasks
(@i{viz.} "Find analogue" and "Judge/Select analogue(s)")
exploit the derived analogy, after it has been generated.
@END(Multiple)
@END(ITEM1)

@SubSection(Analogy Questions)
@TAG(AnalQuests)

This section has mentioned two of the dimension of an analogy --
its distinct senses and its task types.
This subsection shows the "cross product" of these axes,
summarized in the "table" below.
Each entry includes the canonical question for this case, a possible paraphrase
of that question, and one (possibly naive) way of express this in terms of
the two primitives shown above --
generating reasons within an a set of constraints,
and deducing/conjecturing properties based on such reasons.
It also includes a few examples of questions which require this operation.

Note that the most interesting and useful analogy tasks require both generation
of an analogy connecting two analogues,
and a use of this result.

(Note each of the questions is worded as a find or prove query.
We might "invert" the question, to ask, for example,
whether A is like B for reasons @G(b)@-<AB>.)

@B(NOTATION:)
@BEGIN(ITEM1)
The phrase "within the constraint @G(a)" 
refers to an
@i<a priori> pre-specification for the analogical match.@*
(Note this restriction will not always be 
explicitly mentioned in the problem statement.) 
When the restriction arises because we are trying to match proportionally,
with respect to x, we will write @G(a)@-<x>;
and when both proportional terms are known, we will write @G(a)@-<xy>.

"for reasons @G(b)" indicates the nature of this analogical mapping.@*
@G(b)@-<AB> will usually denote the reasons why A is like B.
@END(ITEM1)

@BEGIN(ITEM1)
@B(<Tasks which require the analogy to be generated.>)
@BEGIN(ENUM1)

@i(Similarity:)
@BEGIN(ENUM1)
[Find analogy]@*
Given analogues A and B, and some constraint @G(a), explain how B is like A.@*
@i{Algorithm:}
Derive the reasons @G(b)@-<AB> why B is similar to A.
Note these reasons must be within @G(a)'s specification.@*
@i{Ex:}
Should Law-Case#47 be a precedent for the current case?@*
Did the Quaker song `Simple Gifts' evolve from the same theme as
Copland's "Appalachian Spring?"@*
Was Brittan's `A Young Person's Guide to the Orchestra' derived from Purchel's `Abdelezar'?

[Find analogue]@*
Given A and some constraint, @G(a), find (or construct) some B which is like A.@*
@i{Algorithm:} 
Use the constraint @G(a) to deduce/conjecture suitable properties for this B.
Then use this intensional description to find an appropriate B.@*
@i{Ex:}
Has anyone designed a chip like this before?@*
Compose a Bach-like piece.

[Judge/Select analogue(s)]@*
Given A (and some constraint, @G(a),) find the B in {B@-<i>} which is most like A.@*
@i{Algorithm:}
Derive first the reasons why A is like each B@-<i>, @G(b)@-<Ai>.
Then rank these, to find the best B@-<i>.@*
@i{Ex:} This current case is like both Cases X and Y.
In Case X, the defendant was acquitted, whereas the defendant was convicted
in Case Y.  Which precendent should be followed?@*
Of all recent composers, whose works are closest to Bach's?
@END(ENUM1)

@i(Proportional:)
@BEGIN(ENUM1)
[Find analogy]@*
Explain how x is to A in the same manner y is to B.@*
(Find a relation which takes <x,A> and <y,B>.)@*
@i{Algorithm:}
Find the reasons, @G(a)@-<AB>, why A is like B,
subject to the constraint that this is used as a proportional analogy,
@G(b)@-<xy>.@*
@i{Ex:}
Explain why Cow : Calf :: Ewe : Lamb.

[Find analogue]@*
Find a y which is similar to B in the same way x is similar to A.@*
(Given x in some relation to A, find a y which has a similar relation to B.)@*
@i{Algorithm:}
Find the reason, @G(a)@-<AB>, why A is like B,
subject to the proportional constraint @G(b)@-<x>.
Use these reasons to deduce/conjecture a new fact (conjecture) about B:
the value of the slot which, in A's case, was filled with x.@*
@i{Ex:}
Who is the first lady of England?

[Judge/Select analogue(s)]@*
Given x in some relation to A, find the y in {y@-<i>} which has
the most similar relation to B.@*
@i{Algorithm}
Find a set of reasons, @G(a)@-<Ai>, telling why A is like B,
subject to the proportional constraint @G(b)@-<xy@-{i}>.
Then rank these to find the best y@-<i>.@*
@i{Ex:}
Does it make more sense for the next term in the progression
@QUOTATION(1 2 6 ?)
to be 24, or 30?
@END(ENUM1)

@i(Familiar Relations:)
@BEGIN(ENUM1)
[Find analogy]@*
Given {A@-<i>}, find how they are all similar.@*
@i{Algorithm}
??? Think more about this case ???
1. Find the most specific unifying reason which explains each
@G(b)@-<ij> = the reason why  A@-<i> is like A@-<j>, (given ? constraint --
perhaps of all of the A@-<i>s?)@*
2. Sequentially find the reasons A@-<i> is like A@-<j>, @G(b)@-<ij>,
subject to the constaint of the preceeding pair.
The final set of reasons will be the most specific general reasons why
these things are similar.@*
3. This requires recursion: Find the @G(a)@-<ij> as above.
... <<<HERE>>>@*
@i{Ex:}
How do all Bach pieces resemble each other?@*
Explain problem 12 in @Cite(Baumgard).@*
What do all character "a"s have in common? (From preface to @Cite(Inversions).)@*
Why are all the different senses of "game" denoted with the same word?@*
Is there any common element to this collection of
examples of familiar resemblences?

[Find analogue]@*
Find a new B which should be included in {A@-<i>}.@*
@i{Algorithm:}
First find the reasons why these {A@-<i>} are similar, @G(b)@-<i>,
subject to the constraint @G(a)@-<B>, that we will want to consider B's
admission to this group. ... THIS DOESN'T WORK!@*
@i{Ex:}
Is this an authentic Rodin sculpture?

[Judge/Select analogue(s)]@*
Which B in {B@-<j>} fits best in {A@-<i>}?@*
@i{Algorithm:}
?
@i{Ex:}
Which (of the following) contemporary interpretations of Shakespeare's "King Lear"
is closest to what the original Elizabethian audience would have seen?
@END(ENUM1)
@END(ENUM1)

@B(<Tasks which use the given analogy.>)
@BEGIN(ENUM1)
[Exploit analogy - Deduction]@*
Given A is like B for reasons @G(b), derive that P'(B) is true, when P(A).@*
@i{Ex:}
Russ is like Jock in that both are people; as Jock has two arms, so does Russ.

[Exploit analogy - Conjecture]@*
Given A is like B for reasons @G(b), conjecture that P'(B) is true, when P(A).@*
@i{Ex:}
Given that water systems are like electrical systems,
there should (could) be some electrical component which serves the role
of a dam (i.e. which stores the charge, rather than water).
@END(ENUM1)
@END(ITEM1)

@Section(Analogy Applications)
@Tag(Applications) 

We now consider why someone might want to generate or use an analogy.
Analogy seems to serve two basic functions:
linguistic and deductive/predictive.
This subsection presents a quick summary of these two applications.

@SubSection(Capsule Summary of Applications)
@BEGIN(Itemize)
@BEGIN(Multiple)
"Linguistic"@*
@BEGIN(Itemize, Spread=0)
@b(Motto:) @i(To communicate a lot quickly, using common "ground".)

@b(Scenario:)
A speaker, S, describes B to the hearer, H.
S does so by telling H that B as being like A (possibly for reasons @G(a)).

@b(Preconditions:)
H knows a great deal about B and relatively little about A;
while S knows a lot about both A and B.
(S must also know of H's knowledge of A.)

@b(Purpose:) To quickly relay a bundle of facts about B to the hearer
(these facts are A's B-ness).

@b(Subcases:) Explanation, (exegesis),
teaching a new idea,
elaborating (filling in) an unfamiliar case.

@b(Near miss:)  If H already knew that both A and B were instances of the
same abstraction, (and hence he would know of "real" nature of commonality as well,)
the deductions implied by this analogy would reduce to straightforward
guaranteed inference. (See footnote number @Ref(LingCats).)

@END(Itemize)
@END(Multiple)

@BEGIN(Multiple)
"Deduction/Prediction"@*
@BEGIN(ITEM1)

@b(Motto:) @i<There is (probably a solid) reason why A is like B.>

@b(Scenario:) A reason, R, knows that A and B are similar.
R can then conjecture hypotheses about B, based on corresponding facts about A.
[Worded another way, R, knowing that B is like A in some ways,
asks "why not in this other way?".]

@b(Purpose:) To @i{deduce} or @i{conjecture} some assertions about B.

@b(Preconditions:) R knows that A is similar to B, and a body of
facts about A.
In general R knows a lot about B, and a little about A.  
It is helpful to know in what ways A is like B.

@b(Subcases:) Establishing previously unknown facts about A,
deciding that A cannot have property X because B does not,
realizing that you don't know about something about B (or about A),
B satisfies some of the equations A satisfied,
there may be subcases of B which correspond to known subcases of A,
anything which is like A should be similar to B, ...

@b(Near miss:)
The "strength of the conviction" of the similarity can vary tremendously.
The deductive case, when R has enough facts to positively conclude something
about B, seems only borderline analogy.
It is distinguished from vanailla deduction in that
we insist that
R must actually store the "analogy" pointer from B to A, 
(probably labelled with the reasons,)
rather than simply the results of the deductions performed by virtue of this
connection.
Compare: "Fred eats the same quantity of food as Polly Parrot" vs "Fred eats 
3 lbs of food a day, where Polly Parrot is known to eat 3 lbs a day".
(The @G(b)-structures mentioned in @Cite(Merlin) is a simple example of this.)
@Foot{
Notice what this implies about analogies in general:
their function is subsumed 
by simple algorithmic methods (such as inheritance or instantiation)
when enough is known.
In general analogy is a largely heuristic method,
most useful when you lack that "deep knowledge".}]

@END(ITEM1)
@END(Multiple)

@END(Itemize)
@Subsection(Elaboration of Applications)
@Tag(AppElab)

Notice first that someone must be able to understand the analogy in both cases --
H, the hearer, in the first case, and the reasoner R in the latter.
Only in the linguistic case does anyone have to generate the analogy 
-- the speaker S has this task.

Let's now consider these applications in more detail.
The "Deductive/Prediction" case can be considered representational --
a single "reasoner" uses an asserted (usually underspecified)
connection to infer new facts about an object.
"Deduction" differs from "Prediction" only in how certain R is of his conclusion --
that conclusion may be a valid, well justified deduction,
or merely speculation, depending on how much R really knows of the connection
joining the analogues.

That distinction is needed to fully understand the linguistic sense of analogy.
Note that S and H perform complementary tasks --
in the vocabulary defined in SubSection @Ref(Primitives),
S is responsible for generating the metaphor,
while H has the task of interpreting this message.
S's task is relatively easy 
-- in that he already knows the connection between A and B.
[Note he need not actually communicate this to H.  If not H has even
more work...  See below.]
Hence, in general, he can therefore use a "Deductive" analogy.
(This need not be the case:
Consider the "electricity resembles fluid flow" example.
Even if S did not realize the actual connection between these systems
he might still issue this statement,
confident that the physicist H will 
-- i.e. that H will still "understand" the intended meaning,
even though S's message only suggested it.
There are still other cases --
e.g. when S is technically wrong, but H can still get the intended message ...)

Consider now the role of the person on the "receiving end" of this statement.
For the analogy to "work", H must do a great deal of inferencing.
@Foot{
@Cite(Reddy) provides a nice description of this distribution of labor.
This article, which explains the usefulness of metaphor in communication,
itself uses an excellent example of an illuminating metaphor in communicating
this message.
Also relevant in that article was its description of the distinction
between signal and meaning.
Understanding this distinction provides a "leg-up" to understanding
how the pragmatic meaning of an utterance differs
form its semantic interpretation.}
Continuing with the "electricity resembles fluid flow" example,
H must first "look up" his corpus of facts about fluid flow,
then to decide which properties should carry over to the domain of electricity,
and finally he must map these fluid-related features to corresponding features
in the electricity domain.
In general this mapping step can only be done heuristically --
H can only guess the desired properties of B.
(Hence he is "using" the third goal of analogy listed above.)
(While rare, S might relay enough about the 
connection between A to B that H can logically (i.e. conclusively)
infer B's new properties -- i.e. allow H to use the second goal.@Foot{
@Tag(LingCats)
I considered seperating the "Linguistic" function into two categories,
based on how confidently H can determine B's new property.
There were three reasons I rejected that further partitioning:
First, this potentially opens the entire issue of pragmatics
-- perhaps H could claim that he knew enough
about S that he knew what S was really thinking,
to a sufficient detail that this metaphor was transparent.
If possible, I would prefer avoiding this can of worms...
Second, one important reason S was using a metaphor in the first place
was efficiency -- to avoid using a lot of words.
It seems, in general, self-defeating to spend those saved words by giving
the long, detailed description usually required to make his meaning "inferable".
Third, there are several other criteria I might also use for splintering
this linguistic case.  
(See the last sentence in the prior paragraph of the text.)
There seemed no (non-arbitrary) way I could allow this seperation,
but not those.}

It is quite surprising how well people do at each of these steps.
The first and third seem, epistemologically, quite straightforward.
The second, however, is next to unfathomable.  
Why is it obvious that electricity is not wet, but that its quantity should,
like the fluids, be conserved across a junction?
Or that computer should be able to perform deductions, but not be
composed of neurons.
This issue, of deciding which parts to place in correspondence,
is the crux of intelligent analogizing.

One additional note: 
There can certainly be analogy-like behaviour between a pair of machines,
as well as between humans.
The only requirement is that the same basic purpose be served
-- relatively few bits of information must be used
to transmit a complex concept.
This is only possible if the recipient
can actually infer (or guess) the additional facts needed
to flesh out the bundle of facts actually sent.
(Note we are here using an abstraction of the idea of analogy --
i.e. we are using an analogous definition of analogy, attained
by stretching/extending the standard definition of analogy.

@Section("Dimensions" of Analogy/Metaphor)
@Tag(Dimensions)

The previous sections mentioned various ways of cataloguing an analogy,
based on various features inherent to this instance of an analogy,
and of the situation/problem in which it was posed.
Ths sectio will enumeration various dimensions which can be used
for discriminating amongst different analogies.
The first subsection will summarize the dimensions which were
suggested in earlier sections of this report.
The next two will proffer addition axes, relating to the analogy
@i(per se), and to the problem statement surrounding it, respectively.

@BEGIN(Itemize)
@Subsection(Dimensions Already Covered)

@BEGIN(Multiple)
@TAG(Threesenses)
@B(Senses of Analogy)@*
There are three distinguishable types of analogy:
similarity, proportional, and familiar.
While all seem to use the same type of underlying mechanism, 
each uses the result a little differently.
(See Subsection @Ref(TaskTypes).)
Examples:
@BEGIN(Itemize)
@i(Similarity:) Cattle are like sheep.

@i(Proportional:) Cow:Calf :: Ewe:Lamb

@i(Familiar:) All Bach works resemble each other.
@END(Itemize)
@END(Multiple)

@BEGIN(Multiple)
@TAG(Tasks)
@B(Analogy Tasks)@*
There are four basic types of analogy tasks,
discussed in @Ref(TaskTypes).
@BEGIN(ITEM1)
Find the analogy

Find an analogue

Judge/Select analogue(s)

Use the analogy -- to deduce/conjecture new facts about one of the analogues.
@END(ITEM1)
@END(Multiple)

@BEGIN(Multiple)
@TAG(Functions)
@B(Analogy Functions)@*
There are two basic reasons an analogy may be used:
(This was discussed in @Ref(Applications).)
@BEGIN(ITEM1)
Linguistic

Deductive/Conjectural
@END(ITEM1)
@END(Multiple)

@Subsection(Other Dimensions)
@BEGIN(Multiple)
@B(Model vs Instance)@*
An analogies are map from an abstract model or instance,
to either an different model or an instance.
"Electricity is like Water Flow" is a model to model analogy.
Instance to Instance mappings are often called learning from example(s).
(Consider here generating a new program, modelled after an earlier one.)@Foot{
In his @Cite(SimonFriends) lecture,
Simon pointed out that human experts are familiar with between 50 and 100 thousand
particular instances or examples of their domain of expertise.
Understand some new phenomenon involves
matching that new instances against one of these "friends".
This instance to instance pattern matching works so nicely because
things in any domain are basically similar to one another.}

Instance to model mapping has the basic flavor of induction,
in the same way the Model-Instance case seems simple instantiation.
While there can be the same looseness of fit associated with analogies
in general, these cases seem relatively straightforward -- or at least
the complexities are NOT of the "what does it mean to say X is like Y"
variety.

We have (intentionally) not defined model, nor indicated precisely how it
differs from an instance.
Intuitively, an instance is a "ground case" --
consisting exclusively of fully specified terms,
as opposed to quantified variables or other manners of intensional objects.
This definition readily leads to a full continuum of models, ordered by
number and nature of variables.
The mappings which seem most like analogies are where the "model-ness" of the
analogues are roughly the same.  Hence Fred might be comparable with George,
or "typical man" with "typical zebra", but matching Fred with "typical zebra"
seems strange -- that is, not an instance of an analogical mapping.
@END(Multiple)

@BEGIN(Multiple)
@B(Degree of specificity)@*
For example, examine the transition from metaphor to simile to equation,
seen in the progression:
@BEGIN(ITEM1)
People are birds.

People are like birds.

John is like a bird.

John eats like a bird.

John eats as much as a bird.

John eats as much as a small, full bird.

John eats as many sun-flower seeds as most birds eat.

John ate as many sun-flower seeds on June 24 as Polly parrot ate that day.
@END(ITEM1)
Realize this also holds for non-similarity analogies.
@END(Multiple)

@BEGIN(Multiple)
@Tag(OpenP)
@B(Openness vs closeness)@*
Some analogy connections seem quite bounded 
-- the analogy can be used to answer but a single particular question.
Compare
@BEGIN(ITEM1)
@i(Closed:) Cow:Calf :: Ewe:?@*
What could possibly fill ? but lamb; and what else can one do with this analogy?

@i(Open:) Cognitively, people are like computers.@*
Many assertions can be generated from this comparison --
the usefulness of this analogy does NOT end once one fact about computers has
been transfered to people.
@END(ITEM1)
(This point is discussed in
@Cite(Boyd) pages 363-372, who talks about "inductive Open-Endedness", and
the description of "openness to explication" given in
@Cite(Pylyshyn) pages 430-431.
@END(Multiple)

@BEGIN(Multiple)
@B(Uni-directional vs Bi-directional)
This point is similar to @Ref(OpenP), in considering how useful an analogy is.
Some analogy map only from the vehicle to the topic, while others can be
considered two way -- facts from each analogue carry over to the other.
@BEGIN(ITEM1)
@i(Uni-directional:) John is a pig.@*
Note this describes John -- one's definition of pig is totally unaffected
by John's behaviour/weight/consideration, ...

@i(Bi-direction:)  Genes are like chromosones.@*
As @Cite(Interfield) points out, once the two sets of practicioners of both
domains realized this interfield connection, both sets were able exploited it
to conjecture new facts.
@END(ITEM1)
(See @Cite(Searle).)
@END(Multiple)

@BEGIN(Multiple)
@TAG(Seren)
@B(Serendipity vs Causally connected)@*
This is one of the major issues involved with any study of analogy --
whether the analogy @i{really} has a physical interpretation/reality,
or is just serendipity.  
This is clearly closely tied in with the openness issue raised in @Ref(OpenP).
When there is a meaningful correspondence between the pair of analogues
(e.g. when they are "two perspectives of the same physical object" or 
represent "two models whose behaviour is dictated by the same set of equations"),
an entire class of facts about one analogue MUST correspond to facts of the other.

In the other case the "analogy" may simply be full
of (pleasant?) coincidences.
@BEGIN(ITEM1)
@i(Serendipidy:) Speech is @i(laced) with metaphors.@*
Most of the corresponding features, (usually found @i(a posteriori)),
will be just curious coincidences.
Consider ... if too much, seems sour, or ... <<from DBL>>@Foot{
Different senses of the same word may or may not be in this category.
@Cite(Lakoff) would point to the 
classes of terms which collectively are transfered, systematically, 
from one domain to another, to argue that these are more than coincidental
correlations
in most cases.}

@i(Causally connected:) Genes are like chromosomes.@*
They really are the same thing.
@END(ITEM1)
(@Cite(Boyd)'s talk about cutting "nature at its joints" is closely related to
this.)
@END(Multiple)

@BEGIN(Multiple)
@B(Obscure (strained) vs "obvious", direct, natural)
This is related to the @Ref(Seren) case above -- analogies which are based
on meaningful connections will (or at least should) seem natural;
which cuteness will be strained.
(Of course the meaningfulness of a connection will depend
on the reasoner's background,
and there may be a strong cultural bias towards one type of connection
and away from others.)
@BEGIN(ITEM1)
@i{Obscure:} John is a zebra.

@i{Obvious:} John is a packrat.
@END(ITEM1)
@END(Multiple)

@Subsection(Dimensions of Analogy PROBLEM)
@Comment(This is a statement of the problem statement, not of the analogy itself.)

@BEGIN(Multiple)
@TAG(Explicit)
@B(Explicitness of difference)@*
In some simple cases, one can find map from one analogue to the other
by simply substituting one value of a parameter for another.
@BEGIN(ITEM1)
@i(Explicit:) "@i(5p) is like @i(3p),
except it moves the cursor to the fifth page, not the third".@*
(Taken from the domain of editor commands.)

@i(Implicit:) Many of the equations for water flow
can be used to describe electrical circuits.@*
Here the "parameters" which distinguish these two cases are 
by no means obvious.
@END(ITEM1)
Of course the best any computer program (or person?)
can do is find syntactic matches.
While this ability facilitates parameter substitution,
it is hard to see
how this could handle anything beyond the most superficial of analogies.
Realize this "explicitness" measure is strictly an artifact of the
representation used for the problem.
This research is strongly based on the assumption that any analogy can
be described as a modification of some parameter (or something isomorphic to
that, in predicate calculus), @B(when the problem is stated in the correct
representation).
Appendix @Ref(Reform) elaborates this point, as does @Cite[ThesisProp].
@END(Multiple)

@BEGIN(Multiple)
@B<Refined/precise vs sloppy>
This is related to the above @Ref(Explicit) case, and deals with the nature
of the problem statement.
@END(Multiple)

@BEGIN(Multiple)
@B(Bounded vs Unrestricted)
Closely tied with the constraint given in the problem statement, on what
will qualify as an acceptable analogue (or, in general, analogy).
Consider the proportional case of 
@QUOTATION(circle : square :: sphere : ?.)
Compare the cases when
@BEGIN(ITEM1)
@i(Bounded:) ? @G(e) {line, tetrahedron, octogon, <x,y,z>, "90 degrees"}@*
[Tetrahedron -- as it is, like a square, a regualar figure, with four sides...]

@i(Unrestricted:)  When ? can be anything.
For example, ? might be cube.
@END(ITEM1)
@END(Multiple)

@i{Pre-existing vs de nuvo creation of an analogy}
What can I say here?

@END(Itemize)
@Section(Conclusion)
@Tag(Conclusion)

The analogy dimensions listed in the previous section
provides a first step towards answering the question "what is an analogy".
While it leaves open the entire issue of how to build an analogizer,
it does provide some ideas about what an analogizer (or more likely,
an army of diverse analogy programs) should be able to do.
For example, Section @Ref(Primitives) demonstrated that one needs but two types
of operations to perform any analogy related task -- but unfortunately
it said nothing about how to solve those two problems.

In my case, this analysis has helped me to decide what type of analogizer
I should construct, by suggesting which tasks are the leaves of a dependency
graph of analogy operations.
It has also lead to a rich and diverse body of examples --
good test cases for this, and subsequent analogizing modules.

@COMMENT{
This report has, up to here, totally avoided many of the philosophical
and psychological issues associated with analogy
Let me conclude a few final remarks addressing such points.

The idea of a "deep structure" is at the heart of this whole process.
Clearly this phrase implies some "semantics" to our body of facts --
and no one knows how to instill that.
Any theory used will only as good as the terms it uses.
And all of the operations will still be "syntactic", obviously --
all we can do is twiddle symbols.
As such I strongly agree that no machine (and certainly no
sensor-less device) is capable of @i(de nuvo) creation.}

@Appendix(Properties of Analogy)
@Tag(Properties)

This appendix will discuss some of the obvious properties of analogy,
emphasizing those which have not been covered earlier in this report.

@AppendixSec(Analogy is like Similarity)
@TAG(LikeSimilar)
In a nutshell, we consider two objects to be analogous if
they share some characteristic.
Given this model,
simple feature matchings seems sufficient to explain analogy 
-- A is analogous to B whenever enough of A's properties match the
corresponding features of B.
What makes this task of analogy non-trivial is determining which set of features
to use for this comparison, given that there will be many distinct ways of
representing both A and B.
As Appendix @Ref(Reform) mentions, all feature spaces are not the same --
and in only certain ones of these is the analogy readily found (syntactically).
(The overall thesis may be regarded as a collection of "tricks" useful for
finding an apt feature space -- one which explicates the desired,
previously hidden features.)

@AppendixSec(Other Properties)
Let's jump down from this overall characteristic to examine some more
specific facts about analogy:

@BEGIN(Itemize)
@BEGIN(Multiple)
@TAG(Subjective)
@i{Subjective}@*
Metaphors and analogies are very subjective -- that is, different people
will generate different "answers" to the same analogy questions
(i.e. another analogue, or a different explanation of how X and Y are analogous...)
Note that any given analogy will still be understandable.
(Consider how often a person, on hearing the explanation of a metaphor, remarks
"Oh yea, I wouldn't have thought of it, but now that you mention it...")

(Diversity of feature spaces can be seen in some analyses of major works,
such as Shakespeare's sonnets.
@Cite(?) noted that the same octet of words were used in some sonnet,
as were found in some passage of the Bible;
and inferred that Shakespeare was (either consciously or unconsciously)
connecting his work with the biblical passge.
<<here>>
Different salient features to different people.
Can be culturally based -- consider Shakespeare in Bush monograph,
and their respective interp of the Hamlet, esp wrt ghost.
@END(Multiple)

@BEGIN(Multiple)
@TAG(ContextDependent)
@i(Context Dependent)
Analogies are also quite context dependent 
-- not only will different people propose
different metahors in the same situation, but even the same person
will find other "connections" in different connections.
(@Cite(Rumelhart) and @Cite(Searle) each gloss over this point.
@Comment{ Huh? in terms of being a pragmatic pursuit}
@Cite(Black2), on page 39, refers to this context as a "frame".)

Consider
@BEGIN(Verse)
Washington : 1 :: Lincoln : ?
	5 if dollar bill with his likeness,
	10 if ordinality of presidency.
@END(Verse)
@END(Multiple)

@BEGIN(Multiple)
@TAG(Easy)
@i(Spontaneous and Easy to Generate)
Metaphors are generated spontaneously -- easily and without conscious relection.
(Not only on the linguistic (metaphoric) level, but also on the level of concepts.
Consider the frequency of our "... that reminds me of ..." experiences.)
Further proof comes from realizing how prevalent various metaphor-like
tropes are -- ranging in explicitness from obvious comparisons
(rather dull analogically) to similes to metaphors.

As @Cite(GEB) commented, people will notice almost any prominent similarity,
even when they are NOT looking for it.
(In that book, there is a page in which a set of words appears repeated several
times, in corresponding positions in adjacent lines.
Sure enough, even though no reader was actively expecting that, few would
miss this obvious repetition.)
@END(Multiple)

@BEGIN(Multiple)
@TAG(Inexplicable)
@i(Hard to Explain)
Even after we have "understood" an analogy,
it is often difficult to explain our reasons.
Justifying the connection "felt" seems far more difficult
than generating it.
(This seems especially true for proportional analogies --
where the connection is already known to be via the proportional parts.)
(See @Cite(Darden), @Cite(Schon), p260, and @Cite(Boyd), p357.)@Foot{
This may be simply an interesting property of consciousness.
For example, we might theorize an elaborate "spreading activation" type of
mechanism might be responsible for generating and testing a variety of
matches in parallel.  If and when any of these exceeds some threshold, we feel
that those objects are indeed analogous.  The justification is lost at
this point -- and an effort is required to "reconstruct" the connection.
([Dietterich, personal communication]).}
@END(Multiple)

@BEGIN(Multiple)
@TAG(Asymetric)
Analogies, as used, are often not symmetric.
Note "<A> is like <B>" is distinct from "<B> is like <A>" --
probably because A probably has a different set of @i{a priori} categories
(and ordering of them) than B does.
(@Cite[Miller], p217 discusses this point of different sets of salient features.
See also @Cite(Ortony1).)
Compare
@BEGIN(ITEM1)
Rock climbing is like walking.@*
[ie Rock Climbing is trivial, quickly and thoroughly learned, ...]

Walking is like Rock climbing.
[ie walking requires balance, and is done by moving various appendiges, ...]
@END(ITEM1)

(Or try
@QUOTATION(Rock climbing is like the American economy.)
both ways.
[One proceeds cautiously and tentatively, ready to back up ...]
@END(Multiple)

@BEGIN(Multiple)
@TAG(NotXferable)
@i(Not Transferable)@*
Analogies are not transitive, for same basic reasons mentioned above:
different axes, and sets of salient features may be chosen for each analogical
connection.
Consider
@BEGIN(Example)
Religion is like mythology.
Mythology is like science.
@END(Example)
@END(Multiple)

@BEGIN(Multiple)
@i(Categorizational)
Examining the familiar sense of analogy, note that some categorizing is considerably
simpler than others.
For example, Bach's work seem quite easy for anyone (even musical novices like
the author) to discern; whereas other composers do not has so distinctive
a signature.
[The same might be said for larger categories -- e.g. in the Baroque style.]

Ask why this is true -- why are some artists easier to recognize that others?
Perhaps identifiability comes from some distinguishing mark,
or from being a larger delta from the others?)
@END(Multiple)

@END(Itemize)

@Appendix(Analogy = Reformulation + Match)
@Tag(Reform)

We claimed above that the trick to performing a non-trivial analogy, 
(by computer at least) is reformulation --
that is, rewriting the problem given into an equivalent one,
from which the desired analogy mapping "falls out" as a simple parameter adjustment.
Consider again the water flow/electrical curcuit analogy.
One might view each of the variables in the
various equations associated with the water case as a unit,
with a slot which pointed to the instantiation of this term --
as in junction, or resoviour(sp), or pipe cross-section.
In this representation, changing from water to electricity requires nothing
beyond substituting wire contacts,
battery, and resistivity for the values of those "instantiation slots".

In some cases the formulation given was sufficient.
Realize this is NOT a function of the problem, @i(per se), only of its encoding.
Recall the "5p to 3p" analogy.
This problem was given using a descriptive language in which
each command is regarded as a string of sequential characters.
In this language this analogy is but a simple parameter value change 
-- that is, no (non-degenerate) reformulation is needed.
However, imagine now these commands were each viewed as a bit vector,
or if the editor required roman numerals.
If those cases seem still too straightforward,
we could make this silly editor use one command format to visit
any of the first 4 pages, and another for any other page --
e.g. @i(3p) would still move to the third page, but @i(P+1p) would be required
to reach page 5.
There might still be some relevant, usable analogy -- but it's now much harder
to find.@Foot{
The artifactual-ness of the domain of editors makes any problem
which requires a major reformulation seem silly.
Given that editors were designed for people to use,
their linguistic front-ends should conform to natural ways of viewing the problem.}

<<<Note: what about disjunctive cases 
-- eg role of C-U, meaning either <factor of 4>,
or <the following is a number>.>>>

One of the major limitations of
many of the existing AI analogy programs is their commitment to their
particular set of domain terms and representation scheme.
In @Cite[ThesisProp] I propose some simple reformulations which
permit new terms to be defined, (based on existing terms,) which begin to
remove this dependency on the user's initial set of
pre-defined terms.@Foot{
This approach works via a "common partial theory" which links the two analogues
to one another. <Put more here>.}

There is are several other strong reasons why any competent analogizing
must be able to dynamically reformulate an input problem.
First, solving some problems will require the use of a variety
of analogies, which each contribute a part to the overall solution.
(See @Cite[Darden?].)
In general, any given representation for the problem will facilitate at most
one of these analogies.
Hence no one representation of the problem, alone, will be sufficient --
solving this single problem will involve transforming it into
several different forms.

Introspecting while problem solving will reveal another powerful argument
for reformulation: people often deduce a new commonality while solving
the problem -- one previously not realized.
<<HERE>>

---
Note this blurs the similarity/proportional distinction.
After the reformulation, one has an explicit handle on the nature of the
difference between the two analogues 
-- as mentioned above, it will often in the form of a different value
for some particular slot or parameter, on the two units
It is easy to then assert that this new value, V', is the V of the other
analogue.
Eg. ...

---
What is reformulation:
@BEGIN(Enumerate)
change of representation -- a new description for the same problem.
Consider @Cite(Amarel)'s Missionary and Cannibals.

addition of features -- to the same basic structure, add a few new slots.
Consider how biological trees are like corporate hierarchies.  If the
tree is represented using only the branches-from relation, we'd need to
define its transitive closure, and notice its acyclic nature, and single
maximal element.  This maps onto the Underling* relation.
Another example: addition of new lines, points, ... which are used in a
proof -- ie new things NOT MENTIONED IN PROBLEM STATEMENT, which render
the problem solvable.
---
Must be derivable from facts given -- but bringing it to prominence useful.
Consider the key signature of a peice 
(which relates to its "mode" as major or minor -- emotional content.)

similar problem -- one which has a nearly isomorphic solution.
Consider the <<two points above a line>> problem -- which is readily
solved once that second point is reflected below the line.
(See @Cite(Polya1), p143.)
<<<I call this PROBLEM TRANSFORMATION above>>>

?? Meta or Similate??
@END(Enumerate)
@Appendix(Analogy Vocabulary)
@Tag(Analogy-Vocab)

Here we consider what consitutes a natural description for 
constraining how two things are analogous.
Of course, on any given problem,
these need not be stated explicitly -- in many cases this additional
information may be understood implicitly.
These descriptions can also be used to (partially) define the analogy
(that is, it can indicate how the analogues map into one another).

Fred is [just] like Jill
@BEGIN(ITEM1)
@i(in that both are) students.@*
Or: expressed as @i(as both are), @i(regarding both to be), ...@*
I.e.: both are in the same category or class; and this comparison should
be based on this perspective.

@i(in terms of) profession.@*
Or: "<Adjective>ily, A is like B", as in "Cognitively, people are like computers."@*
<<here>>
I.e. based on this perspective.

@i(except that) Jill is female.@*
Or: @i(were it not that...), @i(disregarding the fact that), ... @*
I.e.: Some implicit, unstated slot [here gender] has a different value.

@i(ignoring) gender.@*
Or: @i(excluding), @i(disregarding), ... @*
I.e.: The value of their respective X slots [here gender] are different.

@i(considering only their) height @i(property).@*
Or: @i(based only on), @i(considering just), 
@i(where) X @i(are the most salient attributes.)@*
I.e.: The value of their respective X slots [here height] are the same (or similar).

@i(after substituting) CalTech @i(for) Stanford.@*
Or: @i(except in) X @i(context rather than) Y, ...@*
I.e.: There is a strong (?causal) proportional metaphor of@*
Fred:Jill :: CalTech : Stanford@*
-- i.e. some relation joining Fred to CalTech holds for Jill and Stanford.

@i(as) Prince Charming @i(is to) Cinderella.@*
Or: @i(ala) Prince Charming @i(with respect to) Cinderella, ...  @*
I.e.: (At least) one of the relations joining Prince Charming to Cinderella
links Fred to Jill. Perhaps "loves enough to search for"?
(This is much like the case above.)

@END(ITEM1)

It is pretty easy to see how this maps onto the similarity case.
The trick mention in footnote @Ref(Unify) discusses how this
relates to the proportional case.
@NewPage
<<This has to be merged in, at some point...>>

For example,
<<HERE - close to abstracting process>>
consider two people's rendition of the same musical peice.
One person might be more rhythm conscious, the other more concerned with tonal
contour.
The first would "abstract" the pitches out of the peices, by percussively
tapping the notes.
The other person might more or less ignore the rhythm, and hum the melodic
contours...

finding the representation of a spoon and a shovel which demonstrates their
common PTRANSing function.
In the feature space which high-lites(sp) functionality, it is easy to
find a partial match of these objects; whereas if the only facts represented
the situation,
there are no obvious correspondence.

Another factor is that analogies often seem to establish the connection:
one might never have realized that 
<<<here>>>
before hearing the analogy that ...

MUSIC

	Some people tend to remember Rhythm, others Melodic Contour
(cite paper on that)
-- that is, how to decide what to abstract away, when condensing the information?
The instrument? or frequency range?
What does it mean to be the SAME PIECE?
same recording (even that different each time)
different conductor and instrumentalists change it considerably
What of Abdelezor vs YPGO?
Is it more like other works of Purcell, or YPGO?
Consider how Schoenberg would "understand" this peice -- in his serialistic
style the notes themselves were important, not "feeling"s, etc.

Consider learning Viola, from knowledge of Violin.
How to play the "same piece"?  Same fingering, or same notes?
perhaps other arrangement, which emphasized viola's characteristics.
Consider renaissance/modern renditions -- now more "Masceline" (sharp,
articulated) vs earlier "feminine" - soft flowing 
(Note those terms are metaphoric!)

General range - to see what new instruments.
Or based on difficulty to articulate -- which varies from instrument to
instrument (consider cello, w/4ths not 5ths between adjacent strings;
or flute, whose "grace note" is just by half-wholing)
Loud forceful.  Musical.

bowings, tuning
(Piaganini's trick: tuning down 1/2 step to play peice in Db- as if in C+)