THESIS.MSS[RDG,DBL]1 - www.SailDart.org

perm filename THESIS.MSS[RDG,DBL]1 blob sn#642962 filedate 1982-02-20 generic text, type C, neo UTF8
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C REC  PAGE   DESCRIPTION
C00001 00001
C00003 00002	@Device[DOVER]
C00006 00003	@Begin[Center]
C00016 00004	The I/O of the components are shown below in Figure @Ref(Components).
C00018 00005	The rest of this report will elaborate on this figure.
C00020 00006	@Section(New Ideas)
C00026 00007	@Section(Example - "Tree")
C00029 00008	@SubSection(I/O Behaviour)
C00038 00009	@Subsection(Internal Process)
C00047 00010	@BEGIN(FIGURE)
C00050 00011	@Subsection(Other inputs)
C00066 00012	@Subsection(Use of this answer)
C00070 00013	@Section(Salient Features of the Analogizer)
C00078 00014	@Subsection(Static Input)
C00081 00015	We now consider the other things the analogizer must know about before
C00087 00016	@Subsection(Output)
C00092 00017	@Section(Scope of Analogizer)
C00103 00018	@Subsection(Limitation of Scope)
C00105 00019	@Section(Research Programme)
C00117 00020	@Subsection[Programme steps]
C00124 00021	@Subsubsection[Step 4]
C00133 00022	@Subsection[Evaluation/Validation]
C00144 00023	@Subsection[Details]
C00148 ENDMK
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|H≠⎇.$≤|⊃,\zλ∩.P60qYr⊂;t]4⊂⊃2↑8∧ended" meafiNgs$ Basedon metaphors
we had not realized we were using.
Wa, For examplE, coNsider time a pRecious commodity,
("he didn't havE the time tk give"),
indicate emotional makeup in terms of spatial (up versus down) position,
("he was feeling down this morning")¬
and even cOnsider speech to be "laced".

Such evidence supports the view that the ability to generate and comprehend
an analogy is a tremendously useful and powerful tool
-- not only forallowing efficient communication,
but also for (increasing our) "understandingα of parts of the world.  
This research has two goals.
We will firsT present A formal cognitive model Which
accounts fkr/explains much of the phenomenon of analogy,
The second goal is the implementatiOn of a computer program which uses
(a SubseT kf this) this analodπSGC0AGCa¬ESYSQr\
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when the only correspondences are at a "surface" level -- 
e.g. when the relevant properties to match
are things like number of deaths, or particular steps in a proof.
It runs into quite a bit of trouble mapping "deeper structures"@Foot{
While related, this should NOT be confused with the psycho-linguistic usage
of "deep structure".  We are NOT pretending to have captured some deep semantic
information, either by virtue of using predicate calculus terms, or because
of the ease with which the system can generate new relations.}
-- e.g. facts not readily apparent in the attribute-value pairs of some
aspect of the analogues.  A good example of this was the "embedded" hierarchy
in common to both types of trees.
Unless the descriptions of both types of trees already happened to have
this specification, (e.g. as appears in @Ref(Use-Hier),)
it is hard to see how (a naive)
feature matching could have notices, much less discovered, this relation.@FOOT{
Indeed, I am prepared to claim that
this common subtheory approach will in fact subsume this other model of analogy.
However, the purpose of this proposal is not to compare these
approaches @i(per se),
but rather to demonstrate considerations of other attempts to deal with
this problem of analogy.  Much more will be said in the thesis itself.}

Neither of these definitions, by themselves, is not terribly useful:
Notice neither tells why a given analogy is proposed,
and not any of the others which are still acceptable by their respective
definitions.
What makes a given analogy appropriate?
The criteria involved in this decision are quite subjective --
different people will propose quite different analogies to answer the
same question.
(There is no single answer to the question "Who is England's first lady?"
-- based on gender, many would insist it has to be Margaret Thatcher,
the female prime minister,
while others, seeing the "Spouse of the Head of State" role as more important,
would argue for her spouse, Dennis Thatcher.
With similiar arguments one could justify
Queen Elizabeth, or Prince Philip, or ... See @Cite[Hof81].)
Furthermore, even a single person may propose a variety of different
analogies depending on the current context:
A spoon is rather like a shovel in terms of its PTRANS-ing function,
whereas it is clearly closer to a knife in terms of size and 
general purpose.)

Despite this, the other analogizing systems mentioned above 
incorporated a single corpus of rules which collectively defined 
how to generate, (and/or evaluate) an analogy.
These rigidly defined what the program could produce.
One of the goals of this program, however, is flexibility --
to be able to accomodate different users, in diverse situations.
We achieve this versatility by employing
a collection of heuristics to guide its decisions
The user will be able to modify these rules to tune the program to
his "ideosyncrasies".
In fact, the overall system will 
include a module to help the user to adjust that body of heuristics,
honing it to produce analogies which fit his specifications.
We will return to this point later.

The I/O of the components are shown below in Figure @Ref(Components).

@BEGIN(Figure)

@BEGIN(Center)
@BEGIN[Verbatim]
						/-----------\
	   +---------------> CONTEXT		|  DOMAIN   |
	   |			|		|   FACTS   |
USER - - > |			|		|	    |
	   |			@Z[↑]		    \-----------/
	   |		|---------------|
	   @Z[↑]            |		    |
	INQUIRY	 ---->	|   @i[Analogizer]     |  --->  ANSWER
			|		|	   
			|---------------|	/----------\
				↑		|   META   |
				|		|   FACTS  |
			     /------------\     \----------/
 /	  |-------------| \  |		  |
(USER --> | @i(KB Modifier)   |->) | HEURISTICS |
 \	  |-------------| /  |            |
			     \------------/
@END[Verbatim]
@END(Center)

@CAPTION(Modules)
@TAG(Components)
@END(Figure)
The rest of this report will elaborate on this figure.
The only "required reading" is Section @Ref(NewIdeas),
which discusses, at a very high level, the basic approach we are taking,
emphasizing how this differs from other AI systems,
and why we think this will exhibit impressive behaviour.
Each of the subsequent sections has a less general goal,
and should be read only if the reader wants an answer to the
particular question that section addresses.

The purpose of Section @Ref(Tree-Example) is to convince the reader of
the feasibility of this approach.  
(It presents a representative problem,
illustrating how the individual modules will interact to answer the inquiry.)
Section @Ref(Salient) elaborates the sketch of the analogizer one might
infer from the example.  
This is followed by a description of the scope of this project,
in Section @Ref(Scope).
Here we provide our definition of analogy,
and then list a small collection of instances of analogy
which our system will (eventually) be able to handle.
The concluding Section @Ref(Programme) will outline the research programme
I intend to follow,
and mention the goals (and anticipated caveats and pitfalls) of this work.

@Section(New Ideas)
@TAG(NewIdeas)
Below is a list of how this work on analogy is
different from previous AI work in this area.

@BEGIN(Itemize)
Definition of analogy@*
Rather than view an analogy as a direct mapping between corresponding aspects 
gf the analogues, we take a model-theoritic approach -- defining
an analogy as a shared partial theory.
That is, two objects are cOnsidered analogous if they each satisfy the
same theory.

Our approach@*
We are taking an "Expert System" approach to the problem of analogy,
(with a few Minor twists).
Expert systems are characterized as knowledge intensive systems, 
containing a large "knowledge base" of factq about the domaif.
Here, that domaif iq analogy rather than, say, chemisTpy3 and
this Information is in the form of rules or heurictics.  
Part of Subsection @Ref(KB-Stuff) discusses the typeq of rqles which are required.
(This approach is contrasted with the procedural attitqde, in which
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t≥~→$→<z,⎇H≠yD;H→,lZ8z,]]λ⊂⊂[2⊂87]ry3:[⊂ w0[7st⎇~w3P0[3wy4]46WεB(2y4_x9P:~2w⊂:~4yP9↑yz2vH;tv6λ0{7tY⊂:42H7q9w[2yqb[1rP;Z4qt⊂~0yP1Z0y0q]2y4⎇→rεE0[6⊂89~wy⊂0[0v7sZ⎇4w3H9|yz→vWεE⊂"g"∀∪zv:4\62TFB "g"
$z2vZ⎇2TFBεEβ )ra]4ww∀⊃|0vx≠2P⊗Pλ*92rHα)¬
@TAG(Tree-Exaeple)
This particularexample should both motivate why various componentq 
(shown in Figure @Ref(Componentq))
are included, and illqstrate how they wilh work.
For pedagogic reasoNs,
the fIpst pass od∧AiQ%fAIKMGeSaQS←\~)oSYX↓ieSm%CYSu∀A←dA∃mK\A%O]←e∀AGKeQCS\A¬gaKGQbA←L↓iQSf↓gsgi∃Z\
∃%hAoS1XAEJ↓M←YY=oKHA	rABA
←eeK
iKH@!oKYX0AYKgLAi←i¬YY`%ε3π33∞≠'?W~Iβ[↔↔≠'?9ph*'→πβ?OON∪3∃1π#Keβ&yβOW∨β↔;⊃πK?WI∧∧FO≡,]FN.d∞Vw&≥Dπ&FTVv"
|bπ&
≡2ε∂∞λVv&∨¬`hU>\'≡/≡XVw"↓Q'≡8⎇~-⎇\c"N⎇;≠λM<xp~\yP:4~yP0w_v7st↑2y⊂$[⊂6wy→P22z_tv⊗εB897{~r2P0H1wv6→qz4g[⊂7s⊂_p∧ditional examples,
and discuss the actpCX↓eKgK¬eGPAAe←Oe¬[[JA$AS]i∃]HAi<AM←Y1←n\~(~∃↓R!≥←iJhA∩OZ↓Gkee∃]iYr↓S\Ai!JAae=GKgf↓←LAI∃mKY←AS]NA¬]←iQ∃dXA[=eJ~∃αk↔π;L¬f>7]Dε/F≥↑εf*∞Mrπ⊗↑
F∞≡T
FFO4∞7'⊗≥≥f."≥f"π↑<Vf/>4ε}vUaPT6\]Bε7,XRπ&tλvf∞l8	$=λ∃
(≤Y.>λ≠qD∞~~<d∞y8p~~ww↔↔∀BEεB !wv[rs:≡βE*ybH0w7j~2y⊂"↑0rx6→P⊗P8→y00x≤P397[P897Yy0vvZw3]εB#t{2[⊂0P 0rogram which sOpps, using a lInked↓YSgh0AG←]MiekGPA←]J↓oQSG ~∃g←Iif\\8@|
∀4∃π←[AkiKd↓`∂∂'.sS'O'→βWO*βS#∃π;?K⊃α∪SK↔*⊃βS<hSK↔≠/⊃βS=ε	β∂↔↔#π'9π#gC∃ε{⊃β∪∂#¬βO'∪W∂S-∪∃8∀T≠3↔π⊗ceβSFKMβ≠⎇∪7πQ∧∧F/≡>-↔π&≥⎇bεO4
f␈"∞Mε*ε-→vf↑⎇≤6∞b]g&OM⊃PW>
≤6Bα.N&.
$	w-9z;L≥≠≡(
\αpw:∧¬
COmputer science has borroWed cEppain particular as@AKGif↓←@→↓↔#K↔∃⊗s↔OLhS'9β&C↔'I¬+Oπ∨*β?→β&C'Mβ&+K *Dλ⊗v"
⎇fgJ∞Mε/≡UaPPH*|Rε≡L\↔⊗g∀λ


9Z`⊂≠q⊂:4→yrP!TP:92YyP∀!TTz12YyTP !s analogous
to natura`_XAEC←1←OSG¬XAie∃Kf@Q8[)eK∃bR\~)¬`↔Q∧¬ε␈;t∧¬.F≤=αππ-}ε/↔M≤W4≤z≠n]→λ⊂l≡\↑(
}Y<OaQP;↑$∞z≡(∞M≠|y$9Y
m⎇λ≠nM→<\gq"C"@↓H∀⎇0⊃∀p¬ction(I/LεA¬K!CmC←β+@∩Hβ"U

<h⊂∀\β a @Ek@↔O&K?9β&C∃βπv3 >⎇≠&/∩8λ-d≥≤↑$∞≠h⊂⊂[9{r`2,∧⊂∀*Mε*∧α3T*Y4V(∞⎇⎇;⊃∧Y(⊂⊂H80t`2 h∂LAβ##↔}-⊗/~ββ"LL<xp→~q0w3H!iVz≤αees and N-Pre@∃`
βK,ε7ε.>M↔6.M∃`hUMRπ∂\XnM9{@⊂~vx6 )citl@dACgW∃HASF@
#?8βπK∃∧εFF/∀λL]_=→,GhKC!↓αA$`.dkrmalhp∩X↓KCGP↓∞M⊗'∪↔¬βLε2ε&\h
-l9λ⊂∀[⊂:2`2ms oF a s@∃hA←L↓]P∨∪-→1α≠|∧F/~AQ&∞vDλ∩ε⊗≥l↔πJ∞,Vf∂M_meλ∪⊂∀[5rr*≠Tε  
The constant Root IpεABAβ≠C↔∂L1β7,k↔HhS?2∞MεO~	mv&/4
6/"βλ⊂∩→q0	ned ac its Maximal el@∃[@↔nAPG>≡Mαπ⊗↑:ε.∨D
FzπMR∧f\≤G
&tλL]_=~-⎇Kλβ!∞z→0→→P eAdsTo is the transitive c@1←gke∃↓
∨∨Qv~∃⊃∃eJA)IC]gSβ#'[⊗≤¬F␈∨↑,RεO4F.⊗≥lV"ε≡4π&FT
VvN⎇`λ
|H≠sLT⊗sSjD≤Y4M{(≠p→βE0
o@IJACAAYSGCQS←]f↓←DAi!J@EMU]GiSα{9	↓jiβ#↔⊗)1β←*β←?Wf!βOπHh"α
,:&:@8Z↔.∂M_mk!"JλJ⊂ ∀H<⊗0→. (LiNiadTo x y) @Z(g)  Lea`sTo x y))
∀Q↓0QαR↓pYbYh\@PQ1KCIgQ↑A`AdR@L@!→KCIM)↑Ar↓tRRA↓0QNRQ→KC⊃g)↑A`AtRR8~∃↓9	7cathematical Structure
  Domain:	Nodes
  Relations:	{}
  Functions:	{LinkedTo}
  Constants:	{Root}
  Facts:	{(@Z(E) LeadsTo. (TransitiveClosure LinkedTo) = LeadsTo  & 
			(AntiReflexive LeadsTo) & (PartialFunction LeadsTo) &
			(@Z(A) x. ((x @Z(B) Nodes) & (x @Z(=) Root)) @~
@Z(g) (LeadsTo x Root)))
@Comment{ or maybe just "the LeadsTo relation is quite important"}
  Refinements:	{BinaryTrees, BalancedTrees, 2-3Trees, ...}
  UsedInField:	{ComputerScience}
  UsedFor:	{SortingRoutines, ...}
  RefersTo:	{DataStructure}

@End[Equation]
@BEGIN(Comment)
(@Z(A) csTree. (csTree @Z(B) (Extension CS-Tree)) @Z(g)
	(@Z(E) Nodes, LinkedTo, Root. (Uses csTree Nodes) & (Nodes @Z(B) SET) &
		(Uses csTree LinkedTo) & (BinaryRelation LinkedTo) &
		(Root @Z(B) Nodes) &
		(@Z(E) LeadsTo. (TransitiveClosure LinkedTo) = LeadsTo  & 
			(AntiReflexive LeadsTo) & (PartialFunction LeadsTo) &
			(@Z(A) x. ((x @Z(B) Nodes) & (x @Z(=) Root)) @~
@Z(g) (LeadsTo x Root)))))

(CS-Tree @Z(B) DataStructure) & (UsedInField CS-Tree ComputerScience) &
(UsedFor CS-Tree SortingRoutines) & ...
¬
(SubTypes CS-Tree {BinaryTrees, 2-3Trees, BalancedTrees, ...})
@END(Comment)

@Comment{ <<HERE>.
The actual "Inquiry" part of the input is ...
In addition,  ... context ...
The resultis ...>

We'll now discuss a few nuances of this I/O specId¬SGCQS←\\4∃
CeMhXAaIKISG¬iJAG¬YGkYUfAMC9CiSGLAgQ←UYHA]=hAEJ↓ISgCAa←S]QKHAi<AgKJ↓iQSF4∃eKaIKgK]QKHAkMS]NA∧AckCMR[Me¬[JAgegiKZαq↓αOd¬w'~≡&*π↑8	,D≥≠h∞>≠|Y,D
≠p↔≠<TP*~7yrFB9x2qZs0qP→0qz9H;t4`#h ind¬CGh↓kgJA∧AES]¬erAe∃YCiS=\XAgUGPACLA	←[¬S\A←β⊃αWO,"':≠L∧Vf"aQ$␈&↑"εNlhn
8=~-⎇Kλ~-d≤_<NM8⎇0⊗_y⊂92[0z4w→β to the characteriStics o`Ai!J~¬→α+πβN&yβK↔fS'?paβπK*β↔;∂|#↔⊃βLqβS#*β7?K*β;πS,ε&∞b∞
&.&≤<↔&*8λ-L⎇;≥.∀_{⊂.↑y(⊂∪≠y2pz∧A!<H⊃1wv\0y0∀mentalizing" thiq inforeation We can the@8ACgg=GSCi∀A[Ki∧[YKm∃X∩+6∂SM∧?W ∧εNvM~fN'\≥Bε≡L≡W≡
↓Q"jJ
_
∧
<h⊃,≡|∧V⊂→4πr Exaeple( to ifdicate how Important @∧AOSm∃\AMC
hASf↓iP≥β&C∃β↔≥≠↔;O(h+ >dλ


<h⊂m⎇Xy<∞Eλ≠tD
≠⎇h
M8y0⊗≡P4z⊂~yP:7H12P&Xz1t2Y↔αE+YP32`%l thipεAS]→←e@7∂#'?9∧εvNfD∞εf∂∀λ∩εN≥)w$≤[p⊗→P;t2[⊂9r`\αchiNe fo@HAC@;∞c?∂'-→0 (%λf␈∩X∞≥<≠⊂∩K⊂:44\P4yP≠2rr2Y⊂:7P_ww;"↑P30q]9P0q≠zz⊂ 4he sAlience of car`)¬S\4Tεπε@|→0→≥4ryWλ⊂)r`% @Ci@QK7?α.F}w⊗≠R`% εEεBαSec@=]HHAβ##∃β&+K /4λ5~@5≤Y,Uλ⊂Z-l<Xj≤2rV⊂⊗Yj9→rV⊂ ~T2z c`iKβ∪¬%0hSK↔≠,ε"π&tλ

(⊂⊂∀J1ww1Yx:∀P≠pε these @Q`↔↔~p4*SF+'Iβ⊗+@∨ε\8

≡Y(⊂∩↑8∧ensions wi`→X↓KCGPαβ∃β&C∃βO,εBε@yH⊂-Mα⊂9`5ch tre`f\4⊃≠≥l⊗fGα+λ
\8π<P≠pε c@=]Uk]
ifASαqβC#*α~ε∞'→βπ⎇3∃βπα,Rπ≡\8mlλεw`2der.
λ¬/QSαc∃β7∞seβ∂|εVf",Rε/∞λL↑|y0∩λ4p∞ @→Seghαk?K∪,ε"`@ ¬0∪@∃G←]Hαk?K∪,ε"ε6βx[.P0q2H8¬se@⊂AEKGαWO∃∧εFF/∀λl\8εFE≠wy2P≤2y1`0eci`+=kf@Zαh4+↔∞≠!βK,επ⊗/<XNNh_ ⊂≠wy2P≤zqqd[1z⊂ !nd natpeC0A`∨πβ∀ε}H→0≤≤92y`3ing @M←[JAα3π∂PhSπ|¬0~λ9wvrH92v ]4p∂n9↓≠⎇x
∂1 QP↔\α example,
the preVious fh∂←I9←iJA≥SmKf↓C]@?&C↔Iβ,πππ⊗↑8m≥{H⊃M}@εE∀∃90w9Zz4s %@πY←GUaJA→αK;/↔%#=%↓h∧∧f.≤N5&ZaQ%&F↑8	$X∧y9]⊂:7P≤p¬co@9HA←e⊃KdAGα{@↔⊗↑8
⎇Y→3L<αyP !re not lo@Mh@JZαβS#↔Hh#πK*βGS?⊗+⊃β'p∧π&FT	V/&∀λd≥J4εNv|⎇F.→y(,<y+D¬∀y0∩CE)za≤p¬ction @Ref(KB-Stu@→H	%9Mp4(Q)v2ε=x
.∞y(⊂-o(≤|\z9P∀XP:<`0e of AS ≠)IK@∃β>K3 "
↔6*
x

<H⊂∪_qz9]λ30∂r↓KqC[AYJX~)g←[J↓GQC@⊗∂S↔⊗KkπSL¬vrε|dεO'4∧&↔⊗≥l6FNβY0→\β" o@HAIKaβ#!1β|ε"ε
LW<|Z<∞M8πw of¬
@QQJA←β##↔IαC;?9oβ?';&+A%β≤¬vwε]nG4≠yH
≡≤h∪M|α2yWλ⊂εE$≠{r{"\⊗⊂0yH844`3 is A desCp¬Sai%←\A←α1βS#*β7?O β⊂≡∞βY0→_v⊂8 /pπgSE1JAπ&5)eKJ0~∃@' β';S,sS'?v33@∀λm]=_h≥≠λ≤n\zλ⊂→→q0	nem`MiL←EkeQQKdAβ≠C↔∂L3'∂π&K?;Mph($
&C∃β∪}kπ'e\F∂&∀λl\⎇~0↔[⊂7s⊂≥42P"≠vptgλ! aj∀P nowledge Base
∀Q¬KCS\αaαOW↔≠↔∂SL¬vr∧
,V %⊂¬aεTz:s3
P22iXy0q2\β thipεAS\αβ∪↔S∞K11∧hSG']_≡Xε0→ qtorepεABAα∪?∪e∧¬v $→X8nNh_8M}=λ⊂∩Xqt⊂!~wv7cZqpv⊂≥92rWβE vw[3P7`4her @QQS@;?→1β↔∞≠!α→m#@⊗.T
ε∂~.&∞v=W
@λ≠⊃,≡Y<k∧8π2 a
∃gαK;∨3*βSKWvY`~ε\≥gJε.,⊗v≡Xd
8>(∞≡≤Z;LT→\[mT≥~~.∀≥≤]-miλ⊂⊂[2εE"Xqt⊂ "pa`≥GαAβ'πJβS#↔p∧π>≤[⎇.D→]4NM→<@⊂_90w1Z2yW ⊃'gj=CE*42\2P6p↑P12P≤wP2f→pπant a theory dh∂dAMkGPA9Cike¬XAie∃Kf@8hR';O&+π"∞|RεN∨∀εF∂lTπ=8εx6≡P0P #olle@
iS←\αβ?2hλ,>≤h⊂,-⎇=⊂_py20Zw
@ACeiS
kYCDαβSK↔-→1↓α4ε&}j∞MεO~∞|Rπ>}YF"ε≡f*πMtεNvNX6*πM
w≡(→y-l<X;↓QXz_.0qz2\4yz4XyP∞  Jqst↓QWnAβ##'M∧εv␈.L@λ,(→≠ml(~0→H4s⊂&↑P⊃;w\9<P X5zzεB40z2\⊂4s⊂∩P40{→P:7Qλ1pp∀egory.}	α(There are, ob c@=kegJ0ABAQ=chA←_AWiQ∃`AMC
ifHAα?W βS#∃ε∪πK4λ⊗v"
~G
ε=x
}Kβ"M≡≤h∀l\<{p↔_v4z<K⊂4z9H392`)t, and various habitats.@∩∃/∀AoSYαaβ'∨v{K∃β&C↔O∃∧∧f␈∩
mw;Z≥f"Q(
m≥≠λ⊃
≡x⎇4n∀≠_=↑H
~-d∀⎇0N≤8⎇~-⎇H⊂∀L\J⊂p↔[82|:
TBE (ow the prodπeCZαβπ3Oz↓⊗↑lZr$≥≠h
≤{[tLT≥~→-Uβ∀FEβE(7d[82y9H:7P 4hese tp∂↑Aβ##↔∨⊗K↔Mβ≤¬vw∨M~G/&Tλ

(λR)@(jdi⊗Q⊂4`.pqt @Q↑AiQ∀AC@;∞c?∨'V+A84U##∃β∞sπ3?>Kk↔I?→β?W'βWQβ>K3 ",Rε
∧.7.↔MV␈↔∀ λ∞⎇~8z∧
<h⊂⊃[vrw`. to @	←iPAα{⊂∩πMW≡(β"N∞Yx∪np¬d @¬]CI←≥k@↔M`h#@&||W&F↑ λ∞⎇=~∀≤_:.⊂7s⊂λ4¬ap@AS]NDα↓# ?$λ
-n⎇_;NM8=~-⎇\k
!QY\[mT≥~~.∀_{p⊗[ww⊂(_y:4`[⊂:42[y<P 4o each of thoce ifputs ≤~(~∃¬CMKHA←αqβS#Lε2ε&↑8λn4x:4[w⊂8 2ovided,
One p@=`∂O'⊗c∃β∂|¬Vn}d
ε∂↔M_⊗bπMV␈↔∀λ
.P0P∀≥42w`2y of a) hierarchy.
Thad↓Sf@1∧∧&␈&∧λ5~J≥f"∧ePλ
NY94d<Y(]8[q
≥9;]∞P7s⊂~4ry0\αchi@∃`
  Q('.Nβ_
→αr`∨Zαβ; >LXd
9H⊂~~2P3 )p¬ghA
CgJX↓C]HA→` >Tλ',8π1`(es iN the othe@HXλ (*Mε*ε=⊗f@→;YlT~8h∞MβP3 )nd that cOmmgn↓aCe@ ∧ε .Y{ ⊂≥42P )nfgrmation proVi`	KHαp4(Q d¬>XλN≤αqz4[w∀$g≥2y70[⊂(97Xp¬ss)
@@)¬∞Qβ]QKe@;∞a$4*MεO~∞Yf&/-K⊗Nvtλ⊗v∞Mx	m∨X;Yd∞≤[p⊃YyyP$\β base` o@8ABAo∃CV@1∧∧v.v↑,⊗bεlXλ.Nαy2P≠pz1d→y↔εE∃42P+_y0	oUq de@
SgS←9b@@5h∧εNr∞λ↔π~8p~[0y⊂ 7hat @Q↑A[CQGP@1∧∧⊗v"

w:πMtε/6≥JV∂&QQ'&FT∧&>}|Mf/∨4 λ
|β⊂0@ matcH -- @]SYXA	JAECMK@A←αqβS#*β#↔W⊗KGS'∨→βOS|ε&.α
_AQ]~→$	→=0→~yz4`#pεA/v{←3↔&;∃α∂≠∃04RC←#'≤Aβ#π4∧Rε⊗\XD
;\≥.D_↑(∞M→(⊂∀[24{4Y8pv⊂≥yry9JW
For e@aC[aYα)1β←F+9β∂|¬Wε∂-→f 4≥≥{d∞⎇≤],>≥<Y.∀λ≥~≡λ~0→K⊂:;wH:42g\αies)
`∨]∀AWE@4¬⊗␈/4
'.fTλ
.P:7P≤p¬arch fo@HAgS[%YCeSβ#'πM∧∪↔S←,∧Vrπ,YF∂&≥⎇g4≠yH∞M→#"N<8εrP_y0	`)d\∩*&C'Mα≤{7Cπ⊗*O']_≡TY3≡~;{NP42`5pistic( 
(p∂QK8ASMGαcW&\@λ
≥H≥~T⊂⊃5*)4p
$PβS KnowleDg`
A¬¬`∂∃1Hh#'≠≤εG.αqz9H:42P_w0v /dπSuKβ⊃βS=∧∧6}o≡&*πMRε⊗α;X.∂(≤Y-L=~3mnhβ"IM9Zy,J≠h⊂-lλ⊂|∞
⎇=≤i≥]≠kAQU~→${{<≡X∧qt[w⊂;w]v2⊂ "e based o@8~∃iQ∀AaeKMK]GJ↓←dAC	gK@;≤∧Rε↑dλ6/↔L≥⊗rα.N&∞w<h	.0q62Hα properties oF the
relations in qu`gi%←\\@↓)QSf↓o←kY⊂A]←i∀AiQCPAEO@&Aβ?→¬##↔O*βSO=¬∪↔3π&K?;Mh*α
,:&:@8↑↔.∂M≥vuhQ(¬$u
"l&\ibHh(λ$,<→k4.w]\W⊗∂LUB¬∨∞,V∞αεPhV≡
εfN\Dπ&@h_ ∞:8H
|H≠9-\Y<\d
yH∃
(≤x-\(≤|≤y0
AQJ≠[lL<h∩-d≠{Y$x<p∩K⊂190[1t2iH4w⊂:~2P7`4her)

both were anti-refhexive@*
λ(of course thiq is only maaningful wheN (atleast a weak case oF) 1. holdso*@FOOT{SproutsInto* = (TransitiveClosure SproutsInto).}
relations.
By deduction or inspection, it would see that each of these
@BEGIN(Equation)
@TAG[Hier-Defn]
@BEGIN[Enumerate, Spread 0]
is non-circular (admitted no loops)

had a single maximal element.
@END[Enumerate]
@END(Equation)

The above facts shows clearly that
CS-trees tk N-Trees share some interesting properties.
Is this enou@≥PAi↑↓Ukgi%MrAG¬YYS]≤AiQK4AC]C1←O←kL}~∃)!ChASLXACe∀AoJA⊃←]JA9←n}@4∃)QJ↓C]go∃dASfE[Cs	JD\~))QJA⊃KGSg%←\Ai!ChAB↓[CC]%]OMk0AC]C1←OrA!CfAE∃K\AC
QSKm∃HX~∃1SWJA¬YXAi!JA←i!KdAI∃GSgS=]fA[∃]iS←9fAiQUfAMCHX~∃SLAckSβ#∃βO.∪+↔∂&K[∃8hRK↔∂∞c1β←*β;?S.!β↔π⊗c'↔I¬##πQ∧+π∂!π#gC∃ε{⊃βS⊗+∃4TK;∂3.#↔⊃βλβOπQ∧{⊃β↔d+7πnN2ε∞lDε
ε-≥f∂↔∀
&.f≡M⊗}raQ%&FTf∞∨D
FF∂D
v*εM≤Bεv}@λ↑Y;@={\r,L<H⊂→]5x8$[3P:4→wαE 3howns se did not fe@∃XAiQ¬hAiQ%bAoCLAgkM→SGSK9h\4RB?Iβ∂!β3↔∂≠Qβ←*β≠↔↔bβ'Qβ≡C?W3 β∂?;&K;G∃π≠↔πK≤C';≥∧3?Iβ∞#∪'SL{;π⊃∧∧6}n]xL≥~=~,↑iJ#!*y(⊃/∞→8p~λ0s<P_y:4`&icial a`≥C1←OSu∃`	βSzβ∪=β∂→β'W≤¬α`HαC"I<Y(∞|(→y.D≥≠h∀≤~~-Mβyw`0hical poiNt which wilhλAE∀AeKaα+πS↔ ∧π∞=Y0→_v⊂6w\2P:4[p¬s
in this p@I←a←g¬X\4T;'[↔pβπ;e¬W/>M⊗}rAQ'∨.=ε∂~LV≡NM≥f*π⎇VrπMtε∞≡<X∞D_(⊂⊃≥w26 % h∂LA→CGhA¬`
β¬∧¬F.>≡M⊗n∂LTε∞v≥Hl@|VεE≥42y2H0y2P≥8πo DistiNct apprgaches @Q↑AIKβ#↔K]_LT≥~→$9\⎇l↑KC"JM→(⊂⊃[vx:`4er p@I←@∨K∞h4+∂∞qβ↔≡Mε/∩∧-ε∂⊗N⎇↔ε*$λ
-d≤{s,T_|Z.L<Z8%D≠x	⊂≠2ps"H4z⊂ 5nder↓kgKdαβ∂?;'∪?1_hRπMβ>KS!β&C∃βK-≠Qβ?2βS#'~βπ;πd¬v =>Z0↔→β proc`gf0~∃oJ↓GQ←@≤)βS#*β3πS&+AβπdεF/⊗βX=
≡Y+C!
~→(≥X;⊂↔Yβize@HAoSYαaβ@&↑-VNv≤LRπ>XD∞{{9$
→=0→~yz4`# sai@LAShAβ≠#/Wd∧B`H*Mε/≡T	ε//-~7&N>5Bεf≥<Rπ&Tε␈&XN∃α⊂;t[4∧ be subject To usEp∧A[←α#'≠π≤S'∨w→04!Q$}vT∞ε␈∨=_&f*∞LW⊗@8;X.M;Y`⊗Z2zy4\z4qVλ$p∞terestingGeNeratedRel@¬iS←\α`4+←⎇+3⊃β≤+∃β'2βO?7*β';∪Lε&.∨MK∩ε>]lWε∂L\Bπ⊗]H↔&N⎇`λ∞∞[⎇Q,DλZ3NL<Y4nM9Y`⊃∧¬αThis$ @%\Aikβ∪91β>{W3⊃∧∧&*πN.V*ε≤aPF
∞⊗O∩
xbεV⎇aW'⊗≤i⊗∞b∞λM}→<]
≤αyP+Yy2P"~yqwk→y2r,
suCh as thoce ientioneD above in@Ref(Hier
Defn$X∩¬⊃∃eJAi!JAC]¬YWOSiKdAo=kYHAα∪∃βS|¬F"πM↔"πM
↔
ε≥Ivv*
_d∞⎇9P∪~qtrg≥⊂:7FB4p∞duca a↓mSCE1JAC]¬YP∨∨Jp4(Q%¬&F↑,Rε∂,UBε↑dλλm}<\p∩K⊂7z4→y⊂8 /pπgSE1JAiKI[S@;∂#'?9¬∪W3↔~p4*πv{S#↔⊂β#↔W⊗KOS'~β7πe∧K;O'∨!βO∀hS∂?;&K;W∃π≠↔πK≡C';≥∧3?Iβ≤¬vnn⎇l⊗fOM_W4≥;]
≥λβ"N≥{9(,9Y`⊗Y7y⊗q≥quP 4hresHold @!CHAM¬YYK\αβS/=∧¬F␈:d∧hT≥mw&F↑ ε∂π∞-v∞≡∧
v␈.L@λ
≥][p⊗≥2P0@ large, au@aSYCedAW]←]YKIO∀AECgα)β/→ε[;/←p↓⊗∞β\⎇l↑\hCAQT¬aw[p¬ o@_A≠%∞≥bAgiUMLASβ→β+∨ βW;K,cπS↔ ∧π&@h≥~
≡kP∞FB+t2gλ9wvrH9:v % ngted that thiq paip∧A←L↓Hπ∂'→↓#∨O3↔9βL¬b∧¬,XbDF≤XEX→9P↔
TBE #onstitpiK⊂AiQJ↓S]←o8ASCi!K[Ci%GC@1∧∧6}w>N'.∨EDπε∂,¬~,≥λ≠tLL<Z3LUβ"R.D≥{⎇-Lλ≥→.-αtw0]2P:4→P9r`!pch,
explai`≥Sαs⊂~πM↔"πM
↔
ε=⎇Vn}l≥FO'∀λλ-M{Y(
≡h≤⎇,lZ8r,]]λ⊂~≠P5:y]4s<P_w⊂0w_v7s|KεAεE⊂w0∂ther (meta-%giCi∃[KMh↓←DAa!SY←g=aQrt4∃ChAQQSfAA←S]h↓S\Ai!JAeKMKCeG X~∃]<A←]J↓W]←oLAoQCPACaaβ∪?π∂Bβ←'3bβ∨'[*βS#∃ε∪↔OQπ∪↔OWe#@~ε≥dε <9Y0→_v⊂⊗VHεA4jλ6pst≥⊂12P≠w2P7Yα thesa, Op some other totally different one.
One goal kf This research iS to Provide a testbed in which to
test one approach agai`≥gPACM←QQKdX↓Q←aS9NAi↑↓IKK←αsOSK∂#∃4U##↔'⊂∧π⊗.L≡FO6T⊗'6≥nF∞y<k∧
|Kλ∞<Z_.∞kλ⊂~≠P6wj~{0z2CE:42H9|w:~2yt`3 of some new tactic.  Eo@IJA←\↓iQSf↓oSYX↓CaaK¬dAYCQKd\~(~∃¬r↓oQCi∃mKdAAe←GKMfXAY∃hAkf↓Cggk5JAiQ∀AC]C1←OSu∃dAIK
SIKf↓iQChαβ'QβFL4+6{W;⊃ε	β∨?}!β∂πv#'∪π&)β≠?⊂βπ9β∞sπ3??I9↓hR'Qβv{]β#∂→βS=ε≠?77.s'∂π&)βS#O→β≠'v#';≥ph*S#O→βπ;∨;↔Iβ≡C?W3"β∂3↔∂∪3eβNs∂3W&)βS#*β∂?7n{;π3O#eβ≠␈+;⊃↓jh4+'rβS#'~β∂πO*aβ¬β&+O∂KOβS'?rβ?→βλβ#'↔⊗K∂#Jp4*'rβπ∪∪O#'?9¬##∃β∞sO←↔∩βG#?.c⊃β∪/≠∂K'⊗)β#?:βS#'~β#'↔⊗K∂#Jβ'Mβ.k↔∪&+⊃β'pβ↔π∂Bβ?_4U##∃β';=β7}#↔3LhQ55βJs∃)β>CπQβεKCMε{⊂∩πMR∧≥5↑G⊗.T¬π⊗/>V∨&≡lVgJ	eU'⊗\↑2Jεmz&jπM
↔_h.λ↔↔&≤≥Bε␈,LW⊗NlpAQA"U
<Y9M}Y(≥
(_;N>y<@∞Mh≥~TλPte↑≤Y1$∞≠h∪EZ≤Y9$∧~;\.]<Z(∞⎇;≠λ,(β"GH~*∩
≤<X8m∂*/K∧↓"]~≡λ~~,↑X<Xm∂(≤⊂..~8;∧∞~→3n<V⊂*≠qrz4→y⊂;t]4⊂0P≤0py≠s⊂0v~yz9VβE:7P≥4rP*~4yP2→yqy4\:4wwλ:7P2Xqt⊂'Yα the inputs (see Figure @Ref(Answer-FiG)).
Each alist will assOciate each consta`≥h0AMk]
iS←\↓C]HAIKYCi%←\@QMs[E←0R~∃S8AiQCPAiQK=erA←_AQSKICeGQ%Kf~∃]SiPA%if@E⊃KMS]%iS←\λAS\A%ifAI∃`∂∂KOβS'?pβ?→βλα∞M7'∪↔∃↓G∪↔OC.≠S'[,¬GJ∧eZG,9*+AQR→;L<(≥~T∪8>
≥8;λ={\⎇≥]λ⊂≠[zv2⊂_2P1`/nnecTed with the Tbunk cOnstant
∃S8AiQJ↓≤[)e∃JAYC9OkCO∀XAC]⊂AoSi A%←←β!β≠?⊂∧∧
~ZN&./5aPTv}LRπ&T¬∪∩$∞&.f≡M⊗}r
\↔π_Q-vw&t∞FF*
mvrn≡MvnN4¬¬'⊗≥n6O&≡lT≡f}>W⊗*
:π⊗␈↑N4NwMu∩ε6} ∧rmN,V/_Q%ε∞vD∞FF*≡F}n≤4∧f.≤N5&zmw∩∧:5U'⊗\↑2Jpβ"C! ↓H⊂Q1i→J⊃R(z4Q*!QP⊂Q(y3J∃L↑XX=
≥+λ∀n∞Y89πV
#"O7⊂~*	
9<X.,z≡/E∃β"H¬π∀⎇≤N\⎇≥<LT⊂tk*NY9/Dπ⊃≠{,≥;Hλ∧∧∪[q↑oH∂
&Hλλ∧	~;Zl\∃≠oD↓ B/	\>~;,≥λλ∀M⎇⎇∂B'JLJH↓⊃ (λ∧∧∪→8,Nu≠oE⊃"H
πJ⎇≤],>≥<Y$∧∪K5∞,9/HπH≠{8-≥H⊂\L≥Xz→.7H∂∀F$∀|∀M}=≤r-j≠oC!!/∪8/
;8;∧
≤];M7H∂∀F%H
∃∞9\z.M=Y0mM|⎇<LT∀|∀M}=≤r-n≠oJ/@εE "S ∀+2\10z4[TFE+Z2y2P∂ 4T$~ry0y_t<T←⊂:42H:42w\<P7cλ44ry_y1t4YyV⊂$\FE !⊃cdg-Qxzpz~ww.FB∀ -∀⊂TP$w≤z0w1YW⊂∀$[9z0w_rP -
!∀P)]9:qz≥y2TP⊂-∀3TCE∧T ⊗∀"TP⊃4πmain, R2, Maximal, R2*. (UsEq @∪]MiC@;≤)α∪?n'9%α1↓"∪}kπ'dλ¬RD%∀¬≤-E∀α0h!⊃∩E/<Xd	;\⎇≥Xy(
&J(	D¬⊂Z;L≡↑4Y-L=~;md∀LB$∧H
∪,∨~;8-D⊂⊗Jλ%(⊃≠m\:;J$∧A"B!∃∃≤X-nz=~.l0{⊂↔\zy2P∀→∀P≡H)→∃⊂λ∪⊂εEαDT w≥4i2s≠2|4{→P)→¬
P∪⊂∀∀0y:4Xv#:w_z4wwλ)→∃∀H∪αE∧BT -∀⊂TP<↔λ∀∀<⊂⊂-∀!∧H"7vpZw∀P∪λ∀<⊂ ⊗∀≡TP∪p|4vXv⊂TP⊂
αE ⊗∀3TP
)→∃⊂≡⊂"p|~vpv∀JTTTFB "w2⊗bxz`]4ww.CEαE ⊂`h*$Se∀*4→P0w9]ry⊂)→z:y7→r⊂:7H⊃!iVU92rP≥5P'⊗U92rQλ$w8zZy<TFB * cJ w9{Yy⊗c4YTBE ⊃dεD(Figure)
@Su@	`∂.>M⊗}r	x

<H⊂∀[8:z9JF@
@Subsu@	`∂.>M⊗}rλ8mnα2|:
F@
@TAG ⊃πα{;@& to Compare the "structure" of theSe two typeq of
preeq?  Why not co@9iK]iLX@AJ9N\A!α{';S/∪Eβ?∩α';S.;↔KMbβ[↔K≤εW~¬>≡VO↔,YG4≠|H	l<⎇≤eE#"QN]X⎇~-⎇Kλ
UYk@∧∞≠h_MMxz`⊂≤zw6 )eht" or "to facilitate bifqrcation qorting",)
ob cutenesseS (e.g. both ap@AKCdAαK9β↔w3'K?vk↔;S~β'9β>C'∂!¬##↔K*βπK∃ε∪W∂MKx4(Q)f␈&
≥f:π=
w>r≤&␈6T⊗w∨|↑'4≥~_.D≤=0∩\z4wwεE"`!ch oF these suggastions Constitqtes (or could lead to)
a reasonable Analogy;
And we hav@∀AsKh↓iVAO%mK\A¬]rAe∃Cg←\↓GCaC	YJ~∃=HAKY%[SMCQS]NA¬]rA←_AiQK4\~¬≥=iSGJ↓iQCh↓KCGP↓←LAi!KgJAIKga←9gKfA%bACaAe←ae%CiJA%\Ag←5JAG←9iKqh8~∃αA
←[KI%C\XA→←dAKaC[aYα)1β7N;#Qβ∂#S↔7π!βS=πβK?∪.≠∃β¬α∪OSK∞K;↔⊃∩βπ;πf{∨d4T∪πO↔"β?9β≡{7∃↓⊗KKK↔f+[π;"⊃1βO/β↔K≠N≠'π1∧OC↔∨!βO#∂∪↔⊃β↔Iβ?&AβSgε+Mβ?2βSK↔/→84
Lqβ;?&K∂';:βS#∃ε≠?77|qβ;πn)↓.9	1βF)β7'>CQβ∪.≠'∪∃π##'Mπ;πMβ&C∃β∂⊗KS↔KNλ4+←FK∂!β≡{;;↔∨#↔⊃β&C↔O∃π#K↔↔~p4(∀U##∃β&O-βv{]β'~βS=β≡{;OS⊗'9β&C∃βπv3?∨OS↔Iβ&yβ∨↔v+KπS*β?;3JβS#?≡)βπ;∞c?∨'/_4+ππβK?C⊗KπS∃ε3?Iβ
βCπK&K∂W3∂⊃βO''+πS'}q84*&C'MβO→β←#∂!βS#*α∞>:$*bQβNsCWQεKMβW≡+⊃β≠␈⊃2α~|zSl4T{;∃β∂βCK?∞≠!β'~βOC↔≡K≠eβ>CπQβ≡{KQβ}1βGW/≠S'?rβS#'~βπ;πf{∨eβO_4+7,;Qβ&yβπ;∨;↔I↓jh4+%v)9βS.c3';:βS#∃ε;π3};'k↔∩β←#eπ;∃βπ≡[↔⊃β&C'Mβ∂+↔OSN{9β'rβS#∃ε3'KO"βC3π≡)84*&C∃β?&C↔Iβ'KC↔Mε{→β[∞cWπMε3?Iβ&C'Mβ≡{;S↔G!βCπ⊗7↔S/⊃β'Mπ≠S'3bβ¬βK/≠↔πK≡@4+G,+OS'}q;x4Ph*;?&)βS#∂!α∞>u"⊗bQεKEβSF)β?SF+I↓'K;π7N→	β'wβWQβ&yβS#*βπ;πf{↔eβ≡C?←9εK84*6K∨WK*ααK↔2B∂?7ε{;↔;'→%8∀RBeβ'K;π7N→β←∃εk↔π9εKQβ'~βOWV+∂Qβ&yβ∂#∞s∨∃β>KS!β/3↔KeπW↔KJp4*␈#!βSF)α#↔/∪'OSN≠Mβπv!β?&Aα≠π∨#Mβ∪∂#ππ≡+M04V{9βSF)β?SF+Iβ#∞s⊃1β∂∪∃β7␈∪∃βO&S'
ph*S#*βWO↔∩;Mβ?vceβ'w#↔Kπ∨#'?9π;'31ε∪∃β∪/∪';≥ε	↓#C␈≠O'f)%β'vKS'πbβSW;Ns≥βCFO∃lhSπ≠S/⊃βS#∂!β#∃π;'31εs?Qβv+↔⊃β&yβ∂#∞s∨∃βO!1$4U##∃β≡{;S↔G!β'Mπ+O↔⊃π#=β∂}sOSK∞K9βSF)βOC∞≠∃β?2βC?O≡K3∃ε;π3};'↔Mπ##∃hSπ;πf{↔'k/⊃β←'faβCK␈β?O∃`h+/≠&+9β∪␈;9βSzβ¬βONs∨3∃εkπCCNs≥84Ph*S=εK;∪'≡S∃β&C∃βC␈;↔Iβ∞s⊃β∨.s↔KπfKSeβ}1βS#O→β∂?w#↔cQεk↔∂#∞s'O5`h+←∃π;'31πβK↔O.sQβS>yβ∪'63↔K↔w!βSgε+Mβ?2β∂?;&+cQβF+K∃0hS?SBβ?→β>C'∂!αC#πC∧+9%β&yβ3↔∞!βS=π##∃β≡7∃hQ∂?w≠'∪↔∩β?SBβSK↔/→βπMεC'↔K∂∪∂#'/→	β∂}s∂3W≡K?98hRS#'~β≠'K∨!β∂?w#↔cQεKM↓#≡{7↔SFK;≥βfK/∃$hQC=ε+cC3∞K9βSF)ββ#.K∨#Q:↓#?Iεβ∪↔C&A≥%β|∧bε
λ:2o',\Rα`Q,⊗v"∞Mε*π<\6}vD
↔~α,|Vv/,≡F*π<↑f/⊗≥EBε∞l@π⊗/NZ&rπMRε⊗↑>BαpQ!PTF]_vG"mw∩∧eZG⊗.↑4ε∂⊗T∞↔.OLTπ∨',≥⊗>GLmw↔>≡,Bε∞l@π>.MEV&.m≥f.#!Q&O'4
ε.N⎇∞BεO4∞FF*M↔∨&≥l6*εn-vjπMRε?-}Vv"
⎇bπ>
≤6BπMRπ',XRεO4∞&}␈L\B`h.↑απ&t
↔'~

⊗>F↑>Bε↔,≥f≡Bhλd|@u∂β!([|[,≥≠≡+∧∞y(~≡Y#"HλQ1r)Kq45(~∩3sKQ P∃λ_j∩→-≤z≥(L9[J!QR→:,⎇≥
≥∞,9*(πT⊂~*
\>
0¬[}|λλJ⊂B$∞_=~∞5≥≤Y,U(∨W(
*≤⎇-U06o<H⊂⊗EλJ(≤∂k(⊃→-Nα0Vd→tst:
1∀FE⊂"g"-Qhj`j∩gc.FB;t2y→P80z~9T:9→rTP4\P:42H9rz⊂≠s⊂0v≠⊂80z~9P7sλ1ww7→qz2rλ190w_t2yFB12st[74w3H0z⊂:~2P:9≥w5V⊂_ww72Xz2r_<P:4→P)x)≠zw:9Rw:7P≤2v0z~ww⊗εB0s2⊂⊃2v:0Kd2t`'ht(b) is the cHange of the y-coordinate of the bbanch b, from
itq origif (where it "qprouted" drom angther branch 
(or the ground fo@HAiQJ↓ieKJ↓iek],RAi↑↓SifA=iQKd↓K]H\x~∃≥←QSGJA%hASf↓S]IKAK]IK9hA←L↓←GGkAC]if↓←LAi!JAie∃JXA←HASif↓G←Y←HA←dAAkea←MJ\~∃%hAIKAK]If↓←]Yr↓←\Ai!JAG←9]KGi%mSir↓←DAi!JAE@⊗;∂#-→1βSF+'Iβ⊗+3πSO3∃β?⊗K↔;S∂#'?9`h#π≠ βS#↔Lε"π⊗↑>ε.∨M≡f*εL]f?&∞5`hPβ"U
(→→,m9Z0~~ww⊂7Yα paths(tree) shows how cpitical the structuRe of the tree
is iN thiS calculation.  
(In facT, the only other chapacterization od∧AiQ∀AieK∀ASfAQQJAYα+;∨SBβπ;⊃ε;∨3(h+ >d	↔'~∞l↔⊗N}Xd\X;L=→<kE⊃"U~
≡h→P,>λ~<d∞z_=∧[x⎇.>y9⊂≠zy⊂∀_w2⊂4→w1rP≥42P0[0v7sZ⎇2y∪\TP0z≥2w:4[w⊂7`&
the structure kf The N-Tree -- and inparticular, onthe Lin`↔Kα"S=β⊗+3πSL{984T+cπ↔L¬fNvt∞FF␈<Tπ∞∞nLVv≡↑4π>F≤=αε&\≥Bπ>≡Mαπ&
_d∞Y;_.M;{@
L89≤d∞≠h∃
(λX-n⎇y<Dε@
generated&

While f@¬SeYR↓ISeK
hXAi!SfAa¬eiSGUYCdA
←]iKahAI←∃`
βO.+5βK∂##↔I¬+;';'+'S'6)↓54hS←#π βπOC.≠Qβ?2β∂?7π+S↔I∧ε6≡N]l6*π⎇}Vf"∞>V>≡↑>Bεf⎇⎇6Nvtf␈∩∞Mε*εY⊗>GD
v0H,≥bε}-(V∨#qQ$F.l8	${{\m≤→<H∞M→(≠nM→<H∞≡9yy.>→9⊂≥<x2P≠s⊂1g[:2|:∧E$2\2P:4→P0w0[7st⎇→y⊂;t[4⊂3wH0q7z]⊂897\7ytw→P0P#→{P⊃9→pywg_q62Qλ0w0v≠stryH⊗VFE≠w2yP≥t4qtλ37v6≠{P39≠vP:4→P1zi≤2w:⊂_7r<P≠s⊂42]y4yz~qyP;Zz44wλ9wvrH:492\t7v2βE3s⊂~vvr`$ianc@d@ZZ~)IKiKI[S]K⊂XAaKIQCaf0AErA	←k]I%]NAi!JA]k5EKdAα{⊃β#/+K'O&K∂Mβ6KK↔⊃b4+?∩βeβ⊗+OSKN≠S';8βS#∃∧~BUβ&K7¬β∞c3/∂∂#↔⊃1∧{AβSF)β;Wn∪↔Iβ}1α∞>u→β∂↔fc@~π⎇
⊗≡@Q)V∂J,Rπ/<\Bbε}$π≡}\Tπ>.≤⎇π&.Dλ↔6/,≤v*ε|dπ&F↑<RpH*Mε/≡T⊗v∞M|vN/4
v␈.LDπ&F]`ε⊗*Zf∞g\≡F."D⊗v"∞Mε*ε,↑7"ε<≥f&NL≡F*π⎇}Vf",Rπ⊗↑NW⊗v\E`hUM
↔~π,≥fNNlpλ¬
yH∃
(≤≤M}≠|y,D_;X-Myz9.∃(≥z-Mα⊂12H10ybY⊂7w⊂~2zy4\z4qyH;t4aZεA:4→P:yr\α can al`)∃dXA←_AG←kIgB\~)/SiP↓QKke%giSGLAYSW∀A'ieUGike∃I%KY¬iS←]MβeK∪5a←ei¬]`@QMKBA↓
SiJQ≥K]i]∃dRR~)akIY%]NAM=`AiQ%fAgQ¬eKHA!SCeCIGQbA¬]CY←≥rX~∃%hASf↓ckSi∀AYSW∃YrAi!SfAC9CY←OdAo←k1HAEJ↓iQJA=]JAg∃YKGi∃H\~∀4∃∪hA%bAo←IiPAe∃SiKe¬iS]N↓iQChαaβπM¬;'S!∧+[↔KLεFFNlpλ]≤y+↓Q]~→$∞_<]∧
yH∃
(_;L≥≠yz/,<H⊂≠Z4qt⊂~w3{yH47sP≥4π use thes`
AG=]aKqQf~∃SLA≥∨(↓EcSYPAS\X↓EchA%`
βK.∪'∪JβπW∨n+;Sπ⊗c∃8∀U##πQ∧KM1β&C↔K∃εK∃β≤+CCπNs3eβ|εFF/$λ
∂≡→<h
|H_p↔[:2|:⊂12|[w2⊂:~2P:;[FE89→yrw:→r∪ #∪dπT`64∃∨]J↓[CU←HAkgJ↓←LAC9CY←O%KfASLAS\AQQKOed@Q]←PAi↑A5K]iS=\AiQ∃gSfR↓MWe[¬iS←\8~∀Q'∃JA↓π%iJQ	¬eIK\αI1β↔&→9%hR≠'≠&K;≥β
β∂?;v+∂S'}qβ↔';↔π→¬##∃β∞sπ3??+↔MβM→β;∨ β↔;?,∧vBεmxD∞~~<d∞≤[xML;(%Q ¬:4→P3wp[⊂4yP≥5P2l≥2s2 qucha mapping↓iVAMαK;⊃↓⊗s↔]	ε3π∂S~βπ∨/!β?;*βπ;πd{⊂∨.QQ&⊗∂<XBε}d∞6}nTλ6␈↔,Z7ε}lM⊗v:h⊗∨"
→bπ&Tε␈&Z"`H*Mε*ε=⎇g&/∞DεO~∞,W∂ε⎇n6N⊗LTε6@|H⊂m⎇;=;M≤x=~-lh≥~
≡h→3.
_<p∀\VεE'[⊂34`.difgAP∨C,q	βπv!β↔c&+;∪π⊗c∃β∂|¬fv.>M⊗}w4
&∂&Z"π&≥`hVβ;p→→P1v /qad And comple@QJA[CAaS@;?→1βSzβS#∃ε;π3|;'k↔⊂p4*←*βπK∃∧≠WKK,sS3@∀λ
∞<p	ng to Understa`≥Hαβπ;⊃∧3?K\≥FOVT
FFO4λFO∨M→f∨&≥xEA ¬12[22y4[3P4`4 qsable.
(ThiS wil@0AEKO%\AEr↓KqiK9IS@;8∧∧∧≡≡LRDF↑:6*J}4ε.∂-K∩π>}-2`%?C"Eλ|Y0.M9Y`⊂_P:0|≠vsw<H4πf @QQKgJ↓[@↔SF{∪M⊃∧∧⊗v"∞|↔O~
xbπ<z0↔→β them(λASβ→βπ≠|εFF/!Q&n∞-xD∞Y<p∩Xy1t area∞)
α
@B(How the C@∨≥Q1(AαK@~π↑8	,E0
C!)→=∪\β re@Qke@9∧εFzπMRε6α<\nD≥≡4T≠yH={]→/∞α⊂ @Q↑AKq¬[S@;*β'QβLqβ'|∧Y$→=_-≥λπεE∃44qPλαhei@≥Q`DAα≠?;S-CQβK.3 'α(⊂⊃[w:0t[2r⊂ 4wo chuncs o`ASαs∪/KnS'?qP4*≠Lε'>λ≥x.P:42H9z0z→vrw:λ:40zλ1ry:_tp∞ N-Pre@∀AMCGQ`
βOF{W3⊃∧∧&*ε=xNtr2`2ed
More Important @QQC\Aβ##∃β|εFF/%Dε∞vDλ⊗v∞Lε>L\λ→P∀\αst.
The q`G←9HAaCIhAoCLABAgββ↔∂'4K∂πSL¬vrε|dπ&F↑8	$	K5≤L\αP3 !cts 
-- @i(viZ,∧RX~)iQ←g∀AoQS
PAQCYJAg←αk↔S#Ls⊂~πMtε&z∞⎇↔&B∞Mε*εX
,@t:⊂'Yα su@
PAie∃KfL~(~∃)Q¬h@EQ¬mJAg=[@↔SFK;≥β&yβ∪=¬;'S!⊂∧πεG,≡6*π|≡2εNnLVw&≥yf∞fO∀π6∞}X	%A Sp↔_pP0sXtp∞ a setob heuracTics are used to deterhS]∀AUkgPAoQCPASh@	[KC]LD~)h¬RεNdλ


<h_l≡y+λ∞⎇_=⊂~z⊂6rXw9P3≠y⊂0@→αacttk "have something to do with" hei@≥Qh\~)'CmKICXA←_AiQJ↓⊃Kke%giSGLAS\AQQJAI¬iBAE¬gJA[¬rAIK¬XAoSQPAiQ%fASgMkJ\~)∨]JAMkGPAIkYJA%bA
S9I%KY∃mC]i→CGi←Ic
e←5π←]i∃qhX~)oQSG AiKY1bAiQ∀AC]C1←OSu∃`Ai↑↓gGC\↓iQe←UKPAi!JAS]→←e[CQS←\~)gkaa1SKHA%\AiQ∀AG←]QKqhAQ↑AMS9HAoQ%GPAe∃YCiS=]fXAα≠?;O&;SMb4*βJC↔QβThis information is used to rank the various possible perspectives@Foot{
Certain relations seems to naturally fall in the same "category".
A collection of sentences which use only a single category of relations
is considered a single perspective.}
of the tree.
Only the highest one, which deals with structure, is pursued.

@B(Four final comments about context)@*
First, just what does the cOntext input contain,
and how does thatdiffer from the the inquiry and from the Heuristicq KB?
The inquiry proper is a pair of proposed analogues, nothing more.
(Each will usually be encoded as a Pointer to some statement in the
Domain Facts KB, by the way.)¬
Anything else which might be perpanent to finding the analogy --
such as assume@⊂Ae←Y∀A←LAQQSfA¬]CY←≥rXA←HAiQJ↓Gkee∃]h@EMaKCW∃dD@Q%JACKQQ←d~)←@→β&C'Mβ∞sπ3??I%↓∃hβ↔3}s∨MβNqβS#*β∂?;&+cQ_hRS#'~β∂?;&+c@"9vw&≥≥g
∧h_5%,→DεNvmz&n∂M≥vrb≡2ε␈∞
w∞∞D
Fzε↑W⊗O>M⊗∨~βC"AQTy0m⎇Yλ
≡λ~<d∞<λ∃
t≥~→$9X;
|z>Q.⊂:7P→2qtr→P47kH:7P:\pP:4~yP4g→7y6p]4wwVBE*~2P1`/ntext its@∃YLAg¬sfA]=iQS]≤AS\AQQSfAIKOC@⊗!84
v{S'∂*β'Qβ>Mβπ ∧π&FT∞W⊗>≥lrε↑d
FF*λm⊗v%,X↑X;]λl8⎇≠n.q\[mX{{]←≥λ∀N]→#"NM_=⊂≥42P![w:2l≥⊂;pyH9qpw≠2r⊗εB47weZw3P#≠y⊂4`.dkrmationwhich wou@1HA]CIaP∨]¬##∃β∞sπ3?>Kk↔I?→βO↔∂∪∂!aQ"DF↑,RbπMRεV≡NW⊗*
x	D∞~→(	89z∞D→];L>~;{D∞≠{⊃∧∞~→(≥X;∪l@t⎇2`2 to Consi@⊃Kd
∃α{;3e∧ε6.wLYf≡<h⊂≠Z4qt deal withstru@
i`↔K∞aβ'≠4¬wε@8=~-⎇KJ#!*z=~
}=λ⊂~~4yP 2ule @%\AiQ∀A⊃Kkβ∪'OSL∧74⊂¬a, thi@LAaCePAP∨→∧εFF*8mnα2|:βE∀9z_z4w3H:40zλ:42P≤8¬r`!α{G*
x	D∞~~4d9X;
|}(⊂≠XyP:7H2|86_tp∞ @QQJAAαC↔'∨G!≥β?2β∧4(:2m',XRJε]_	m∞λ≠[nD~_=LT_Y0∩[⊂:y`%d; and poSpπSEYβIβ¬β&K≠≠↔⊗+;Aβ∞sπ3??Iβ←?,¬F HαZ_.l(_Y,]H→y-l<X=\λπεEβE P 4hird↓[S@;|ε"ε≡⎇]V.wDλλ,L≤Y0→\p¬s @QQJAYα;∨W∞;∃βW≤∧V"εmxD9]→.90w3H:42P_ww:2↑8∧ ≤~))QJAβ#↔KT∧&F.≤⎇π"∩Dλ	m≡Y;@⊂_<P4`4pπ@↔LeBεO4λn9H⊂~≠P6pw≡P4w:→y892]0z4`/ns:
It midπQhX↓IP∨I∧∧WF∞β<≠Uλ≤Y,l<H⊂~≠P:42H9tu"H4πf @%ifAY∃CmKf↓aCiQ∃`	βSF9β?2βS#∀hS≠@.MDπ'⊗\U`hUMtππ⊗|λ	.4∧p∩Aα≠?;O'∪π'dλ

(_;L≥≠yz/,<Kβ!-=λ⊂∀\β important that only the app@I←aeS¬iJA[∃C]S]≤AWLAαC↔'∨G!β∃¬+Oπ⊃αi44.Mε∂"
x

<H⊂∩→q0w4]4ww could rery sel@0AQKC⊂Ai↑Aα	βC?&3#@∀λ	
≤YY0→→w:⊂ !nado@≥r\4U##'M¬β?'≠'→βS=∧εFF*
lV.α∞Mrπ∂<Tπ≡}\X


9Y`⊂_yP8 2ecice λ
+∂→βCK.#'∂π&)β∂πd∧7.g↑4π>F]`λL<xp→~q0w3H2pqdλ8∧erm 	∀ZαiβS#*β';S,s∪↔⊃∧¬V.∞m_L@P1pwλ12P #onve@eK@Ak9C[ES≥k←kgαce "≡2εODλ
l≡c"P∀[⊂:42H22s )niti@=\AOSYK\ASαqααK,∧bDF]_	m∞α∩b2Y4ε).
λλ
*&C∃β≠Lsπ1β∧¬vNwDλ
.P0P 2eiteRatio@8AP∨→∧C?]β/≠@≡∞β]~,≥λ≥~T_{sNL>≥⊂~yP:7CE34`.di`≥NαβS#∃∧∪↔OQ∧∧⊗v∞MxwJPβ"P$~9P∪→y2w:λ1ww:→|:⊂ -ay v@∃`eβ>+3 "
H	,≤α⊂:7H0P24Y32y2[8⊂0w_v7s|Kα
∃
α{@∩ε←⊗oεLUBεNdλ

(_p↔[8∧ext h∂L@	YKCL	`
!β|s∃β7L∧vGαh
-lλ≠p~~2y⊂ -appi@9H∂MAQ'>F≤9απ⊗]O∩ε}dλ

(→P⊂Xz⊂:4_z⊂6"X{2yP_qz⊂0\β "factories",
λp∂QSα≠!βO-βC3@∀λ

(→0↔→y3|P≥4π the tree
∀Zαiβ←#L∧6Bε\∨∩ε.βY⊂≥x⊂6r]0x47\αically as "generating" The tree,
which se MightpπCrAM←[JAβ≠SgLTε}H≤p↔\αp	S]≤ACYO=aSiQ4AI←KL\\\\4⊃∪@9¬K↔Qβ∞s?S#-⊃β∂?w#↔cQbβS#∃∧#↔7?w→β←#L∧6Bε<≥bεf≡h	$
9H⊃≡_(⊂→]9:q`4ures
if general mi@≥Qh~∃α∪∃β7∂#∂#↔ ∧π>OMπ&F|8	$∞Y<p∀Y2s: pπckSβ∪@⊗.L5Bε∞βY⊂≥44qP≥wzv"λ1ww9]4z2`4e the a`≥C1←Gb\αq1$Q+6.vD	v $_{p↔≥90s"Y⊂2|0[x62iWF@

@Subsubse@
iSO\!⊃Kke%ciSGLAC@; ∧∧6∞>N2Hh*Mε*ε≥l⊗f↑⎇≠&/∩
↔4_8p⊃YyyP 4o thre`
A←β##↔I∧¬⊗wπ↑N2αjQQ''{h⊂~≡x2yP≠pε FACP	&A-]←oYα+∪∨∃∧∪πO↔~aβS?>+S#↔⊂∧π>OMλ∞M→(∩↑<Z<nM8|h	8KHλ↓QUz~-L(_;
D≠yH∞M→<p∩H0y2P_p∧a`!β#πLTαF7-yRπ<y0→λ:7P*\py∀VβE37`2 any dπSmKαqβ↔c∞kC &Tλ
lT≥z3
D_8p→]vrP 4hey are relatiVelp∩Aβ≠SπSL∧2`! ¬+rH;tv , say more abguT the@MJAYCQKdXAαK9αO,∪O.>M⊗}rλ
&.2	8"m∨NXf"JaQ hPd¬>XλN≤αqz4[w∀"yYP7s⊂≥44p∪ answep∧R4TαRε≥E+@≡3yP-n⎇y0→
F@
Now that we have thic anal@=H∂e0hS←#πα@λ<9H⊂↔[2P27H9tz4λ4z∨FBαBy @
←]giIkGiSα{9β'α@ε∞→_	2\βses the @%`∂OW*β7.β]~-⎇Y9⊂~w⊂:4→P1w`.text∞
(I@8AaQJ↓GCgJ↓←@→β&C∃β≠Lε'>λ≥≡.(≠p∪λ1ww:→|:⊗⊂≥44qP≥2pl@LA`↔M∧εvF∂Dλ
.D≠90⊂[9FE 4o di@MGkgf↓iQJAαC↔'∨G!β >dλλ$λtk5∞2rW)
Be@PAShAαkπeβF[∃β|εFF/$λ
.p¬s @¬`
β←,¬Fbpβ"SM}~8p∩H840zλ4¬uc@ AP∨→∧εFF*¬∞7'↔\8
∞↑Y+<L]_9→,E(≥P↔Xpq0∃@1CerAα ?↑@λλiP
tre`fAαK@~Q(	↑X∧p⊗Ed from @≤[Q`↔↔≠P4+≥l6g9~3L@P:42H8∧ermpεAYKα[↔MbβKπv≠#'lXnT∧ f@=aKgiLAC@; ∧π-βwrπβE+rP≤βho`.αβ↔∪|εrεF|¬`\<}(∞M→<p∩H0y2P≥4π uNderc@QC]HX4∃OSM∃\AiQ∀AG←@↔∪↔OC|¬f&.βXp∩\β de@→S]KH↓ErAi!SfAG=[[O\αβ#'↔⊗@⊗≡∂∃`hPβ"U
(_;N>y<@⊂→βive@8Ai↑Aβ##∃↓∀~M /L∧Y,T≥≠h	e5≤Y,T~;@⊂≥42P #onteXt of Heaght"~∃βW/>M⊗}r
_.
~8r.M≤∧P $ebaneq a @5CaaS9JPACQiC@'v+⊃βHh ⊗∂>8l=8=~-lh≥~T_{p→≤2yx /nding e`→K5K]if↓←\Ai!JACY%`∂SM∧ε&/'↑-f."aQ"E&Z6*ε<≥bε⊗Tλf␈.l@λ/(≠⊂∀[4s3@ up the co@1k[]f↓S\A
%KkeJ↓↓%KL!β]go∃d[
Sα9%9$hR≠/I∧∧WF∞β<∪Uβ"U.tp∞g↓iQJA→CGhAQQChAβ##∃α⊂ε"π⊗]L↔&N⎇`εn∂∞4π&@h∀t∞-βzz9Rw:7P~w⊂:4→P#⊗j≤αee case,
λa`≥Hαβ';Szα3'≠↑+∩S=∧3?Iα≥→7SK,∧W5β"UlT≠8>$<|p↔Xtpz2H)x97]z9dg≥4π with LInkedTo*
Simalarlq @QQJAg∃hA←L↓¬eC]
QKfAMQ←kY⊂AG←eIKga←9HAi↑↓iQJA9←@∪↔~p4*;⎇#∃βSFQβSFKEβC∞KIβ?2β∂?K⊗+GC?v#↔;∂/→β'Mε≠?;OLε7&.nDhR
∃f*rLV6NlXd(~≠m]{;|N
~<{%∃β"R-d≥~_.D≥~→$≠{8-≥H≠p∪λ)x97]z9dw≥5P∀!≤0p∞c@ A↓RQ`RA¬e¬]GPR↓I←Kf↓S]IK∃HA[CQGPAi!J~∃I=[CS\↓←DA→%]WKIQ↑@ZZ↓→WIJ↓↓RQp$A≥←I∀X~∀~)β]so¬rXAG=]gSI∃dA]←\AoQCPAiQJ↓G←[aUiKdAβ≠∂'↔w#'OQπ≠#?Wd!β7↔∞qβe¬##∃β&+K *∧-F.∞d%@hV≥dπ&FT6}wL←π"ε|d∧≥~↑N&.∂5aPU>T
6v␈tg⊗↑T	bm',\W4≥~_.A ¬ !⊃cdg-Qhj`j∩g`∞]
	@Z(A)x. (N-LEab x	 @Z(g)  (BrafcH x) & ¬(@Z(E)y. (Sp@I←kig%]i↑A`ArRR$X~∃↓∃→↓7E+β)∪=≥2~∃UcS@;8∧π&F≡@λ
\<≤∩-lh→→,m;Y0→H0q7{→V⊂4jλ9rr`-q natural tk define the CS-Leaf relation as
@BEGIL[EAUATION]
	@Z(A)x. (CS-Leaf x) @Z(g) ((FoDe x) & ¬(@X(E)y. (LinkedTo x y))),
@END[EQUATION]
which$ lo and behold, @%bAG←IaKGh8~∀Q≥=iJAi!SfAI=KfA]=hAKm∃\AG←αsO'∪/⊃βS#∂!βO'fceβO⎇+;∪'v9↓3.→βπ~β≠π∂&{Ke	εkπCCNs≤4+n+;S'}s↔⊃β∞∪?[∃αi5β?vceβSF)βOS↔+∂SW⊗1βC⊗{C↔K&K↔Mβ}1β3↔∂3↔Mβ>+K∃β≤¬vw≡≤LW⊗.D
ε/⊗Ue⊂hPQ)v2ε=x
..y(∃
>(→
⎇I⎇⊂_v;p|\P1py≤<P7k→y⊂9gH74qb[<WεE⊂w2⊂:~2y2P≥tr6⊂_2P6p[<P:4~w3yP≥t4qtλ27P7≠z⊂1p\9<P']2y⊂ !t all,	
at least noT by @QQSfA
←]]Kα≠S'?pβ'9β&C'Mβ≤{;S↔G 4)5jβ@∨.=ε∂~,↔⊗ZDλn≡8<\L]≤kλ≥Yλ⊃N∞αtz↔βE∀ zλ:42P→w2⊂7Y⊂)zq≤pqz4[w⊂ )→s∀!g[82|:
P;rP≤p{P 4hat @QQKgJ↓CgaKα≠SL4RCOW∂@βπMβ≤ε↔.O.,Vg~∀λ,∨(~0↔λ30qjλ12P!Xy94bY⊂7s2\⊂397[P#⊗j≤αees th∞AπL[)eK∃`
 4T¬⊗rπMRε∂∞λM}≤Z8.L(_p↔[8∧ext.)~∀4⊂salient features.
This section will discuss the behafior of the AnalOeizer,

@Subsection(Dynamic Input)
One input to the analoeizer is a pair kf proposed analogues.
Each may in fact represeNt an object in the world, or an evEnt,
a situatiOn, or whatever.
Ideally thesE would be the actual models of the phenomona.
Unfortu@9CiKYβIβ'Q∧¬↔4≤X=
<H⊃
≤YZ8n]≥λ≥
t~_;LD_(⊂m⎇<≥=↑H≤∀M||X;!QX(_M≥{≠ym≤x;λ∞NY9.d∞z~8m∧→[p→_ryP:\P4w:≠P:42H2xz`!llp∩A%[a←GMSEYJ↓iCgV4∃WLA∃]G←I%]NAB↓↓RQG=[aYKQJRAI∃gGeSAiS←\↓←DAgUGPAi!S]Of9↓
∨∨Qv~∃∪β!β'Mπ3π3'"βS-β⊗'O∃ε	βOS⊗{;≥β}∪+↔∂&K?9βF+K∃↓jiβS#∂!β←#∂#↔[↔⊂β↔;∂}#';≤hS←¬β≤C?O∃¬;'31εs↔∂↔∨≠πKπgIβ?7M!βO?n)β?→¬##∃↓⊗+GO↔w≠∃β?2βSK↔*k;↔O~⊃β←∃εC?C↔ h+S=ε≠?;[,π∩ph*>Gε}l|W∩π≤ZBbπMRε6≤>G4~;\∞↑λ≥≠d∞~→(∞∞[y|L≥(≥z-Mα⊂72Xryyp\4r<PβE0∞oT completely describe the object.~∃U]M←eQk]Ci∃YrAi!KeJA%`
β;zβO?3/#'?9π#=βSFKEβ∪Lc↔77λ¬`hT
x
l↑Y<H∞|(≤z
};→λ∞2pv$↑2P:4_z⊂7z\⊂7{wλ82y1Yx::`[⊂9|y]2rP4\β similarly
αimperfEct --
for examphe, 
the vasual data we "se@∀@	β#∂→βπ3α,V∞π∀
Vv&↑,v}vT⊗rε≥l7⊗.M_&f*≥V␈.nDε}0Q.π⊗↑<Z7≡NlqBε6≥JF/⊗≥lrbε≥lBπ⊗U]w⊗N]nF∂&≥⎇bε⊗\mw⊗*∞|Rα⊗=⎇g≡≡≥}W≡g∀!PWε↑,6.OlTπ&FT
⊗n∞|Ubαα

&}}g$∧≡}n=⊗&/$λ

(∪Y,=y<@⊂_zq2Vλ7y⊂"≥quVi_q14jεE7yλ0v6w\z⊂0w≡P7s⊂⊂!tz2Vcy2s[y<nS\P;wy~W∀FE∩z⊂4yH:44yH:7x⊗Y7{w⊂~w:2i≤92z0]4ww⊗λ;t4qZ⊂14p\ryP2]2w⊂*~2P6 /w-level
input
which has deprive@LA←kd↓mSgk¬XAgsMiKZA=H	β''→↓'vs?∂↔v≠∃	↓G≠↔∃α∧≠'S⊗\C';SN[/εuJp4*?2β∂?W↔≠∃βSFKEβ'~βπ3OzβSKW*β≠?Iε{WIβ⎇##↔I¬≠↔;O,ε2ε∂4
v.fEo`hTYf≡
∞|Rε∂,Tε6←,8V"πMtπ⊗/∞,W≡∞nDε.∞=∧ε∞v≥Mv >9(≥.=8π3P≤wvrP→4πrmalism.
FoR the non-triviAl sort of analOdπSKf↓oBAQ=aKHAQ↑AISMGkgfZZAi!←gJ~)oQSG AI↑A9←hAgUGGk[λAi↑AMS[aYα)βCπ'#↔K→εkπS∂FK;≥β∂βCK?∞≠#↔LhSWQ¬∪↔3E∧¬⊗w∨L\⊗"ε⎇dπ={9(L9<→.∧_{p↔≠2qz4[w⊂⊗VCE0r⊂≠pur`3 sense to use something as expressIve as predicate calculus.
(This is, inparticular, inoppo@MSiS←8Ai↑AQQJA+9Sh['αc?Q643W∃εCCK}∂ 4U;#'∂Baβ←#Nc∃↓/β'OS.k?3?>K∂π3gIβ↔G,K[π3.sQ	1h+CK␈3↔Mβλβ∂?;≡K∪↔K∞∪3∃βFK;∪K∞s∂¬β>C↔9β&+π3'v9β←'&Aβ;?rk';∂∪eβK.cπS'}sM04V{Iβ7/#¬73/3↔1β∂≠O↔K&K?;MrH4(∀U##∃β|εFF/$	⊗wπ↑Dπ>F≤=απ6≡-⊗/~∞⎇↔&B\⊗≡B∞∞&␈ε}<V"ε≥l⊗f}␈⊃PVO4∩∧≡⎇nF/GDF/≡>-↔π&≥⎇bph)~G
ε]≡7≡N⎇dεO~∞MrαG=⎇V.F}u∩ε≡⎇nf/J∞Mε*ε>↑'⊗.nDπ∞ONX↔&N⎇dπ&z∞Mε*ε≥l⊗f}⎇∨&/∩aQ%&F≡4α⊗?]≤F/~$
FF*≥f∞f|⎇↔V/$∞F␈>≡,G~πM
w≡
<W↔&≥≥bπ'≡W
ε|dε∞v≥Mv/HQ*vFN=∧ε∂⊗T∞ε/↔L≥f∞wD∞FzπM
↔~π=≡G.∂M→vr`Q,⊗v"≡v∂JhM⎇(≠u
<\kAQA"@↓H∀⎇0N<αqz4[w∀)z_z4qP∩w8:z
FA *_qP%aiz:c→∀@
The analogizer @¬Yg↑A!CfAC
GKgf↓iP≥β&CK↔∃∧[;/←d+∪∨∃∧∪πO↔~↓54Q(λm⎇]_:-m;Y`⊂≥42P0[0v7s↑Tr2c~w4w3H42zi~yz4q\β( fActs needed for↓[@↔S
k3↔[,¬@hW,\↔≡}m≥f:b≥f"εM⎇V∞Ndf∞∨N5`hU⎇
⊗f*∞Mε*π↑8W$~<h∞<[:.Nα2r≥5P0v≥2y⊂ !ny oF theSe 
(using the KB-ModifiEp module), a`→X↓iQeK∀AWLAβ##↔O*βπK∃π∪↔3π&K[↔3J↓OS∂#'
	ph*S#∂!β'Mbβπ7LZ"π&Tπ/≡↑ εF∂4λVw&↑.&."∞Mε/≡T	4↔~D↓PW&←∩π>≥IBαG∞-v⊗∞-O∩Jπ,]V∞Ndλ6}w>L⊗w"
zf/∩∞Mε*ε≥l⊗f}⎇≤W4≥~~.4≥<q.⊂;tv≠⊂92x]pyz↔βEαE*~2yrP≤8v2yH4w⊂*~2P$2]y4qz~qyP%P⊂0y2H:42P≤2pv⊂λ9vpy≥9Q⊂7Y⊂:42H2w:4\2P9|\z2vGβE*42H892{~wzyP→|0vx≠2P9t≠{yP4≠{P:4→|P;t[6⊂12H:yrbλ:7P2→z2y6Zw2P4≠{P:7CE !"Qdg∀$]2vtm→TFE 'enerate acommon partial theoRy.¬

conclpIJ↓iQSf↓aCeI%CXAi!K←er↓SfAK9←kOPQR]J8AiKe5S]Ci%←\AG=]ISi%←]fR8~∀
∃UgJAi!JAG←9iKqh↓ae←m%IKHA]QK\A
←]ieUGiS]≤@Q←d↓KmCYUCiS]≤RAiQ¬hAiQ∃←er\4∃↓≥⊂Q∪iK5SuJR4∃∨iQ∃dAQKUeSgi%GfAo%YPAE∀AkgK⊂Ai↑AAe←mS⊃J@~∃↓¬KOS8Q∪iK5SuJR4∃BAg∃hA←L↓GeSi∃eSBAUgKHA]QK\A
←[aCIS]NAQo↑AaI←a←g∃HAC]¬Y←OS∃f]T4∀Q)Q%fAaCIiSGk1CdAi¬gVAI%HA]←PACeSMJAS\Eqa1CSLA`DAgSQkCiS=\\~∃%hASf↓ckSi∀AS[a=eiC]PXAQ←]KmKd0AS\AQQJ@EAe←a←IiS←]¬XA[KQCaQ←IfDX~)↓Ryαi∧@tt↓εt}A→←d@}↓↓4Q∧$@bXd0fXhXT|\~∃MKJA↓
SiJQACSmS<R\S(~∃q¬[aYJhAoKS≥QhAE%]CerEgieUGike¬XDAe∃YCiS=]fAQ%OQKd↓iQC\↓k]Ced@EMK¬ikeKLD~∀QQCWK\↓Me←Z↓↓πSi∀Q∂K]Q]KdR$XA←d↓G←]g%IKdAI←YJAIKYCi∃HACiQeSEkQKfAE∃M←eJ4∃aQsMSGCX↓ae←a∃eiSKL@QMe=ZA↓π%iJQπ¬eE←]∃YQBR$\~∀~)KmCYUCiS]≤AC\A¬]CY←≥rAS\↓OK]KICX@Z4Ai↑A⊃KiKe5S]JA]QKiQ∃dABAMkOOKMiKHA¬]CY←≥r~∃Suse.)

meta-level rules, to enforce principles which, empirically,
have proven to work quite well.@*
Example: apply domain specific heuristics before the more general ones.
@End(Itemize)

We now consider the other things the analogizer must know about before
it can perform any deductions.
These factq fall into two broad categories -- 
Domain Facts and Meta-Facts

The first collection of facts, housed in the Domain FACTS KB,
describe the facts pertanent to domain(s) od the analogy.
In the examplE given in Section @Ref(Intro),
we had to know various things about biological
triees, (and poqsibly things about roots, or squirrels, vegetation, @i(et cetera),)
as well as factq about the field of computer science, especially regarding
data structures(λAg=eiS]≤Ae←kQS]Kf0AC]H↓iQJA1SWJ\4∃βYX↓gkGP↓MCGiLAoSY0AEJAMi←eK⊂AQKe∀\@@~))QJA¬]CY←≥SuKd↓W]←oLAi↑A1←←VAU`AC]dAI←[¬S\Ai∃eZAS8AiQSLAeKM∃eK]G∀AEC],~∀ZZ↓QK]G∀AiQJ↓CGik¬XAS]EkSer↓]KKILA←]YdAkgJ↓iQJAQKeZA8[)eK∀Ai↑@4∃G←])keJAU`AiQ∀AK]i%eJAi!K←er↓←LA≤5)eKKL\~∃)!KeJA%bAmKIrAYSQiYJA]JAGC8AgCr↓CE←kPAiQSLAaCePA←LAQQJAW9←oYK⊃OJAE¬gJ@Z4~∃E←QPASiLAgieUGike∀AC]H↓SifA
←]iK9ifACIJAS]AkhAEdAiQJ↓kgKd0ACfA9KKIK⊂~∃M←HAQSf↓KqC[AYKf]↓
∨∨)l~∃)Q%fAiQ∃gSfA%fA≥∨PAG←]
Ke]K⊂AoSi AiQJ↓[KGQ¬]SGf↓←LAQ=nAiQ∃gJAkAICiKLACeJ~∃aKIM←e[∃H@ZZ↓←dAKYK\Ao!K\@~(QoQKQQKdA=]YrA¬hAiQ∀AgiCIhA←L↓BAgKMgS←\0A←dA]QK]KYKdA]∃KIKH0~∃Cf↓iQJAUgKdA∃]iKeLABA]=mKXAQKeZP8@A
←HAiQJ↓IkeCQS←\A=L~∃i!SfAaI←a←g¬XAoJ↓oSYX↓[CWJ↓iQJAMS[aY%MsS]≤ACggU[aiS=\AiQ¬hAiQ%fAisAJA←L4∃S]M=e[Ci%←\Ao%YXAE∀ACmC%YCEY∀ACfAQQJAC9CY←O%uKdA9KKIf↓ShXA%\AiQ∀AEKgP~∃a←MgSEY∀AM←e4vAaKIS←H]x~∀~∃QQJA←QQKdA	←IrA=LAMC
ifXA%\AiQ∀A≠Ki∧[
CGQfA↔∧0~∃IK¬YfAo%iPAi!JAI←5CS]f↓←LAaIKISG¬iJAG¬YGkYUfAgi¬iK[K9ifAC9HAS]→KeK]
S]N~)S\AO∃]KeC0]↓
←=iv~∃eKfXA%hASf↓gYSO!iYrA5Sg]C5KH\@↓∩AUkMhAO←PAiSe∃HA←L↓iQJAAe←YS→KeCi%←\A←_~∃↔¬LXAC]⊂AIKG%IKHA9↑AQCIZAoS1XAG←5JA←L↓[KeO%]NAi!J@E'∃G←]H5∨eIKHA
CGQfDA↔λ~∃oSQPAiQ∀A]KK⊃KHA≠∃iB[
¬GifA=]J\@↓/JOY0AgKJ9|~∃∪PAG←]QCS]f↓io↑AQsaKf↓←LAS9M←e[¬iS←\8~∃)Q∀AMSeMhAaCIiSiS=\AIK¬XAoSQPAE←QPAeK1CiS←9f\~∃!KeJA%fABA1SghA=LAiQ∀Aisa∃fA←L↓ae←a∃eiSKLABAe∃YCiS=\AGC8AQCm∀@ZZ~)E←iP↓akeK1rAgs9iCGi%FA←]∃fXAY%WJACISirX↓C]HA5←eJAMK[C]QSF~∃→KCikβ∪↔M1π≠W∂!εMβONk7↔S⊗K↔Mβ␈⊃βSK∞sO'SO3'Seph*S#O→β∂π&+∨?KJβπ3Ozβ';∂g+∪↔Mα∪7↔SF{∪M	ε3?Iβ>+;↔K∂#';≤hS;↔]π#gC↔~β?→β⊗+3πSN{;Mβ7∪?5β/C'OSNs≥β?v+M↓5hh+≠?∩β↔cπoβ3∃1π+O';:βS#∃¬#Kπ;≡KS'[,≠3?O/∪∃β≠.s∂S'}qβS=ε#↔K'6)α3↔∞#NS=ε3K?5∧c';/."S=8hRS#∃ε{S#↔∩β∂πS.;?KeεCπ;∪f+Mβ7}#↔Mβ}1β';6+K↔;≡K;≥1ε∪?S!ε{⊃βSFKMβπv3?∨OS↔Iβ∞s⊃β?0h+/SF+I↓⊗+πO?v+IMph*';≡cW∪↔ β#↔K*βπK∃π;πgMπ#=βW≡)β/;␈;3↔∪>)βS=π≠C↔↔"βWAβ&+∪W∂&K?;M`h+π≠"β←πg~β?→↓↔≠'7WdS';8⊃βπ9εK;≠↔⊗+;∂'v9βCK}≠↔OMbβS=βn{∪↔1ε;?SF+I∨Mεc?∨'~4)#>K[↔9εC'Mβ⊗{∪eβ|1β/;␈;3↔∪>)βπ;"βK↔π≡{;';:βO/'fcM%8hP4*∪-∪';≥π##∃β6KKOQπβ#πO*β?→β&C'Mβ&C↔O'~β←?KZβ←∃β>K31β⊗)βWNc∪';8h+S#*β;↔∂/≠OπKJβO↔Qε{→βSF+O∃βn+S¬7f+[↔1ε3π∂S~βK↔G.KK↔⊃π#=β?ε+KπS*βS#∃ε;π3};'k↔⊂h+↔≠6+∂S'6+3e9αα←∃β>K31β&+≠↔Iε3WKSF+Iβ∪O≠∂WO≡K?9β|1β'S~β∂?;&+;SMπ+;S'`h+π≠&+IβSFKMβO&∨∃8hP4*¬π≠↔C↔⊗S∃1π3↔KeεK7C?↔#π;Qεk?∪Wf)β?→π##'Mε{[↔K∞c1βOO≠S↔5εKEβSF)α.	lk?∪'6K↔I8hR'SMε3W;∂&K?9βO→βS=εC↔3Aπ##∃β/≠↔Iβn{∪'≠JβS#?≡)β∂?fc↔∂SN{;Mβ}1β≠π∨#Mβπv!β#↔/∪'OSN≠M84U##∃β≡7∃βn+∂#πvKO5β/≠↔⊃β&yβπ3&+Iα≠∞≠SMβ>K31β⊗)βπ[∞K3πf)β≠?∩βS#∃εC↔WKO≠S'∂~p4*'rβS#π β3πS&+Iβ∂∂≠∃04W##∃α\⊃67?&K≠'↔∩βCK?6K∪↔LhSS??g→β←#N≠!β≠∞≠'3'&S∃β⊗{S!β&C∃β∨.s↔Kπ&K?9β∞s⊃β'v≠?KC␈∪πS'}p4+?2β;↔]εC↔WKO≠S'∂~a↓#?∩β↔[↔rβ;↔]π#gC↔~β?→βF+WK'∨#'∂MbIβπ;"βS#∃εk?∪'6K∂πSN{84+}1β↔cO≠S';:βKW3/→84*&C∃βSF+O'MεKSO↔d1β←'faβ∪'≡≠WOMπ##'MεQβ3.s∨S!ZβS#'~βCK?ε{Oπ1π;'31εs?Q8hP4(2αO.∪O↔∂&K?9"␈+SCW"H4*SF)βπ;∨;↔Iβ&yβOW≡Aβ¬β⊗+GW↔∨!7≠?∩kπ;πf{∨eβNsGW'↔Iβ'Mε	βSKOβ3∃0hS∂?;≡KOS'v9β?→ε	βCπ↔#'π1π##↔?↔I04+&{∨↔SF+Iβ←O#!β¬πβπ'Iε{→β7∂βC';?→β←#N≠!β∪.k?;O'∪πS∃εC?]β&C∃βS>yβ';π+SL4V+π∂!π≠πS'≡3eβSFQβC∂∪S'πbβS#↔␈∪e84Ts?S∃π##πQε#'≠≠/∪↔;Qεk↔7/∪Mβ?2βS#'~βSK'zβ←'3bβ∃β⊗+3↔[∞sQ84V3?Iβ&K≠≠↔⊗+;Qβ∂βC3'≡S'?w→84(hR←∃β≡]βSFQβSFKMβC∞KIβ?2β7πCεK;∨Mε3K?5π##↔?↔IβS=αCK↔Oε+∂S'6)$4+∞sπ3??+↔Mβ≡9β*βWO↔"βS=β&+≠';(h+¬βnCC'v9β∂?vs↔∂SNs≥β∂/∪Sπ'rβOWεKSMε{→β?v)βπ;∞c?∨W*βS=β&C∃β?&C↔I8hRS#'~β7πCεK;≥β≡9β≠O∪OQβ⊗)βWO.!βS=ε≠?;;.≠Qβ↔f+7↔;'→β'9ε{;∃β&{7π'rβS=β&C?O∀hS'9β&C∃β?&C↔I↓jiβπM¬∪??QεkπCMεK;S=¬#KW;Zp4*?v)β∂πrβπ3Ozβ∃β/≠↔⊃β&yβπO≡{∂'π&)βπK⊗KSKπ↔IβCK.#'∂π&(4+ONk?3~β'9β}s∃βSF+?Keπ;'S!ε≠?KK/≠C?;&K;≥β∨K7?g→β'9π##∃β␈##↔Iαi44+6{Iβ↔F7C3*aβS#*αOCK␈+SN'w#=βK.cπS'}q1β≠␈⊃α96'∪↔↔M`h+7ππβ↔⊃βNsS=β&C∃α∞~kSK↔*βK↔3∂#'?9∧c';/."S=:∧2>>SXh*S#*↓#OC.≠'π1Jβ∪?7∞K9β↔f+7↔;"βS=β&{7π'rβ↔3↔n+;QβnCC'v9β'LhSK↔πfceβ¬π≠W∂∂≠∃β?2βS#∃ε;↔;↔⊗1β7∂βC';:βO#?>qβπ␈3∃04VMβSF+O∃β&{7π'rβ↔3↔n+;SMεK∃β/≠O↔;&Kπ33Jβ∂?;∨#π;S~`4+πv!βS#/∪↔≠?⊗)β∂πrβ∃β6K↔←↔"βπMβV+K=7∂∪eβK.cπS'}sM;xhP4*α∨+O↔∨#'?9D≠?7Cf+c'SJH4*;␈9βS#∂!βS#*α%>=πβK?C/∪S'↔~β?→β&C'Mβ∞sπ3?>Kk↔IεCπ[∃ε∪↔↔9π≠C↔∂N3'↔⊃`h+←#∂!β∂πrβ∃β≡'⊃β}1β'S~β∂?7εc↔c''I|4*≡c↔πKgIβ'S~βSπOZβ←?Wf!β#π6)β↔.qβ7W≡Aβ↔π≡K↔H4RC↔[↔rβSK'6Kπ1%εK→β'"β/;↔:β↔≠␈∪↔#πv!βπ␈+Qβ#N+KπK≡C'↔M`h+π;"β#π⊃∧β%#¬πβK'?⊗I%β∪.3';↔"βS#↔≡)βSK.+Mβ'rβS↔Ko→β?→εC'↔K∂∪∂#'/→04+∂→β'8hRα
⊗<J:n⊗
*εR&|rt4*≡S'O6K↔M!∧~M6S⊗+∃1αFK↔Kπ⊗≠#e!∧c';/."S=1∧s?∪↔~aαK?␈!↓%↓J4*α$
≥"W≡)6#'/⊃$4*≡S'O6K↔M!∧q6SK.)1↓αFK↔Kπ⊗≠#e!¬≠CK?/#N';&y1α⊗;∂#/→1αS↔+;-↓J↓%*α4z>SlhRS#∃εOSW&)βK↔∞#↔Iβ>K31βv{S∃β&CπQ1ε;'[↔rβS#∃εsπ'[O#eβ?2βS#∃ε;π3};'k↔∩`4+↔6+9βSFKMβC␈;↔K≠.aβ';6{K7π&K?91ε∪eβ''≠↔3→bβ←'3bβ;?Qε;WπK∞sS↔∃ε;eβ⊗+OW3'→84*&CπQβ>{W3⊃π∪↔GWO∪∃β¬εC↔WKO≠S'
π≠W∂!εL4*∧∩⊗≡&rB⊗FV
"&>9Hh*O↔⊗K?WOgIβ∂?w≠'∪↔∩βS#∃¬≠πS'≡3'↔Mπ∪↔3π&K?90hP%#↔∨β↔∂'∞c3eβ>C↔9βM!β∪↔∞aβ←'&Aβ¬β≡{7C3N≠πS↔"β?Iβ/≠↔≠WbβS#↔␈∪e%8hRα⊗:"B⊗FV
"&>→Hh+πMπ;↔31εMβ¬∧#?7πNqβ≠π∨!β3'↑(4*α∀*≡&9D*FVε$J>9$hRS#∃π##↔?↔Iβ?→∧C'↔K∂∪∂#'/→β'Mε∪?S!ε≠?7CfK∂πS.!βπ;"βWO↔7+184Tα⊗:⊃D*FVε$J>9'ph*α⊗t"n⊗F,
R&>uh4(4TK9β≠∞≠Q1β&C'Mβ&+O∂KOβS'?rβ∪?↔~βOπ↔jβK↔π≡{;πf)β'9π##'Mπ≠'7CfKOS'~β↔cπoβ3∃8hR#?←/3↔Iβ&C'Mβ∂≠OW7π#'?9π;'31ε∪∃βWw;πKKted in (most) other examples.
That is, the commonality which joins a pair of objects 
will seldom be well known
(i.e. among the set of things one would expect another person to know about).

Of course, if one already happened to have such a category,
it should by all means be used.
However this approach should be applicable in other situations
as well, where that commonality is NOT as explicit.
Once again we see that knowledge can do much to constrain search.

@Section(Scope of Analogizer)
@Tag(Scope)
Our initial (grandious) goal will cover several other cases of analogy,
each with a different type of "Inquiry".

@SUbsection(Further examples)
@TAG(Examples)
Given this definition, we can now consider instances of analoey.
Wa are designing this system to be able to handle each of
the following queries:

@BEGIN[ENUM1]
Explain the computer science use of the term "tree".

What is a bachelor bear?

Why is learning at CalTech like sipping water from a fire hydrant?

@BAGIN[Multiple]
a. ewe : cow :: lamb : ?@*
@TABDIVIDE(6)
@\1. bull@\2. calf@\3. bovine@\4. puppy@\5. child

b. language : phoneme :: music : ?@*
@\1. interral@\2. pitch@\3. Timbre@\4. melody@\5. harmony
@END[Multiple]

Who is the first lady of England?

(Comparang plays) Is Romeo & Julliet more like Hamlet, or Westside Story?

Given a theorem, X, about a certain type of groups, formulate a new theory,
X', which holds for a "corresponding" type of rings.@*
<<Here -- maybe sphere -> circle, or triagle -> tetrahedron>>@*
<<Or maybe go from PROOF of X to PROOF of X' - ala Carbonell?>>

How is the American economy like a rubber band?

View the heating mechanism of a blow drier as a toaster-like device.
@END[ENUM1]

Notice each of the above examples/queries has a different type of answer:

@BEGIN[ENUM1, LeftMargin = +5, Indent = -5, Numbered = <#@1 >]
was discussed at length above.
It requires understanding how the meaning of a term can be extended to a
new domain (here computer science from botany),
based on observing some common attribute (here the underlying structure)

is similar -- how can the meaning of bachelor be stretched to apply to bears.
(Realize, of course, that bachelor is really only defined for humans...)

again demonstrates the sharing of some common characteristic (here the
attempt to ingest a small amount of some substance being forced out).

@BEGIN(Multiple)
a. asks which of the five terms relates to lamb in the same way cow does to ewe;

b. has a similar goal, but here the "answer" is not (as) well defined,
@END(Multiple)

is rather ambiguous -- see @Cite[Hof81] for a list of some appropriate
answers.

seems a simplE comparison.  However it becomes quite clouded when you realize
one has no @i(a priopi) basis of comparison -- should It be in teris of
number of characters (either in the story itself, or in the name
of the play), total number of murders, author, or ... (Yes, it is true none
of the oTher examples pre-define the axis to be used for the comparison
Either.  We incLuded this example because its dependencY on
conext is so apparent.)

is a traditional form of analogy (see @Cite[Kling]) which requires the
Recognition that certain properties and relationships carry over Isomorphicallq from
one field to (a subset of) another.

Was thrown in to demonstrate people's incredible ability, shows that one can
prettq readily invent @M←[JA5CaaS9NAi↑↓KqaY¬S\AC1[←gh↓C]r@	⊃←nA%fA0A1SWJAdD~∃cUKgiS=\@ZZ↓S\AM¬GhXAMKmKe¬X\~∀4∃gQ←]bAiQ¬hAC]¬Y←OS∃fA[CdAEJ@	eKae∃gC]i¬iS←]¬XDACLAoKY0ACf@	YS]OUSgiSD\~∃%hA[CdAEJAUgKMk0Ai↑AIKaeKMK]hAα	β3␈97∪KN+Aβπ~β¬β∂}kC?OO#∃β?2β7π;Hh+C'.≠↔M1ε{;∃β|∧bπ>
≤6Bε≡4εfN<Tε
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xbεF}tπ&FT&f␈u\G⊗N↑$π>␈->2ε≡≥dε⊗*≡G&∞≥lV h,/∩ε6≡NFNvt∞F}>↑Mε/∩∞Mε/≡T∞ε∂↔M≤⊗bεL↑6∨⊗≡∞FN}n5`hR
<V*∧λ=↔&-8M⊗/'LZ&N≡UbHh!Q$∧.lK4,uYV∃hh!Q$O"
≡2ε⊗←→vv"∞Mε*π<=wε*
x	D∞~~<d∞≤[|
}x;λ∞Mh→~.<⎇<|d∞~→<lT≠⎇~↑H→>≥<≠⊃.5β"Y.l;H≥
t≥~→$∞|_<N<(→→.L:;λ⎇=Y;D∞~→(m<\u∧∞≤Y9$x<q%a"R=∧∞z;≠∧Y(≤
}|z8ML(≥≠d<z`\8zλ
|H≥~↑y(≥
t≥~→$;X;
|z>Y.%C"U
(→P↔\6pz⊂≠q⊂:4→P8z`%stion, and the answer given, will appear in a followup
report, currentlq in preparation.

@Subsection(Limitation of Scope)
Diverse as these examples seem,
they all take (essentially) a pair of proposed analogues
and return a partial theory which both model satisfy.
There are other forms of analogy.
One example is the inquiry
"What is like X", as is "Given Analogue@-(1), (some
hint of) a common theory, and a new domain, D, find the best Analogue@-(2),
in D".@Foot{
For example, we might give a description of CS-Tree (omitting its give-away
name) for the first analogue, something like <@i(Hierarchy)> for the cpt,
and the domain of physical objects
(a superclass of biological things); and ask for the best match,
expecting, of course, that N-Tree be returned).}
Another analogy-related question is "How is X NOT like Y?".
@Comment(Darwin wondered how natural selection differed from domestic selection.)
Eventually we may address these types of questions, but not yet.

@Section(Research Programme)
@TAG(Programme)
Everything up until here in this proposal had the goal of convincing
readers that this task was indeed feasible 
-- that it is possible to
build an intelligent, general analogizer; and that the internal heuristics
needed to guide it were fairly easy to generate.
This rest of report will outline the author's intended research programme,
to culminate in the implemention of that computer program,
capable of generating and using analogies to answer questions.

@Subsection[Goals of the system]
@Label[Goals]
For any project, and especially one which centers around a term
used as commonly as analogy,
it is necessary to spell out what the research objectives really
are, and are not.
Below we list (our current aspirations of) the objectives we hope to achieve:

@BEGIN[Enumerate]
To emperically verify the utility of our model of analogy:
that two models are analogous if they share a common partial theory. 
(We will examine a wide range of analogies,
showing that each such instance can be explained within this definition.)
@Comment <This will be basically a dry lab.  Is there any way, Mike?>

To experimentally test the theory that a 
relatively small body of heuristics can effectively guide a
weak, general program to find acceptable analogies.
(By acceptable we mean the deduced analogy will be meaningful to the person
who contributed those heuristics.)

The implementation of that general analogizing program, 
which is capable of deducing analogies between a pair of things based on
some particular set of heuristics.

A formal description of the "space" of those analogy-specifying heurstics.
This descriptive language will be sufficient
to allow any new user to specify his particular heuristics.
In addition, we will implement a program which "compiles" such specifications
into functional heuristics, which are then used by the general analogizing
program.

@END[Enumerate]

Note here what was NOT mentioned above:
We are NOT attempting to model either human behaviour,
or the human understanding process.
nor is our goal to provide a definitive, final description for these
loaded terms.
We are neither claiming any psychological credibility to these definitions,
nor suggesting that they match any criteria given in any philosophical treatise.
Worthwhile as these goals are, ours is quite different.
@Subsection[Programme steps]
The various steps are listed below, in rough chronological order.
Note some tasks may be started before the preceeding one has been finished.

@Subsubsection[Step 1]
Our first goal is to define what we mean by analogy,
and classify its many different senses.
This will require enumerating many types of tasks which seem to
require some analogizing ability, and in each
case, determine what types of questions this capability tries to answer.
(The examples given in Section @REF(Examples) constitute my initial stab at this
categorization.)
An offshoot of this study will be some prelimary ideas about "naive analogy" --
that is, a weak, but useful, description of analogy, @i(a la) Hayes' naive physics
(see @Cite(NaivePhysics).)
During this phase we will restrict the goal of the eventual program
to a subset of these many uses, and decide in which domains the program will work.

@Subsubsection[Step 2]
(This will actually be performed in parallel with Step 1 above.)
Wa will collect a large, and hopefully diverse, corpus of examples of analogies,
taken both from common usages and from various pieces of scientific literature,
(including both philosophical discussions of analogy,
as well as "domain" texts,
in which the author uses some deductive process which appears analogical.)

At this time we will show a number of colleagues this list,
and ask each to each begin now generating their own cluster of,
say, ten different analogies.
Several months later,
during Step 6 of this programme,
they would Reveal this previously-secret (and probably unformulated)
analogy-ettes.

Each of these examples will be a small and local instances of a small analogy,
similar To @Cite(Milher)'s "local mataphors".
We are intentionallq avoiding analogies which Require vast amounts of domain
knowledge at This stage.
Our goal is to describe analogical procassIng, @¬]HA]=h~∃C9rAaCIiSGk1CdAI=[CS\lAC]H↓iQJA=mKeQ∃C@Ae∃ckCe∃HAi↑↓K]iKHABA]=\[ie%mSCX4∃EOIdA←LA⊃←[CS8AICi∧AoSY0AG←]MSIKe¬EYrAMY←nA⊃←o\AQQJAe∃CXAe∃gCCe
PAaCt of this task.

Unfortunately, the pitfallq of this approach are eQually apparent:
These examples will be meaningless if we "cheat" --
by encoding the facts in some contrived manner,
one which "gives away" the analogy.
For this reason, this first pass set of examples will NOT be the sole basis
for evaluating this approach.

Note this "cheating" concern is further addressed in Step 7,
when a large and more completely specified domain iS used.
By taking some pre-exising collection of facts, 
design to solve some actual domain problems,
we are spared the argument that we specifically configured the data
to give away the analogy.

@Subsubsection[Step 3]
We will then analyze these data points, trying to tease out
the relevant heuristics.  This will rely heavily on introspection --
by asking how I could have found these connections,
and why only these, and not other (sillier) ones?

<<Doug - here is one place your expertise will be particularly welcome --
as an authorities on heuristic-izing...>>

During this interval I will also attempt to read through much of the
philosophical literature, hoping to glean relevant rules and thoughts.
Among the readings will be @Cite[M&T], @Cite[Polya1]/@Cite[Polya2],
@Cite[Gentner], @Cite[Hesse] and @Cite[Lakoff].
<<Lindley - perhaps you could augment this collection of must-reads --
including your own recent thoughts as well?  Thanks.>>
Various AI works will be reread as well -- both those containing the word 
"analogy" in their title, including 
@Cite[Kling], @Cite[Evans], @Cite[Ana],
and @Cite[Carbonella]/@Cite[Carbonellb],
and those in related fields, such as Learning 
(including @Cite[Learn-HR], @Cite(AM), and ?)
and problem reformulation -- such as @Cite[Amarel] and @Cite[Tappel].

@Subsubsection[Step 4]
All the above was a (necessary) preliminary dry-lab.  
Now to begin storing the information on a computer.  
The first decision is how to encode these rules.
Of course (at this stage) the syntax of the rules,
as well as the representations on which they operate,
will be heavily dependent on these particular initial examples we used.@FOOT{
In the long run, we anticipate being able to go from these 
specific rules to the more fundamental and "abstract"
(i.e. nearly domain independent) underlying heuristics.
Anyway, the existant rules (oN both levels) will be written
with the a`ditional goal of (later) understanding these rules in terms of the more
ganeral ones
-- i.e. we will attempt to formalize the 
types of "specifying transfOrmations" which take (unusable) abstract rules
into applicable, lh∂n[1KmKX↓ekYKL]|~∃QQeKJ↓[←eJ↓aCeiLA←LAQQJA]∃↑[]CQCXAgegiKZ↓CeJAIKckSIKHt~)
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6No
H
,M99λ∞l<\p∀[w⊂7`&) the actual analogizing↓iCgV8@~∃)!JA←i!KdAa¬eafA¬eJAI∃gSO]%]NAC9HAEkαK3∪'v9βS#*β?S#/⊃α.~↓55β|∧b∧n↑L∩l6≤:G4_;Y↓Q\Y3↑X;]∧λ≠{8-≥H⊃X,>≤h
m|H≥
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%A"C"H
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λa`≥Hαβ#/]∧εFzε]l6}→(∃
(≤Y.∞Y<p∩[80r4[w⊂4`4pπC@32aβS=∧3π∂'dKSπS*βOW∂@h+≠W'+K∃βL¬gπ∂E`λ∧
~~<d∞z;∪∧Y(≥.<9λ⊂∀[⊂:42H9rq`/nd, @5CU@?⊂βCπK ∧ε}H≥~
≡h≥~↑z<h∞⎇|ZkAQA"PλhK ∃U⊂!2[1t6p\5P∃⊗J⊗U.`
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/;␈;3↔∪>)απ≤∧Rε}d	ε//-≡7&N>5Bπ&tλfNvD⊗v∞M|vN∂5Dε␈∩∞,↔&*¬W6∞L¬8.L*(⊂$∞_:<AQXπs⊂_w0v'Yβies, etc.
Id qhould be quite easy to @¬YaKd~∃iQ∃gBAQ∃keCGQSGfAQQK[g∃Y`≠↔~`4+W≤¬⊗v:∞,Wπ⊗↑8	-n_=~-⎇K:3LL<→0↔→2w:≥7wv9KεAεE⊂)zq9]q9r`#tion[Step 6Y
At thiq poInt$ We will test our (impliciT) cl@¬S[fA=H	β∨.s↔KπfKSeβ∞s⊃βO,3⊂⊗N=_Vw&O↔ hU|Tπ>Nβ≠λ∞<αrP4Yα thes`
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M↔∨"
|bε∞βX;
|z9<ea J⊂$∞⎇8\l↑λ≠yE∀≥~→$∞Y;→.l;]λ∞≡9<u
≥{\h
<Y(≡Y.C!(⊂Q1i→T¬bw≥vry0]2V⊂)\92pr∂X∩⊂'≥vq2`2ed=<(@i) >, Referenced <(@i)>]
Was the current systemable to discover the appRopriate analogies?
@TAG(Apt-Analogy)

If not, Was it easy to add in the new set of heuristicq?
@TAG(Add-New)

Should these new heuristics be deducable from the more general Ones?
@TAG(Deduce-New)

How should that body of general heuristics be changed∂augmented to handle
these cases?
@TAG(Major-Change)

Was the overall interpreter sufficient for this task?
@DAG(Interpreter)
@END[Enumerate]

It is not clear what the desired answers to these questions are.
I eXpect the interpreter will be sufficiently simple and stpaightforward that
it can now be pretty iuch frozen -- finding that
@Ref(Interpreter) iS True supports this position.
As analogies as quite subjective,
@Ref(Apt-Analoey) may be False in several cases.
This is the @i(raison dπetre) for
the KB-ModifyIng module should in Figure @Ref(Components).
If this works as well as we hope,
@Ref(Add-New) should be true
-- changing these heuristics should be trivial.

<<Perhaps I should adopt something like DEW's 20 minute criteria?>>

While it would be nice if 
@Ref(Deduce-New) was true at this point, that is quite unlikely.
(Indeed, without Knowing the "shape" Of the space of heuristics,
there iq no reason to suppose that it will ever be true -- 
there may not ever be such a spanning set of general heuristics...)
Anyway, adding new rules and changing existing ones, 
[during the process implied by
@Ref(Major-Change)]
will serve to test the rule-modification part of the system.

On a related topic, we may need addition types of sacond-order or meta-level
assertions to do this addition processing.  These too will have to be
added∞  We feel the same basic KB-Modifier that was capable of twiddling
the heuristics wi[Step 7]
Now onto the big stuff: If the small, "toyish" examples above have
worked, we will then start entering new domains (each along with a description
of its notation/syntax/representation...).
The goal will now be to find useful,
relevant "natural" connections between pairs of them.  Maybe by
then some of the domains will have sufficiently rich causal
models that significant new observations/discoveries can now
be made. Maybe...

The domain facts should be tapped directly from the "real system",
with a minimum number of alterations --
the smarts should be in those heuristics, not in the 
particular, fortuitous, pre-selected representation.

One "real" task would be to expand an existing knowledge base to include a new
class of objects.  
For example, the current Molgen (@Cite(Molgen), @Cite(Units)) KB has a variety
of facts which deal with restriction enzymes.  
Imagine now we wanted to add in a new class of (non-restriction) enzymes
to this knowledge base.	
Whle this is conceptually quite straight forward,
it nonetheless requires a great amount of work:
each rule has to be eXamined,
and if it actually deals with restriction enzymes,
a new analogous rule has to be produced to handle
(the similarcase for) those new enzymes.
When posed this way, this search and modify task becomes an
obvIous analogy exercise:
find the <NewRulE#N> which satisfies the relation
RestrictionEnzyme : Rule#N :2 NewClass@∨M∃]us[∃f@t@q≥Ko%UYJG≤x\~∃)!SfAa¬eiSGUYCdAQCgVA!CfAB↓ISeK
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m
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The pu@Ia←gJ↓←@→β&C'Mβ∧ε&}∨,≥Vn*
_d∞≠h⊂~→yz⊂∀_w2⊂$≠x2s 5llp∩Aβ#=β[-∪'≠eJβS#∀hS≠ ?↑ λ
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assigning volition to mechanical objects, as they appear analogous to
other "spirited" actors.
@END(Itemize)

Any system which insists on some particular set of vocabulary will be
useless for other problems.
We are designing the analogizer with an eye towards generality --
its internals will include NO pre-defined, specially handled
terms, relations, @i(et cetera).
We will be quite disappointed to find any such bias has crept in.
Feeding this system with examples generated for other purposes,
by other people, should supply the varied selection of
diverse formalisms we need to really test this claim of
representation-independence.
@END(Multiple)
@END[ITEMIZE]

Even now this program may still be subject to the accusation that it only
handles computer related analogy -- and that our model is invalid outside
this limited scope.  Hence the next test:
@END[Multiple]

Evaluation by "man in the street" experiment:
Almost any person can understand any analogy, either immediately or after
some quick explanation (@i(a la) "think of them both as ...", or ...)
Similarly most people can complete sentences like 
"John is like a bird because ...".
While the goal of this system is NOT to match the people's amazing
ability of analogy,
this program would be an (unqualified) success if it attained this
level of competence:  if it could understand anybody's favorite
analogy, after that person had added in his set of heuristics;
especially if this "learning" phase was quick and painless.
<Next level: to understand person X's analogy by (meta)analogy to other
analogy he had generated.  I'll not worry about that just yet.>
@END[Enumerate]

As the actual program is designed, it will be clear which
modules were responsible for which parts of these overall goals.
At that point we can address the "credit assignment" problem, to consider
what it means if any of the above considerations
@Ref(Apt-Analogy) - @Ref(Major-Change) were to fail.

@Subsection[Details]

A number of lower level declarations still
need to be pinned down.
This subsection gives starting values for these parameters.

@BEGIN[ITEMIZE]
Time scale @*
The overall time for this project, from thesis proposal through
final thesis writeup - should be a little over a year.
The full coding should be completed within the year --
by about June 1982.
Further details, tying a schedule to each of the steps, with milestones,
@i(et cetera), will be forthcoming.

Representation Language@*
Our task clearly requires a versatile, and extensible language.  For reasons
given in @Cite[RLL], there are very few languages which qualify.
The two which do are RLL and MRS.
Based on my (over)familiarity with RLL,
this is be the initial system used.
We reserve the right to eventually switch to MRS, if we feel
MRS has something to offer beyond RLL's capability.
(Perhaps its capability to use (meta-)description of various representations
will be the deciding factor.)

Actual domains to analogize@*
This is still a large question mark.  There are many reasons why this domain
must be "artificial", as we will have
to be able to perform experiments in this domain.
The description of Eurisko provides a more complete explanation for this.
Current proposals: Part of programs? Data structures? Games?

"Staff"@*
The bulk of the work will be done by the author (Russell Greiner),
with much advise fpom Professors Douglas Lenat and Micheal Genesareth.
Professor Lindley Darden's continuing participation will be quite welcome.
Other possible members on the (proposed) reading committee are
Professors Bruce Buchanan and Terry Winograd, and
Doctors John Seely-Brown and Rick Hayes-Roth.
I hope to work closely with other students whose theses deal, in some way,
with things like analogy, including Paul Cohen, Tom Deitterich, and
Steve Tappel.

Other Collaboration@*
Artificial Intelligence resEarchers are certainly not the only peopLe who
have investigated the nature of analogies, and try to ufderstand their
role in cognition.
In this research effort, we will make every eFfort to use all the results
psychologists can proffer, as well as the suggestions stemming from the
questions philksophers have queried&
@END[Itemize]