IT/CS 811 Principles of

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IT/CS 811 Principles of Machine Learning and
Inference
Learning by analogy
Prof. Gheorghe Tecuci
Learning Agents Laboratory Computer Science
Department George Mason University
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Overview
Learning by analogy definition
Design issues
The structure mapping theory
Determinations
Problem solving by analogy
Exercises
Recommended reading
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Learning by analogy definition
Learning by analogy means acquiring new knowledge
about an input entity by transferring it from a
known similar entity.
One may infer, by analogy, that hydraulics laws
are similar to Kirchoff's laws, and Ohm's law.
Which is the central intuition supporting the
learning by analogy paradigm?
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Discussion
Central intuition supporting learning by analogy
If two entities are similar in some respects then
they could be similar in other respects as well.
Examples of analogies
Pressure Drop is like Voltage Drop A variable in
a programming language is like a box.
Provide other examples of analogies.
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Learning by analogy illustration
Illustration The hydrogen atom is like our solar
system.
The Sun has a greater mass than the Earth and
attracts it, causing the Earth to revolve around
the Sun. The nucleus also has a greater mass then
the electron and attracts it.Therefore it is
plausible that the electron also revolves around
the nucleus.
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Learning by analogy the learning problem

Given A partially known target entity T and
a goal concerning it. Background knowledge
containing known entities. Find New
knowledge about T obtained from a source entity S
belonging to the background knowledge.
Partially understood structure of the hydrogen
atom under study.
Knowledge from different domains, including
astronomy, geography, etc.
In a hydrogen atom the electron revolves around
the nucleus, in a similar way in which a planet
revolves around the sun.
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Learning by analogy the learning method

ACCESS find a known entity that is analogous
with the input entity. MATCHING match the
two entities and hypothesize knowledge.
EVALUATION test the hypotheses. LEARNING
store or generalize the new knowledge.
In the Rutherfords analogy the access is no
longer necessary because the source entity is
already given (the solar system).
One may map the nucleus to the sun and the
electron to the planet, allowing one to infer
that the electron revolves around the nucleus
because the nucleus attracts the electron and the
mass of the nucleus is greater than the mass of
the electron.
A specially designed experiment shows that indeed
the electron revolves around the nucleus.
Store that, in a hydrogen atom, the electron
revolves around the nucleus. By generalization
from the solar system and the hydrogen atom,
learn the abstract concept that a central force
can cause revolution.
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Discussion
How does analogy help?
Why not just study the structure of the hydrogen
atom to discover that new knowledge? We anyway
need to perform an experiment to test that the
electron revolves around the hydrogen atom.
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Overview
Learning by analogy definition
Design issues
The structure mapping theory
Determinations
Problem solving by analogy
Exercises
Recommended reading
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Learning by analogy Design issues

ACCESS find a known entity that is analogous
with the input entity. MATCHING match the
two entities and hypothesize knowledge.
EVALUATION test the hypotheses. LEARNING
store or generalize the new knowledge.
Given a target, how to identify a few potential
sources in a very large storage?
Given a potential source, how to identify the
knowledge to hypothesize?
Why and how to test the hypothesized knowledge?
How to learn?
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Learning by analogy Formalization

Given - a target entity T - a universe of
potential sources U - an access function f1
with a threshold value f1 - a matching function
f2 with a threshold value f2. Find - new
knowledge about T (using analogical learning).
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Learning by analogy Access
Find potential sources for T in U f1(Sk, T) gt
f1 This should result in S1, , Sn
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Learning by analogy Matching
Find the best match between one of S1, , Sn and
T. Let Sk A B C, T A' D where f2(Sk,
T) gt f2 gives the best match. A and A' are the
parts of Sk and T that make them
analogous f2(Sk, T) f2(A, A') B, C and D are
the other parts of Sk and T. As a side effect of
partially matching Sk with T (or totally matching
A with A'), one obtains a correspondence
(substitution) list s ( o1 o1', ... , on
on') where oi is an element of A and oi' is the
corresponding element from A'. By applying the
substitution s to Sk one obtains s(Sk) s(A)
s(B) s(C) A' s(B) s(C) A' B'
C'. By analogy with Sk one concludes that T
might also have the features B' C'.
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Learning by analogy Evaluation and learning

• By analogy with Sk one concludes that T might
  also have the features B' C'.
• However, the evaluation phase shows that T has
  the features B' but it does not have the features
  C'.
• Therefore
• B represents the part of Sk that is transferred
  to T because of the similarity between A and A
• C is the part of Sk that is not transferred to
  T
• - D represents the features that are specific to
  T.

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Case study discussion Rutherfords analogy
"The hydrogen atom is like our solar system".
In this case, the fact that S and T are analogous
is already known. Therefore, the access part is
solved and the only purpose of the matching
function remains that of identifying the correct
correspondence between the elements of the solar
system and those of the hydrogen atom. This is
an example of a special (simpler form of
analogy) A T is like an S. This is useful
mostly in teaching based on analogy.
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Case study discussion potential matchings
Which are the possible matchings between the
elements of S and the elements of T?
yellow
yellow
color
color
sun
nucleus
sun
nucleus
mass
temperature
mass
mass
temperature
mass
Msun
Mnucleus
Msun
Mnucleus
Tsun
Tsun
attracts
attracts
attracts
attracts
greater
greater
causes
greater
greater
causes
greater
greater
revolves
-
revolves
-
around
around
Tplanet
Tplanet
Mplanet
Mplanet
Melectron
Melectron
mass
mass
temperature
temperature
planet
mass
planet
mass
electron
electron
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Case study discussion potential matchings
There are several possible matchings between the
elements of S and the elements of T and one has
to select the best one Matching1 sun
nucleus, planet electron, Msun Mnucleus,
Mplanet Melectron, which is supported by the
following correspondences mass(sun, Msun)
mass(nucleus, Mnucleus) mass(planet , Mplanet )
mass(electron, Melectron) greater(Msun,
Mplanet) greater(Mnucleus, Melectron), attracts
(sun, planet) attracts(nucleus,
electron) Matching2 sun nucleus, planet
electron, Tsun Mnucleus, Tplanet Melectron,
that is supported by the following
correspondences greater(Tsun, Tplanet)
greater(Mnucleus, Melectron), attracts(sun,
planet) attracts(nucleus, electron) Matching3
sun electron, planet nucleus, Msun
Melectron, Mplanet Mnucleus
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Similarity estimation issues and sample solutions
1. How to search the space of all possible
matchings ?
Exhaustive search. Other solutions?
2. How to measure the similarity of two elements ?
Two elements are similar if they represent the
same concept or are subconcepts of the same
concepts. In such a case their similarity may be
considered 1 (on a 0-1 scale). Other solutions?
3. How to combine the estimated similarities of
the parts in order to obtain the similarity
between S and T ?
The similarity of two entities is the sum of the
similarity of their elements. Other solutions?
4. How to define the similarity threshold ?
Similarity threshold defined by the designer (a
hard critical issue). Other solutions?
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Case study discussion Matching result
The best matching is Matching1 (because it leads
to the highest number of common features of the
solar system and the hydrogen atom) that gives
the following substitution s (sun nucleus,
planet electron, Msun Mnucleus, Mplanet
Melectron)
yellow
yellow
By applying the substitution to the solar system,
one obtains the following structure
color
nucleus
The features in light color are those that could
be transferred to the hydrogen atom as a result
of the analogy with the solar system
color(nucleus, yellow) temperature(nucleus,
Tn) temperature(electron, Te) greater(Tn,
Te) revolves-around(nucleus, electron)
causes( (attracts(nucleus,electron),
greater(Mnucleus, Melectron)), revolves-around(
nucleus, electron))
mass
temperature
Mnucleus
Tsun
Tnucleus
attracts
greater
causes
greater
revolves
-
-
around
Telectron
Tplanet
Melectron
mass
temperature
temperature
electron
electron
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Case study discussion Evaluation
The evaluating phase shows that The hydrogen
atom has the features revolves-around(nucleus,
electron) causes((attracts(nucleus,electron),
greater(Mnucleus, Melectron)),
revolves-around(nucleus, electron)) The hydrogen
atom does not have the features color(nucleus,
yellow) temperature(nucleus, Tn)
temperature(electron, En) greater(Tn, En)
Which is, in your opinion, the most critical
issue in analogical learning?
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Discussion
Which is the most critical issue in analogical
learning?
What kind of features may be transferred from the
source to the target so as to make sound
analogical inferences ?
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Case study discussion transfer of causal relation
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Case study discussion Learning
Store the new acquired knowledge about the
hydrogen atom revolves-around(nucleus,
electron) causes(attracts(nucleus,electron),
greater(Mnucleus, Melectron)),
revolves-around(nucleus, electron)) By
generalization from the solar system and the
hydrogen atom one may learn the abstract concept
that a central force can cause revolution
causes(attracts(x, y), greater(Mx, My)),
revolves-around(x, y)) Question When to store
the acquired knowledge and when to generalize it?
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Analogy in Disciple
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Causal networks of relations
An important result of the learning by analogy
research is that the analogy involves mapping
some underlying causal network of relations
between analogous situations. By causal network
of relations it is generally meant a set of
relations related by special higher order
relations such as 'physical-cause(ri, rj)',
'logically-implies(ri, rj)', 'enables(ri, rj)',
'justifies(ri, rj)', determines(ri, rj)
etc. The idea is that similar causes are
expected to have similar effects
The basic scheme of analogy
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Overview
Learning by analogy definition
Design issues
The structure mapping theory
Determinations
Problem solving by analogy
Exercises
Recommended reading
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Gentners structure mapping theory
The main claim of this theory is that relations
between objects, rather than attributes of
objects, are mapped from source to target.
Moreover, a relation that belongs to a mappable
system of mutually interconnecting relationships
is more likely to be imported into the target
than is an isolated relation (the systematicity
principle).
See Gentner D., The mechanisms of analogical
reasoning, in J.W.Shavlik, T.G.Dietterich (eds),
Readings in Machine Learning, Morgan Kaufmann,
1990.
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Gentners structure mapping theory (cont.)
Analogy maps the objects of the source onto the
objects of the target s1 t1, ... , sn tn
These object correspondences are used to
generate the candidate set of inferences in the
target domain. Predicates from the source are
carried across to the target, using the node
substitutions dictated by the object
correspondences, according to the following
rules 1. Discard attributes of objects A(si)
-/-gt A(ti) For instance, the yellow color of the
sun is not transferred to the hydrogen nucleus.
2. Try to preserve relations between objects
R(si, sj) -?-gt R(ti, tj) That is, some relations
are transferred to the target, while others are
not. 3. The systematicity principle the
relations that are most likely to be transferred
are those belonging to systems of interconnected
relations R'(R1(si,sj), R2(sk,sl))
R'(R1(ti,tj), R2(tk,tl))
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Literal similarity, analogy, and abstraction
Gentner's theory distinguishes between literal
similarity, analogy, and abstraction. One says
that a target T is literally similar with a
source S if and only if a large number of
predicates is mapped from source to target,
relative to the number of nonmapped predicates
and, also, the mapped predicates include both
attributes of objects and relations between
objects. For instance, 'kool-aid' is literally
similar with 'water' since it has most of the
features of 'water' (both attributes of objects
and relations between objects).
Give other examples of literally similar entities.
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Literal similarity, analogy, and abstraction
One says that a target T is analogous with a
source S if and only if relations between
objects, but few or no attributes of objects, can
be mapped from source to target. For instance,
'heat' is analogous to 'water'.
One says that a source S is an abstraction of a
target T if and only if the source is an abstract
relational structure and each predicate (a
relation between objects or an attribute of an
object) from the abstract source is mapped into a
less abstract predicate of the target there are
no nonmapped predicates. For instance,
'through-variable' is an abstraction of 'heat',
where by 'through-variable' we mean something
that flows across a difference in potential.
Give other examples of abstractions.
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Similarity, analogy, and abstraction discussion
Given that two entities overlap in relations,
they are more literally similar to the extent
that their object attributes also overlap.
Therefore, literal similarity might be seen as a
particular case of analogy. Abstraction may also
be seen as a special case of analogy in which all
the predicates of the source entity are mapped
into the target entity.
What could we conclude from these observations?
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Similarity, analogy, and abstraction discussion
What could we conclude from these observations?
Overlap in relations is necessary for any
perception of similarity, analogy or abstraction.
The contrast between literal similarity, analogy,
and abstraction is a continuum.
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Gentners theory implementation and discussion
An implementation of the Structure-Mapping theory
is the Structure-Mapping Engine (Falkenhainer,
Forbus Gentner, 1989 The Structure-mapping
Engine. Algorithms and Examples, Artificial
Intelligence, 411-63. Also in Readings in
Knowledge Acquisition and Learning). Given the
descriptions of a source and a target, the
Structure-Mapping Engine constructs all
syntactically consistent analogical mappings
between them. Each mapping consists of pairwise
matches between predicates and objects in the
source and target, plus a list of predicates
which exist in the source but not the target.
This list of predicates is the set of candidate
inferences sanctioned by the analogy. The
Structure-Mapping Engine evaluates syntactically
each possible analogy to find the best one.
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Gentners theory implementation and discussion
The Structure-Mapping Engine needs to be given
the descriptions of a source and a target. This
requires the ACCESS problem to be solved
first How do we find potential sources for a
target? MAC/FAC (Forbus, Gentner, Law, 1995
MAC/FAC A model of similarity-based retrieval,
Cognitive Science, 19(2)141-205) is a system
that addresses the access problem. The MAC stage
uses a simple, nonstructural matcher to filter
our a few promising candidates from a large
memory of structured descriptions. The FAC stage
evaluates each candidate using SME to provide a
structural match. MAC/FAC was scaled-up in the
DARPAs HPKB and RKF programs.
What is, however, a problem with Gentners theory?
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Gentners theory discussion
What is a problem with Gentners theory?
Gentners interpretation rules depend only on the
syntactic properties of the knowledge
representation, and not on the specific content
of the domain.
Why is this a problem?
Consider these equivalent representations Book
1-on-Table On(Book1, Table)
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Overview
Learning by analogy definition
Design issues
The structure mapping theory
Determinations
Problem solving by analogy
Exercises
Recommended reading
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Determinations Definition
Instead of giving a general criterion for the
validity of analogical knowledge transfer (high
order relations or causal network of relations),
Russel and Davis propose to specify explicitly
what knowledge can be transferred. The rules for
specifying this are called "determination rules".
P(x, y) gt- Q(x, z) (P plausibly determines Q)
meaning "S, "T If y P(S, y) P(T, y)
then it is probably true that z Q(S, z)
Q(T, z) where P and Q are first order logical
expressions.
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Determinations Definition (cont.)
A determination rule is an expression of the
following form U(x1,...,xn,y1,...,ym) gt-
V(x1,...,xn,z1,...,zp) One says that U
determines V. That is, whenever the arguments of
U have certain values, the arguments of V are
very likely to have corresponding
values. Example Rainfall(x, y) gt-
Water-in-soil(x, z) Rainfall(Philippine,
heavy), Water-in-soil(Philippine, high)
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Analogical reasoning based on determinations
Given Rainfall(x, y) gt- Water-in-soil(x,
z) Rainfall(Philippine, heavy),
Water-in-soil(Philippine, high) Rainfall(Vietnam,
heavy) Conclude Water-in-soil(Vietnam, high)
What is the difference between a determination
rule and a deductive rule?
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Determinations Discussion
A determination rule is different from a
deductive rule. The form of a deductive rule
is U(x1,...,xn,y1,...,ym) --gt
V(x1,...,xn,y1,...,ym) That is, the variables
which appear in the left hand side of a rule also
appear in the right hand side. Therefore, if we
know that 'U(a1,...,an,b1,...,bm)' is true, we
could apply modus ponens to infer that
'V(a1,...,an,b1,...,bm)' is also true. This type
of reasoning is not possible in the case of a
determination U(x1,...,xn,y1,...,ym) gt-
V(x1,...,xn,z1,...,zp) because we do not know the
values of the variables z1,...,zp. In order to
apply a determination rule, one would need a
source entity, as will be illustrated in the
following.
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Analogy based on determinations Method
The basic procedure for answering the query V(T,
?z) by analogy 1. Find a determination such
that U(?x, ?y) gt- V(?x, ?z) (i.e. decide which
determinations could be relevant for T U(T, ?y)
gt- V(T, ?z)) 2. Find 'a' such that U(T, a)
(i.e. find how the facts are instantiated in the
target) 3. Find a source S such that U(S, a)
(i.e. find a suitable source) 4. Find 'b' such
that V(S, b) (i.e. find the answer to the query
from the source U(S, a) gt- V(S, b)) 5. Return
'b' as the solution to the query (U(T, a) gt- V(T,
b))
U(S,a) U(T,a)
b
V(S,b) V(T,?z)
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Analogy based on determinations Illustration
Let us consider the following target Nationality
(Jack, UK), Male(Jack), Height(Jack, 6'), ... and
the problem of answering the following question
by analogy What is the native language of Jack
? (i.e. Native-language(Jack, ?z)) 1. Find a
determination such that U(x, y) gt-
Native-language(x, z) Such a determination is
Nationality (x, y) gt- Native-language(x, z) 2.
Find 'a' such that Nationality (Jack,
a) Nationality (Jack, UK) a UK 3. Find a
source S such that Nationality (S,
UK) Nationality (Jill, UK), Female(Jill) ,
Height(Jill, 5'10"), Native-Language(Jill,
English) S Jill 4. Find 'b' in S such that
NativeLanguage(Jill, b) Native-Language(Jill,
English) b English 5. Return 'English' as the
solution to the query Native-language(Jack,
English)
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Determinations Discussion
Consider the determination rule U(x1,...,xn,y1,.
..,ym) gt- V(x1,...,xn,z1,...,zp)
Should U and V be terms or could they be
arbitrary logical expressions? Why?
What if we cannot find a source S for applying
the determination?
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Determinations Discussion
U and V may be an logical expressions.
Example The rainfall of a flat area
determines the quantity of water in the soil of
the area Rainfall(x, y) Terrain(x, flat) --gt
Water-in-soil(x, z) Rainfall(Philippines,
heavy), Terrain(Philippines, flat),
Water-supply(Philippines, high) Rainfall(Vietnam,
heavy), Terrain(Vietnam, flat) Water-in-soil(Viet
nam, ?t)
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Determinations Discussion
What if we cannot find a source S for applying
the determination? Sometimes there is no source
S such that U(S, a) is true, but one may find S'
such that U(S', a') is true. In such a situation
one needs a way to decide whether a' and a are
similar enough to infer V(T, b). Therefore, even
in the case of determinations one may need a
matching function.
Example Latitude of an area determines the
climate of the area Latitude(x, y) --gt
Climate(x, z) Latitude(Romania, 45),
Climate(Romania, temperate) Latitude(France,
47)
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Overview
Learning by analogy definition
Design issues
The structure mapping theory
Determinations
Problem solving by analogy
Exercises
Recommended reading
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Problem solving by analogy
Analogy means deriving new knowledge about an
input entity by transferring it from a known
similar entity.
How could we define problem solving by analogy?
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Problem solving by analogy definition
Problem solving by analogy is the process of
transferring knowledge from past problem-solving
episodes to new problems that share significant
aspects with corresponding past experience and
using the transferred knowledge to construct
solutions to the new problems.
What could be the overall structure of a problem
solving by analogy method?
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The problem solving by analogy method

Let P be a problem to solve. First, look into
the knowledge base for a previous problem solving
episode which shares significant aspects with the
problem to solve. Next transform the past
episode to obtain a solution to the current
problem.
What questions need to be answered to develop
such a method?
What it means for problems to share significant
aspects? How is the past problem solving episode
transformed so as to obtain the solution to the
current problem?
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The derivational analogy method (Carbonell)

• Two problems share significant aspects if they
  match within a certain threshold, according to a
  given similarity metric.
• The solution to the retrieved problem is
  perturbed incrementally until it satisfies the
  requirements of the new problem.

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The derivational analogy method illustration
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The derivational analogy method discussion
How does analogy facilitate the problem solving
process?
How does the derivational analogy method relates
to the generally accepted idea that the relations
which are usually imported by analogy from a
source concept S to the target concept T are
those belonging to causal networks?
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The derivational analogy method discussion
How does this method relates to the generally
accepted idea that the relations which are
usually imported by analogy from a source concept
S to the target concept T are those belonging to
causal networks?
Intuition The relation between a problem and its
solution is a kind of cause-effect relationship.
Fermats last theorem There is no integer
solutions of xn yn zn for ngt2
What does this example suggests?
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Discussion
Except for the trivial problems, a solution does
not emerge immediately from the problem
formulation, as would be the case in a
cause-effect relation. What other relation from
the problem solving process might be closer to a
cause-effect relation?
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Discussion
What other relation from the problem solving
process might be closer to a cause-effect
relation? The relation between a problem and
its derivation trace (i.e. solution
process). What is transferred from a past
problem solving episode is not a problem solution
but the problem solving process itself, what
questions have been asked, what factors have been
considered, etc. One would try to repeat the same
process in the context of the new problem. With
this interpretation we retrieve the derivational
analogy method
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The transformational analogy method (Carbonell)
Two problems are considered to share significant
aspects if their initial analysis yields the same
reasoning steps, that is, if the initial segments
of their respective derivations start by
considering the same issues and making the same
decisions The derivation of the solved problem
may therefore be transferred to the new problem
by reconsidering the old decisions in the light
of the new problem situation, preserving those
that apply, and replacing or modifying those
whose supports are no longer valid in the new
situation. Derivational analogy gives better
results than transformational analogy. However,
it has the disadvantage to manipulate complex
structures representing derivational traces.
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Overview
Learning by analogy definition
Design issues
The structure mapping theory
Determinations
Problem solving by analogy
Exercises
Recommended reading
58
Exercises
1. Compare explanation-based learning, empirical
inductive learning, and learning by analogy from
the point of view of input information,
background knowledge needed, and outcome of
learning. 2. Define learning by analogy and give
an example of analogy. 3. Describe the four
stages of learning by analogy. 4. Illustrate
learning by analogy with the help of the
following example
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Exercise
Let us consider a learning by analogy system
having the following background
knowledge Facts Economy-type (Germany,
highly-industrial), Location(Germany,
Europe) Population(Germany, 70), Economy-type
(Vietnam, agricultural), Location(Vietnam,
Asia) Population(Vietnam, 70),
Economic-state(Vietnam, poor), Economy-type
(Japan, highly-industrial), Location(Japan,
Asia), Population(Japan, 100),
Economic-state(Japan, excellent),
Determination Economy-type (x, y) gt-
Economic-state (x, z) (the economy-type of a
country determines the economic state of the
country) Write a detailed trace of the reasoning
of the system for answering the following
question What is the economic state of Germany
? (i.e. Economic-state (Germany, ?z))
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Exercises
Provide an example of a successful application of
the transformational analysis method. Provide an
example where the transformational analysis
method does not apply, but the derivational
analogy method does apply. What is the
difference between a determination rule and a
deductive rule? Illustrate the difference with an
example.
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Recommended reading
Gentner D., Holyoak K.J., Kokinov B.N. (eds.),
The Analogical Mind Perspectives from Cognitive
Science, The MIT Press, 2001. Carbonell J.G.,
Learning by analogy formulating and generalizing
plans from past experience, Machine learning I,
1983. Carbonell J.G., Derivational analogy a
theory of reconstructive problem solving and
expertise acquisition, in Shavlik J. and
Dietterich T. (eds), Readings in Machine
Learning, Morgan Kaufmann, 1990. Also in Readings
in Machine Learning and Knowledge
Acquisition. Davies T.R., Russell S.J., A
logical approach to reasoning by analogy, in
Shavlik J. and Dietterich T. (eds), Readings in
Machine Learning, Morgan Kaufmann, 1990. Gentner
D., The mechanisms of analogical reasoning, in
J.W.Shavlik, T.G.Dietterich (eds), Readings in
Machine Learning, Morgan Kaufmann, 1990. Winston
P.H., Learning and reasoning by analogy,
Communications of the ACM, 23, pp.689-703,
1980. Forbus K.D., Exploring Analogy in the
Large, in Gentner D., Holyoak K.J., Kokinov B.N.
(eds.), The Analogical Mind, 2001 Tecuci,
Building Intelligent Agents An Apprenticeship
Multistrategy Learning Theory, Methodology, Tool
and Case Studies, Academic Press, 1998, pp
101-108.