Machine Learning: Symbol-based

1
Machine Learning Symbol-based
9
9.0 Introduction 9.1 A Framework
for Symbol-based Learning 9.2 Version Space
Search 9.3 The ID3 Decision Tree Induction
Algorithm 9.4 Inductive Bias and Learnability
9.5 Knowledge and Learning 9.6 Unsupervised
Learning 9.7 Reinforcement Learning 9.8 Epilogue
and References 9.9 Exercises
Additional source used in preparing the
slides Jean-Claude Latombes CS121 (Introduction
to Artificial Intelligence) lecture notes,
http//robotics.stanford.edu/latombe/cs121/2003/h
ome.htm (version spaces, decision trees) Tom
Mitchells machine learning notes (explanation
based learning)
2
Chapter Objectives

• Learn about several paradigms of symbol-based
  learning
• Learn about the issues in implementing and using
  learning algorithms
• The agent model can learn, i.e., can use prior
  experience to perform better in the future

3
A learning agent
Critic
environment
KB
Learning element
sensors
actuators
4
A general model of the learning process
5
A learning game with playing cards

• I would like to show what a full house is. I give
  you examples which are/are not full houses
• 6? 6? 6? 9? 9? is a full house
• 6? 6? 6? 6? 9? is not a full house
• 3? 3? 3? 6? 6? is a full house
• 1? 1? 1? 6? 6? is a full house
• Q? Q? Q? 6? 6? is a full house
• 1? 2? 3? 4? 5? is not a full house
• 1? 1? 3? 4? 5? is not a full house
• 1? 1? 1? 4? 5? is not a full house
• 1? 1? 1? 4? 4? is a full house

6
A learning game with playing cards

• If you havent guessed already, a full house is
  three of a kind and a pair of another kind.
• 6? 6? 6? 9? 9? is a full house
• 6? 6? 6? 6? 9? is not a full house
• 3? 3? 3? 6? 6? is a full house
• 1? 1? 1? 6? 6? is a full house
• Q? Q? Q? 6? 6? is a full house
• 1? 2? 3? 4? 5? is not a full house
• 1? 1? 3? 4? 5? is not a full house
• 1? 1? 1? 4? 5? is not a full house
• 1? 1? 1? 4? 4? is a full house

7
Intuitively,

• Im asking you to describe a set. This set is the
  concept I want you to learn.
• This is called inductive learning, i.e., learning
  a generalization from a set of examples.
• Concept learning is a typical inductive learning
  problem given examples of some concept, such as
  cat, soybean disease, or good stock
  investment, we attempt to infer a definition
  that will allow the learner to correctly
  recognize future instances of that concept.

8
Supervised learning

• This is called supervised learning because we
  assume that there is a teacher who classified the
  training data the learner is told whether an
  instance is a positive or negative example of a
  target concept.

9
Supervised learning?

• This definition might seem counter intuitive. If
  the teacher knows the concept, why doesnt s/he
  tell us directly and save us all the work?
• The teacher only knows the classification, the
  learner has to find out what the classification
  is. Imagine an online store there is a lot of
  data concerning whether a customer returns to the
  store. The information is there in terms of
  attributes and whether they come back or not.
  However, it is up to the learning system to
  characterize the concept, e.g,
• If a customer bought more than 4 books, s/he will
  return.
• If a customer spent more than 50, s/he will
  return.

10
Rewarded card example

• Deck of cards, with each card designated by
  r,s, its rank and suit, and some cards
  rewarded
• Background knowledge in the KB ((r1) ? ?
  (r10)) ? NUM (r) ((rJ) ? (rQ) ? (rK)) ?
  FACE (r) ((sS) ? (sC)) ? BLACK (s)
  ((sD) ? (sH)) ? RED (s)
• Training set REWARD(4,C) ? REWARD(7,C)
  ? REWARD(2,S) ? ?REWARD(5,H) ?
  ?REWARD(J,S)

11
Rewarded card example

• Training set REWARD(4,C) ? REWARD(7,C)
  ? REWARD(2,S) ? ?REWARD(5,H) ?
  ?REWARD(J,S)
• Card In the target set?
• 4? yes
• 7? yes
• 2? yes
• 5? no
• J? no
• Possible inductive hypothesis, h,
• h (NUM (r) ? BLACK (s) ? REWARD(r,s)

12
Learning a predicate

• Set E of objects (e.g., cards, drinking cups,
  writing instruments)
• Goal predicate CONCEPT (X), where X is an object
  in E, that takes the value True or False (e.g.,
  REWARD, MUG, PENCIL, BALL)
• Observable predicates A(X), B(X), (e.g., NUM,
  RED, HAS-HANDLE, HAS-ERASER)
• Training set values of CONCEPT for some
  combinations of values of the observable
  predicates
• Find a representation of CONCEPT of the form
  CONCEPT(X) ? A(X) ? ( B(X)? C(X) )

13
How can we do this?

• Go with the most general hypothesis possible
  any card is a rewarded card This will cover
  all the positive examples, but will not be able
  to eliminate any negative examples.
• Go with the most specific hypothesis
  possible the rewarded cards are 4?, 7?, 2?
  This will correctly sort all the examples in the
  training set, but it is overly specific, will not
  be sort any new examples.
• But the above two are good starting points.

14
Version space algorithm

• What we want to do is start with the most
  general and specific hypotheses, and when we
  see a positive example, we minimally generalize
  the most specific hypotheses when we see a
  negative example, we minimally specialize the
  most general hypothesis
• When the most general hypothesis and the most
  specific hypothesis are the same, the algorithm
  has converged, this is the target concept

15
Pictorially

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boundary of G
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boundary of S
potential target concepts
16
Hypothesis space

• When we shrink G, or enlarge S, we are
  essentially conducting a search in the hypothesis
  space
• A hypothesis is any sentence h of the form
  CONCEPT(X) ? A(X) ? ( B(X)? C(X) ) where, the
  right hand side is built with observable
  predicates
• The set of all hypotheses is called the
  hypothesis space, or H
• A hypothesis h agrees with an example if it
  gives the correct value of CONCEPT

17
Size of the hypothesis space

• n observable predicates
• 2n entries in the truth table
• A hypothesis is any subset of observable
  predicates with the associated truth tables so
  there are 2(2n) hypotheses to choose from
  BIG!
• n6 ? 2 64 1.8 x 10 19 BIG!
• Generate-and-test wont work.

18
Simplified Representation for the card problem

• For simplicity, we represent a concept by rs,
  with
• r a, n, f, 1, , 10, j, q, k
• s a, b, r, ?, ?, ?, ?For example
• n? represents NUM(r) ? (s?) ?
  REWARD(r,s)
• aa represents
• ANY-RANK(r) ? ANY-SUIT(s) ? REWARD(r,s)

19
Extension of an hypothesis

• The extension of an hypothesis h is the set of
  objects that verifies h.
• For instance,
• the extension of f? is j?, q?, k?, and
• the extension of aa is the set of all cards.

20
More general/specific relation

• Let h1 and h2 be two hypotheses in H
• h1 is more general than h2 iff the extension of
  h1 is a proper superset of the extension of h2
• For instance,
• aa is more general than f?,
• f? is more general than q?,
• fr and nr are not comparable

21
More general/specific relation (contd)

• The inverse of the more general relation is the
  more specific relation
• The more general relation defines a partial
  ordering on the hypotheses in H

22
A subset of the partial order for cards
23
G-Boundary / S-Boundary of V

• An hypothesis in V is most general iff no
  hypothesis in V is more general
• G-boundary G of V Set of most general hypotheses
  in V
• An hypothesis in V is most specific iff no
  hypothesis in V is more general
• S-boundary S of V Set of most specific
  hypotheses in V

24
Example The starting hypothesis space
G
S
25
4? is a positive example
We replace every hypothesis in S whose extension
does not contain 4? by its generalization set
aa
na
ab
The generalization set of a hypothesis h is the
set of the hypotheses that are immediately more
general than h
nb
a?
4a
n?
4b
4?
Generalization set of 4?
26
7? is the next positive example
Minimally generalize the most specific hypothesis
set
aa
We replace every hypothesis in S whose extension
does not contain 7? by its generalization set
na
ab
nb
a?
4a
n?
4b
4?
27
7? is positive(contd)
Minimally generalize the most specific hypothesis
set
aa
na
ab
nb
a?
4a
n?
4b
4?
28
7? is positive (contd)
Minimally generalize the most specific hypothesis
set
aa
na
ab
nb
a?
4a
n?
4b
4?
29
5? is a negative example
Minimally specialize the most general hypothesis
set
Specialization set of aa
aa
na
ab
nb
a?
4a
n?
4b
4?
30
5? is negative(contd)
Minimally specialize the most general hypothesis
set
aa
na
ab
nb
a?
4a
n?
4b
4?
31
After 3 examples (2 positive,1 negative)
G and S, and all hypotheses in between form
exactly the version space
ab
nb
a?
n?
1. If an hypothesis between G and S
disagreed with an example x, then an
hypothesis G or S would also disagree with
x, hence would have been removed
32
After 3 examples (2 positive,1 negative)
G and S, and all hypotheses in between form
exactly the version space
ab
nb
a?
n?
2. If there were an hypothesis not in
this set which agreed with all examples,
then it would have to be either no more
specific than any member of G but then it
would be in G or no more general than some
member of S but then it would be in S
33
At this stage
ab
nb
a?
n?
Do 8?, 6?, j? satisfy CONCEPT?
34
2? is the next positive example
Minimally generalize the most specific hypothesis
set
ab
nb
a?
n?
35
j? is the next negative example
Minimally specialize the most general hypothesis
set
ab
nb
36
Result
4? 7? 2? 5? j?
nb
NUM(r) ? BLACK(s) ? REWARD(r,s)
37
The version space algorithm

• Begin
• Initialize S to the first positive training
  instanceN is the set of all negative instances
  seen so far
• For each example x
• If x is positive, then (G,S) ?
  POSITIVE-UPDATE(G,S,x)
• else (G,S) ? NEGATIVE-UPDATE(G,S,x)
• If G S and both are singletons, then the
  algorithm has found a single concept that is
  consistent with all the data and the algorithm
  halts
• If G and S become empty, then there is no concept
  that covers all the positive instances and none
  of the negative instances
• End

38
The version space algorithm (contd)

• POSITIVE-UPDATE(G,S,x)
• Begin
• Delete all members of G that fail to match x
• For every s ? S, if s does not match x, replace s
  with its most specific generalizations that match
  x
• Delete from S any hypothesis that is more general
  than some other hypothesis in S
• Delete from S any hypothesis that is neither more
  specific than nor equal to a hypothesis in G
  (different than the textbook)
• End

39
The version space algorithm (contd)

• NEGATIVE-UPDATE(G,S,x)
• Begin
• Delete all members of S that match x
• For every g ? G, that matches x, replace g with
  its most general specializations that do not
  match x
• Delete from G any hypothesis that is more
  specific than some other hypothesis in G
• Delete from G any hypothesis that is neither more
  general nor equal to hypothesis in S (different
  than the textbook)
• End

40
Comments on Version Space Learning (VSL)

• It is a bi-directional search. One direction is
  specific to general and is driven by positive
  instances. The other direction is general to
  specific and is driven by negative instances.
• It is an incremental learning algorithm. The
  examples do not have to be given all at once (as
  opposed to learning decision trees.) The version
  space is meaningful even before it converges.
• The order of examples matters for the speed of
  convergence
• As is, cannot tolerate noise (misclassified
  examples), the version space might collapse

41
Examples and near misses for the concept arch
42
More on generalization operators

• Replacing constants with variables. For
  example, color (ball,red) generalizes to
  color (X,red)
• Dropping conditions from a conjunctive
  expression. For example, shape (X, round) ?
  size (X, small) ? color (X, red) generalizes
  to shape (X, round) ? color (X, red)

43
More on generalization operators (contd)

• Adding a disjunct to an expression. For
  example, shape (X, round) ? size (X, small) ?
  color (X, red) generalizes to shape (X,
  round) ? size (X, small) ? ( color (X, red) ?
  (color (X, blue) )
• Replacing a property with its parent in a class
  hierarchy. If we know that primary_color is a
  superclass of red, then color (X, red)
  generalizes to color (X, primary_color)

44
Another example

• sizes large, small
• colors red, white, blue
• shapes sphere, brick, cube
• object (size, color, shape)
• If the target concept is a red ball, then size
  should not matter, color should be red, and shape
  should be sphere
• If the target concept is ball, then size or
  color should not matter, shape should be sphere.

45
A portion of the concept space
46
Learning the concept of a red ball

• G obj (X, Y, Z)S
• positive obj (small, red, sphere)
• G obj (X, Y, Z)S obj (small, red,
  sphere)
• negative obj (small, blue, sphere)
• G obj (large, Y, Z), obj (X, red, Z), obj (X,
  white, sphere) obj (X,Y, brick), obj (X,
  Y, cube) S obj (small, red, sphere)
  delete from G every hypothesis that is neither
  more general than nor equal to a hypothesis in S
• G obj (X, red, Z) S obj (small, red,
  sphere)

47
Learning the concept of a red ball (contd)

• G obj (X, red, Z) S obj (small, red,
  sphere)
• positive obj (large, red, sphere)
• G obj (X, red, Z)S obj (X, red, sphere)
  
• negative obj (large, red, cube)
• G obj (small, red, Z), obj (X, red, sphere),
  obj (X, red, brick)S obj (X, red,
  sphere) delete from G every hypothesis that is
  neither more general than nor equal to a
  hypothesis in S
• G obj (X, red, sphere) S obj (X, red,
  sphere) converged to a single concept

48
LEX a program that learns heuristics

• Learns heuristics for symbolic integration
  problems
• Typical transformations used in performing
  integration include OP1 ? r f(x) dx ? r ? f(x)
  dx OP2 ? u dv ? uv - ? v du OP3 1 f(x) ?
  f(x) OP4 ? (f1(x) f2(x)) dx ? ? f1(x) dx
  ? f2(x) dx
• A heuristic tells when an operator is
  particularly useful If a problem state matches
  ? x transcendental(x) dx then apply OP2 with
  bindings u x dv transcendental (x) dx

49
A portion of LEXs hierarchy of symbols
50
The overall architecture

• A generalizer that uses candidate elimination to
  find heuristics
• A problem solver that produces positive and
  negative heuristics from a problem trace
• A critic that produces positive and negative
  instances from a problem traces (the credit
  assignment problem)
• A problem generator that produces new candidate
  problems

51
A version space for OP2 (Mitchell et al.,1983)
52
Comments on LEX

• The evolving heuristics are not guaranteed to be
  admissible. The solution path found by the
  problem solver may not actually be a shortest
  path solution.
• The problem generator is the least developed
  part of the program.
• Empirical studies before 5 problems solved in
  an average of 200 steps train with 12
  problems after 5 problems solved in an average
  of 20 steps

53
More comments on VSL

• Still lots of research going on
• Uses breadth-first search which might be
  inefficient
• might need to use beam-search to prune hypotheses
  from G and S if they grow excessively
• another alternative is to use inductive-bias and
  restrict the concept language
• How to address the noise problem? Maintain
  several G and S sets.

54
Decision Trees

• A decision tree allows a classification of an
  object by testing its values for certain
  properties
• check out the example at www.aiinc.ca/demos/wha
  le.html
• The learning problem is similar to concept
  learning using version spaces in the sense that
  we are trying to identify a class using the
  observable properties.
• It is different in the sense that we are trying
  to learn a structure that determines class
  membership after a sequence of questions. This
  structure is a decision tree.

55
Reverse engineered decision tree of the whale
watcher expert system
see flukes?
no
yes
see dorsal fin?
no
(see next page)
yes
size?
size med?
vlg
med
yes
no
blue whale
blow forward?
Size?
blows?
yes
no
lg
vsm
1
2
sperm whale
humpback whale
bowhead whale
gray whale
narwhal whale
right whale
56
Reverse engineered decision tree of the whale
watcher expert system (contd)
see flukes?
no
yes
see dorsal fin?
no
(see previous page)
yes
blow?
no
yes
size?
lg
sm
dorsal fin and blow visible at the same time?
dorsal fin tall and pointed?
yes
no
yes
no
killer whale
northern bottlenose whale
sei whale
fin whale
57
What does the original data look like?
58
The search problem

• Given a table of observable properties, search
  for a decision tree that
• correctly represents the data (assuming that the
  data is noise-free), and
• is as small as possible.
• What does the search tree look like?

59
Comparing VSL and learning DTs
A hypothesis learned in VSL can be represented as
a decision tree. Consider the predicate that we
used as a VSL exampleNUM(r) ? BLACK(s) ?
REWARD(r,s) The decision tree on the right
represents it
NUM?
True
False
BLACK?
False
False
True
True
False
60
Predicate as a Decision Tree
The predicate CONCEPT(x) ? A(x) ? (?B(x) v C(x))
can be represented by the following decision
tree

• ExampleA mushroom is poisonous iffit is yellow
  and small, or yellow,
• big and spotted
• x is a mushroom
• CONCEPT POISONOUS
• A YELLOW
• B BIG
• C SPOTTED
• D FUNNEL-CAP
• E BULKY

61
Training Set
62
Possible Decision Tree
63
Possible Decision Tree
CONCEPT ? (D ? (?E v A)) v
(C ? (B v ((E ? ?A) v A)))
KIS bias ? Build smallest decision tree
Computationally intractable problem? greedy
algorithm
64
Getting Started
The distribution of the training set is
True 6, 7, 8, 9, 10,13 False 1, 2, 3, 4, 5, 11,
12
65
Getting Started
The distribution of training set is
True 6, 7, 8, 9, 10,13 False 1, 2, 3, 4, 5, 11,
12
Without testing any observable predicate,
we could report that CONCEPT is False (majority
rule) with an estimated probability of error
P(E) 6/13
66
Getting Started
The distribution of training set is
True 6, 7, 8, 9, 10,13 False 1, 2, 3, 4, 5, 11,
12
Without testing any observable predicate,
we could report that CONCEPT is False (majority
rule)with an estimated probability of error P(E)
6/13
Assuming that we will only include one observable
predicate in the decision tree, which
predicateshould we test to minimize the
probability of error?
67
Assume Its A
68
Assume Its B
69
Assume Its C
70
Assume Its D
71
Assume Its E
So, the best predicate to test is A
72
Choice of Second Predicate
A
F
T
False
C
F
T
The majority rule gives the probability of error
Pr(EA) 1/8and Pr(E) 1/13
73
Choice of Third Predicate
A
F
T
False
C
F
T
True
B
T
F
74
Final Tree
L ? CONCEPT ? A ? (C v ?B)
75
Learning a decision tree

• Function induce_tree (example_set, properties)
• beginif all entries in example_set are in the
  same class then return a leaf node labeled with
  that classelse if properties is empty
  then return leaf node labeled with disjunction of
  all classes in example_set
  else begin select a property, P, and make it the
  root of the current tree delete P
  from properties for each value, V,
  of P begin
  create a branch of the tree labeled with V
  let partitionv be elements of
  example_set with values V
  for property P call
  induce_tree (partitionv, properties), attach
  result to branch V
  end endend

If property V is Boolean the partition will
contain two sets, one with property V true and
one with false
76
What happens if there is noise in the training
set?

• The part of the algorithm shown below handles
  this
• if properties is empty then return leaf
  node labeled with disjunction of all
  classes in example_set
• Consider a very small (but inconsistent) training
  set

A classificationT TF FF T
A?
True
False
False ? True
True
77
Using Information Theory

• Rather than minimizing the probability of error,
  most existing learning procedures try to minimize
  the expected number of questions needed to decide
  if an object x satisfies CONCEPT.
• This minimization is based on a measure of the
  quantity of information that is contained in
  the truth value of an observable predicate and is
  explained in Section 9.3.2. We will skip the
  technique given there and use the probability of
  error approach.

78
Assessing performance
79
The evaluation of ID3 in chess endgame
80
Other issues in learning decision trees

• If data for some attribute is missing and is
  hard to obtain, it might be possible to
  extrapolate or use unknown.
• If some attributes have continuous values,
  groupings might be used.
• If the data set is too large, one might use
  bagging to select a sample from the training set.
  Or, one can use boosting to assign a weight
  showing importance to each instance. Or, one can
  divide the sample set into subsets and train on
  one, and test on others.

81
Explanation based learning

• Idea can learn better when the background
  theory is known
• Use the domain theory to explain the instances
  taught
• Generalize the explanation to come up with a
  learned rule

82
Example

• We would like the system to learn what a cup is,
  i.e., we would like it to learn a rule of the
  form premise(X) ?? cup(X)
• Assume that we have a domain theoryliftable(X)
  ? holds_liquid(X) ? cup(X)part (Z,W) ?
  concave(W) ? points_up ? holds_liquid
  (Z)light(Y) ? part(Y,handle) ? liftable
  (Y)small(A) ? light(A)made_of(A,feathers) ?
  light(A)
• The training example is the followingcup
  (obj1) small(obj1)small(obj1) part(obj1,handle)
  owns(bob,obj1) part(obj1,bottom)part(obj1,
  bowl) points_up(bowl)concave(bowl) color(obj1,re
  d)

83
First, form a specific proof that obj1 is a cup
cup (obj1)
liftable (obj1)
holds_liquid (obj1)
light (obj1)
part (obj1, handle)
part (obj1, bowl)
points_up(bowl)
concave(bowl)
small (obj1)
84
Second, analyze the explanation structure to
generalize it
85
Third, adopt the generalized the proof
cup (X)
liftable (X)
holds_liquid (X)
light (X)
part (X, handle)
part (X, W)
points_up(W)
concave(W)
small (X)
86
The EBL algorithm

• Initialize hypothesis
• For each positive training example not covered by
  hypothesis
• 1. Explain how training example satisfies
  target concept, in terms of domain theory
• 2. Analyze the explanation to determine the
  most general conditions under which this
  explanation (proof) holds
• 3. Refine the hypothesis by adding a new rule,
  whose premises are the above conditions, and
  whose consequent asserts the target concept

87
Wait a minute!

• Isnt this just a restatement of what the
  learner already knows?
• Not really
• a theory-guided generalization from examples
• an example-guided operationalization of theories
• Even if you know all the rules of chess you get
  better if you play more
• Even if you know the basic axioms of
  probability, you get better as you solve more
  probability problems

88
Comments on EBL

• Note that the irrelevant properties of obj1
  were disregarded (e.g., color is red, it has a
  bottom)
• Also note that irrelevant generalizations were
  sorted out due to its goal-directed nature
• Allows justified generalization from a single
  example
• Generality of result depends on domain theory
• Still requires multiple examples
• Assumes that the domain theory is correct
  (error-free)---as opposed to approximate domain
  theories which we will not cover.
• This assumption holds in chess and other search
  problems.
• It allows us to assume explanation proof.

89
Two formulations for learning

• Inductive
• Given
• Instances
• Hypotheses
• Target concept
• Training examples of the target concept

• Analytical
• Given
• Instances
• Hypotheses
• Target concept
• Training examples of the target concept
• Domain theory for explaining examples

• Determine
• Hypotheses consistent with the training examples
  and the domain theory

• Determine
• Hypotheses consistent with the training examples

90
Two formulations for learning (contd)

• Inductive
• Hypothesis fits data
• Statistical inference
• Requires little prior knowledge
• Syntactic inductive bias

• Analytical
• Hypothesis fits domain theory
• Deductive inference
• Learns from scarce data
• Bias is domain theory

DT and VS learners are similarity-based Prior
knowledge is important. It might be one of the
reasons for humans ability to generalize from as
few as a single training instance. Prior
knowledge can guide in a space of an unlimited
number of generalizations that can be produced by
training examples.
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An example META-DENDRAL

• Learns rules for DENDRAL
• Remember that DENDRAL infers structure of
  organic molecules from their chemical formula and
  mass spectrographic data.
• Meta-DENDRAL constructs an explanation of the
  site of a cleavage using
• structure of a known compound
• mass and relative abundance of the fragments
  produced by spectrography
• a half-order theory (e.g., double and triple
  bonds do not break only fragments larger than
  two carbon atoms show up in the data)
• These explanations are used as examples for
  constructing general rules

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Analogical reasoning

• Idea if two situations are similar in some
  respects, then they will probably be in others
• Define the source of an analogy to be a problem
  solution. It is a theory that is relatively well
  understood.
• The target of an analogy is a theory that is not
  completely understood.
• Analogy constructs a mapping between
  corresponding elements of the target and the
  source.

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Example atom/solar system analogy

• The source domain contains yellow(sun)
  blue(earth) hotter-than(sun,earth)
  causes(more-massive(sun,earth),
  attract(sun,earth)) causes(attract(sun,earth),
  revolves-around(earth,sun))
• The target domain that the analogy is intended
  to explain includes more-massive(nucleus,
  electron) revolves-around(electron, nucleus)
• The mapping is sun ? nucleus and earth ?
  electron
• The extension of the mapping leads to the
  inference causes(more-massive(nucleus,electron)
  , attract(nucleus,electron))
  causes(attract(nucleus,electron),
  revolves-around(electron,nucleus))

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A typical framework

• Retrieval Given a target problem, select a
  potential source analog.
• Elaboration Derive additional features and
  relations of the source.
• Mapping and inference Mapping of source
  attributes into the target domain.
• Justification Show that the mapping is valid.