Mapping models of analogy

1
Mapping models of analogy

• CS/ISYE/PSYC 7790
• Fall 2003

2
What weve discussed

• General format for analogical reasoning
• Feature-based models of analogical comparison...
• Today
• Using a mapping model of analogy for problem
  solving

3
Outline of talk

• Mapping models of analogy
• The Structure Mapping Engine
• PHINEAS, a system for modeling physical systems
  via analogy

4
Stages of analogical reasoning
target description
base description
ACCESS
MAPPING
possible inferences
Casebase of descriptions
Revised inferences
TWEAKING
5
Mapping models of analogy

• Descriptions are given in a form of predicate
  calculus, or as semantic network
• Mapping involves finding set of correspondences
  between nodes of the network
• Emphasis on preserving system of structural
  relationships

6
Structured representations

• Evolved out of early work by Schank at Yale and
  by the LNR group (Levin, Norman Rumelhart) at
  UCSD
• Entities given as nodes of tree
• Predicate calculus representation with fixed set
  of predicates
• (In some systems) hierarchies of nodes can be
  built

7
Simple water flow representation
8
Types of predicates

• Entities
• Simple objects, like PIPE1, SUN, ICE-CUBE
• Attributes
• Properties of a single object, such as RED, TALL,
  LIQUID
• Relations
• Properties that exist between two or more objects
  or expressions, such as TALLER, COLOR-OF, FLOW

9
Limitations of other metrics

• In feature list matching, cannot account for
  relationships between matched items
• Cat chases dog very different from dog chases cat
• In frame matching, predefined level of
  carry-over.
• Always modulo frame size

10
Mapping models of analogy

• Descriptions are given in a form of predicate
  calculus, or as semantic network
• Mapping involves finding set of correspondences
  between nodes of the network
• Emphasis on preserving system of structural
  relationships

11
Making graph matching tractable

• General graph matching is hard
• Limited version of graph matching can be much
  faster O(n ln(n))
• Limit the nodes that can match
• predicate identicality
• similarity table, giving matchable predicates

12
Initial descriptions...
13
Mapping between descriptions
14
Constraints of structure mapping

• Identicality
• Only items with identical predicates, or
  functions that are arguments of identical
  predicates may match
• One-to-one mapping
• Parallel connectivity
• Systematicity

15
Generating candidate inferences
16
Generating candidate inferences

• Carry over parts of the base that intersect the
  mapped part of the base and target
• Once the inferences are proposed, they must be
  tested (can still be invalid)

17
Algorithm for Structure Mapping

• Quickly create sets of potential matches using
  identicality constraint
• Assemble matches into connected sets, called
  kernel mappings
• Assemble global mappings as sets of consistent
  kernel mappings
• To understand this, lets dissect a mapping...

18
Anatomy of a mapping
19
(No Transcript)
20
Advantages of structure mapping

• Complexity
• Local matches O(n ln(n))
• Kernel mapping constructure ??
• Global mapping construction O(n)
• Run best when
• Small numbers of identical predicates in each
  domain
• Lots of higher order structure (order 3 or higher)

21
Systems using structure mapping

• PHINEAS, developing theories for novel physical
  processes (Falkenhainer, 1988)
• SEQL, abstracting over a sequence of geometric
  shapes (Skorstad, 1988)
• MAC/FAC, a model of analogical access (Forbus,
  Gentner, Law, 1995)
• I-SME, an incremental mapping engine (Forbus,
  Ferguson, Gentner, 1994)
• MAGI, symmetry detection for stories and
  perceptual data (Ferguson, 1994)
• JUXTA, a model of diagram understanding
  (Ferguson, 1995)

22
PHINEAS

• Domain Physical systems described using
  Qualitative Process Theory
• Model Verification-based analogical reasoning

23
Example problem
Hot brick in water
24
Representation of problem
(Solid brick) (Volume-solid brick) (Liquid
water1) (Contained-liquid water1) (Container-of
water1 bucket) (substance-of water1
water) (Immersed-in brick water1) (Contained-in
water1 bucket)
25
Representation of problem
(Meets (Situation 2-obj-hf-sit0) (Set
(Decreasing (Temperature-in brick))
(Increasing (Temperature-in water1))
(Greater-Than (A (Temperature-in brick))
(A (Temperature-in water1)))))
(Situation 2-obj-hf-sit1 (Set (Constant
(Temperature-in brick)) (Constant
(Temperature-in water1)) (Equal-to
(A (Temperature-in brick))
(A (Temperature-in water1))))))
26
(No Transcript)
27
Access and mapping stage

• Two stages
• Finding new behavior in hierarchy of observed
  behaviors
• oscillation
• dual-approach finish
• Create an SME mapping for each candidate
• In this case, chooses 2-container fluid flow

28
Generated candidate inferences

• Theory is generated as part of candidate
  inference creation
• Inferences require two kinds of tweaking
• Filling in skolem values
• Modifying incorrect theories

29
Resulting mapping
30
Filling in skolem values

• Somes the mapping does not provide an identity
  for some values
• (Physical-path container1 container2 pipe) gt
  (Physical-path brick water ?pipe)
• Fix it using an abstraction hierarchy
• (Physical-path brick water
• (common-face brick water))
• Simply assume it is a new value
• (skolem water) gt sk-water-1 (caloric heat)
• Must also assume new phase for sk-water-1 (since
  it is not a liquid)

31
(No Transcript)
32
Verifying the new theory

• Run the new theory through QPE (Qualitative
  Process Engine), and get an envisionment
• In this case, the theory correctly predicts the
  observed behavior

33
Revising the theory

• Use past experimence to revise the theory
• Never fully implemented by Falkenhainer
• After revision, run through verification again
• If unsuccessful, try another analog

34
Final theory
35
Final results on PHINEAS

• Runs on over a dozen examples, which are
  variances of nine basic explanation tasks
• boiling
• liquid flow
• osmosis
• floating
• oscillation

36
Limitations of PHINEAS

• Inaudequate model of access
• may be corrected in new version which
  incorporates MAC/FAC
• Always proposes a theory
• Revision mechanism ad hoc
• Cannot merge multiple analogies

37
Summary

• Mapping models depend on creating hard limits on
  local matchability
• SME is a model of mapping which uses identicality
  to limit search
• PHINEAS uses SME to flexibly generate physical
  theories of novel physical behaviors