1. Conceptual Graphs(1) A CG is a finite, connected, bipartite graph.
The nodes of the graph are either concepts or conceptual relations.
Concept nodes represent either concrete or abstract objects in the world.
Conceptual relation nodes indicate a relation involving one or more concepts.
Figure 6.14, Figure 6.15 (p. 219)
Using bipartite graphs rather than using labeled arcs cause us to simplify the representation of relations of any arity. (Figure 6.14)
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2. Conceptual Relations of different arities (6.14)
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3. Graph of “Mary gave John the book.”(6.15)
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4. Conceptual Graphs(2) Types, Individuals, and Names
In CG, every concept is a unique individual of a particular type.
Each concept box is labeled with a type label (i.e. class, e.g. dog, Fig. 6.14) or the names of the type and the individual(Figure 6.16), or the names of the type and the marker(Figure 6.17).
Each concept node indicates an individual of specified type.
Individual is the referent of the concept and is indicated by either an individual marker(Figure 6.18) or the generic marker (Figure 6.20).
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5. Conceptual Graphs(Fig.6.16, 6.17, 6.18)
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6. Conceptual Graph of the sentence “The dog scratches its ear with its paw.” (6.20)
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7. Conceptual Graphs(3) Type Hierarchy
type hierarchy is a partial ordering on the set of types indicated by the symbol (  ).
Figure 6.21
t  s (t is a subtype of s, s is a supertype of t)
t  s, t  u (t is common subtype of s and u)
s  v, u  v (v is common supertype of s and u)
type hierarchy of CG is a lattice, a common form of multiple inheritance system.
Minimal common supertype, maximal common subtype, universal type, absurd type.
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8. A type lattice illustrating types (Fig. 6.21)
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9. Conceptual Graphs(4) Generalization and Specialization
CG has four operations(copy, restrict, join, and simplify) to form new graphs from existing graphs.
copy rule allows us to form the exact copy of a new graph.
restriction allows concept nodes in a graph to be replaced by a node representing their specialization.
e.g. replace the generic marker by an individual marker
replace a type by one of its subtype
join rule lets us combine two graphs into a single graph if there is a identical concept node.
If a graph contains two duplicate relations, then one of them may be deleted by the simplify rule(Figure 6.22).
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10. Examples of restrict, join, and simplify operations (6.22)
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11. Conceptual Graphs(5) Propositional Nodes CG and Logic
CG includes a concept type, proposition, that takes a set of conceptual graphs as its referent and allows us to define relations involving propositions.
Figure 6.24: “Tom believes that Jane likes pizza.”
CG and Logic
Negation: using propositional concepts and a unary operation called “Neg” (Figure 6.25)
Disjunction: using a relation OR which take two propositions
Existential quantification: generic concepts are assumed to be existentially quantified.
$X $Y (dog(X) Ù color(X,Y) Ù brown(Y)) – Fig. 6.14
Universal quantification: using negation and existential quantification
"X "Y (Ø(dog(X) Ù color(X,Y) Ù pink(Y))) – Fig. 6.25
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12. CG of “Tom believes that Jane likes pizza.”(6.24)
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13. CG of “There are no pink dogs.”(6.25)
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14. Conceptual Graphs(6) CG and Logic (Continued)
Changing a CG(g) into a predicate calculus
1. Assign a unique variable x1, x2,…,xn to each of the n generic concepts.
2. Assign a unique constant to each individual concept in g.
3. Represent each concept node by a unary predicate with the same name and whose argument is the variable or constant assigned to that node.
4. Represent each n-ary conceptual relation in g as an n-ary predicate whose name is the same as the relation.
5. Take the conjunction of all atomic sentences formed under 3 and 4.
Figure 6.16
$X1 (dog(emma) ^ color(emma, X1) ^ brown(X1))
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15. CG and Predicate Calculus
$X1 (dog(emma) ^ color(emma, X1) ^ brown(X1))
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