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) Knowledge Representation

Conceptual Relations of different arities (6.14) Knowledge Representation

Graph of “Mary gave John the book.”(6.15) Knowledge Representation

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). Knowledge Representation

Conceptual Graphs(Fig.6.16, 6.17, 6.18) Knowledge Representation

Conceptual Graph of the sentence “The dog scratches its ear with its paw.” (6.20) Knowledge Representation

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. Knowledge Representation

A type lattice illustrating types (Fig. 6.21) Knowledge Representation

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). Knowledge Representation

Examples of restrict, join, and simplify operations (6.22) Knowledge Representation

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 Knowledge Representation

CG of “Tom believes that Jane likes pizza.”(6.24) Knowledge Representation

CG of “There are no pink dogs.”(6.25) Knowledge Representation

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)) Knowledge Representation

CG and Predicate Calculus $X1 (dog(emma) ^ color(emma, X1) ^ brown(X1)) Knowledge Representation