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From a semantic point of view, a conceptual graph corresponds to an assertion about some individuals that exists in the particular application domain to be considered. Accordingly, the graphical representation of Figure 13A corresponds to the assertion: "A (generic) pretty lady is dancing gracefully." The arrows on the arcs are used to distinguish the first and the second arguments of the predicate ("dancing"). Very often, conceptual graphs are expressed in a sort of "linear" representation; the representation in linear form of the assertion: "The (specific) pretty lady identified as the entity #1520 of the system is dancing gracefully" is then given in Fig. 13b.

An important formal property of conceptual graphs concerns the possibility of their exact mapping into FOPC formulas thanks to the operator [capital phi]. The mapping produces a conjunction of predicates, one corresponding to each node of the graph; generic concepts map to variables, individual concepts map to constants. Therefore, if u is the graph represented in Figure 13B, its translation in predicate calculus gives:

[capital phi]u = [inverted E]x [inverted E]y [inverted E]z (Lady(#1520) [wedge] Attr(#1520, x) [wedge] Pretty(x) [wedge] Agent(y, #1520) [wedge] Dance(y) [wedge] Manner(y, z) [wedge] Graceful(z))

To give, however, to conceptual graphs the full power of first-order logic, it is necessary to introduce some "second-order" extensions of the notions above. These extensions make use mainly of two particular concept-types, PROPOSITION -- concepts of the type PROPOSITION are used to represent contexts: PROPOSITION can take, in fact, one or more conceptual graphs as referent -- and SITUATION. The difference between the two is explained in this way by Sowa: "When a conceptual graph is the referent of a concept of type PROPOSITION, it represents the proposition stated by the graph. When it is the referent of a concept of the type SITUATION, it represents the situation described by the proposition stated by the graph." Combinations of the two are then used to represent second-order structures like those corresponding to the "subordinate" or "completive" constructions of natural language. We reproduce here, Figure 14. In the figure 14, John believes a proposition: Jane, on the contrary, wants a situation; we can note that the "subjects" of the "states" represented by BELIEVE and WANT are represented by "experiencers" (EXPR) rather than "agents" (AGNT). As Sowa states, in general, negation, modalities, and the "patients" (objects), PTNT, of verbs like "think" and "know" are linked with contexts of the type PROPOSITION; times, locations, and the patients of verbs like "want" and "fear" are linked with SITUATION contexts. The way contexts are nested determines the scope of the quantifiers. In Figure 14, DUCKLING is existentially quantified inside the situation that Jane wants, which is in turn existentially quantified inside John's belief. Figure 15 is the linear form of Figure 14: as we can see, the graphical form is surely more legible in order to show the nesting of contexts.

[PRETTY] <-- (ATTR) <-- [LADY: #1520] <-- (AGNT) <-- [DANCE] --> (MANR) --> [GRACEFUL]

FIGURE 13B Conceptual graph in linear form.


FIGURE 14 Graphical representation of second-order constructions.

We can conclude these few notes about conceptual graphs by giving some information about an important CG construction, the "lambda abstraction." [slashed Lambda]-abstraction can be defined as a conceptual graph with one or more generic concepts identified as formal parameters (variables); the relationship with the analogous LISP construction is evident. As in LISP, the main function of [slashed Lambda]-abstraction is that of allowing new definitions: monadic abstractions are used to define "concept" types, and n-adic abstractions are used to define n-adic "relation" types. Moreover, an important, specific function of [slashed Lambda]-abstraction is that of allowing the definition of "single-use" types: instead of inserting in the type field of a concept a label corresponding to a concept-type defined in a permanent way, this type field can contain a [slashed Lambda]-abstraction that is expressly created for a single use. This possibility is typically used to represent restrictive relative clauses (see below). Other possible utilizations of [slashed Lambda]-abstraction concern the possibility of defining schemata used to represent defaults and expectations, of defining new individuals as aggregations of parts, and of defining typical individuals as prototypes. We give now, in Figure 16, a simple example of use of [slashed Lambda]-abstraction to define PET-CAT as a type of CAT owned by some person -- CAT is then a supertype of the newly defined concept PET-CAT. The equals sign that appears in the definition means that the type label (PET-CAT) is a synonym for the [slashed Lambda]-expression that defines the type, i.e., any occurrence of the type label PET-CAT could be replaced by the [slashed Lambda]-expression.

[PERSON: John] <-- (EXPR) <-- [BELIEVE] --> (PTNT) --> [PROPOSITION:
[PERSON: Jane*x] <-- (EXPR) <-- [WANT] --> (PTNT) --> [SITUATION:
[*x] <-- (AGNT) <-- [MARRY] --> (PTNT) --> [SAILOR]]]

FIGURE 15 Linear representation of second-order constructions.

PET-CAT = ([slashed Lambda]x) [CAT: *x] <-- (PTNT) <-- [OWN] <-- (STAT) <-- [PERSON]

FIGURE 16 Defining a new concept using the [slashed Lambda]-abstraction.

As a second example of use of the [slashed Lambda]-abstraction, we give in Figure 17 the representation of the sentence "some elephants that perform in a circus earn money." A [slashed Lambda]-expression is used here to define the "single-use" concept "elephants that perform in a circus"; the BENF relation indicates that the elephants are the beneficiaries of the earning. In this conceptual graph, the (global) single-use concept has a "generic plural referent" represented by the symbol "{*}" that, in this case, indicates some unspecified elements of the type "elephants that perform in a circus." In CGs theory, plurals are regarded as generalized quantifiers, and represented in the same way as other quantifiers. For example, the representation of the sentences "every elephant that performs in a circus earns money" and "many elephants that perform in a circus earn money" is still that of Figure 17, with the symbols "[inverted sans serif]" and "{*}@many" in place of the generic plural referent "{*}"; "@" is a "qualifier," and the expression "@many" indicates "many of them."


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