Handbook of Knowledge Representation

A conceptual graph (CG) is a graph representation for logic based on the semanticnetworks of artificial intelligence and the existential graphs of Charles Sanders Peirce.Several versions of CGs have been designed and implemented over the past thirtyyears. The simplest are the typeless core CGs, which correspond to Peirce’s originalexistential graphs. More common are the extended CGs, which are a typed superset ofthe core. The research CGs have explored novel techniques for reasoning, knowledgerepresentation, and natural language semantics. The semantics of the core and ex-tended CGs is defined by a formal mapping to and from the ISO standard for CommonLogic, but the research CGs are defined by a variety of formal and informal extensions.This article surveys the notation, applications, and reasoning methods used with CGsand their mapping to and from other versions of logic. 5.1 From Existential Graphs to Conceptual Graphs During the 1960s, graph-based semantic representations were popular in both theoret-ical and computational linguistics. At one of the most impressive conferences of thedecade, Margaret Masterman [21] introduced a graph-based notation, called a seman-tic network, which included a lattice of concept types; Silvio Ceccato [1] presentedcorrelational nets, which were based on 56 different relations, including subtype, in-stance, part-whole, case relations, kinship relations, and various kinds of attributes;and David Hays [15] presented dependency graphs, which formalized the notationdeveloped by the linguist Lucien Tesnière [40]. The early graph notations representedthe relational structures underlying natural language semantics, but none of them couldexpress full first-order logic. Woods [42] and McDermott [22] wrote scathing critiquesof their logical weaknesses.In the late 1970s, many graph notations were designed to represent first-order logicor a formally-defined subset [7]. Sowa [32] developed a version of conceptual graphs(CGs) as an intermediate language for mapping natural language questions and asser-tions to a relational database. Fig. 5.1 shows a CG for the sentence John is going to 2145. Conceptual Graphs Figure 5.1: CG display form for John is going to Boston by bus. Boston by bus. The rectangles are called concepts, and the circles are called concep-tual relations. An arc pointing toward a circle marks the first argument of the relation,and an arc pointing away from a circle marks the last argument. If a relation has onlyone argument, the arrowhead is omitted. If a relation has more than two arguments,the arrowheads are replaced by integers 1, . . . , n.The conceptual graph in Fig. 5.1 represents a typed or sorted version of logic. Eachof the four concepts has a type label, which represents the type of entity the conceptrefers to: Person, Go, Boston, or Bus. Two of the concepts have names, whichidentify the referent: John or Boston. Each of the three conceptual relations has atype label that represents the type of relation: agent (Agnt), destination (Dest), orinstrument (Inst). The CG as a whole indicates that the person John is the agent ofsome instance of going, the city Boston is the destination, and a bus is the instrument.Fig. 5.1 can be translated to the following formula: (∃x)(∃y)(Go(x) ∧ Person(John) ∧ City(Boston) ∧ Bus(y) ∧Agnt(x, John) ∧ Dest(x,Boston) ∧ Inst(x, y)). As this translation shows, the only logical operators used in Fig. 5.1 are con-junction and the existential quantifier. Those two operators are the most common intranslations from natural languages, and many of the early semantic networks couldnot represent any others.For his pioneering Begriffsschrift (concept writing), Frege [8] adopted a tree no-tation for representing full first-order logic, using only four operators: assertion (the“turnstile” operator ), negation (a short vertical line), implication (a hook), and theuniversal quantifier (a cup containing the bound variable). Fig. 5.2 shows the Begriffs-schrift equivalent of Fig. 5.1, and following is its translation to predicate calculus: ∼(∀x)(∀y)(Go(x) ⊃ (Person(John) ⊃ (City(Boston) ⊃(Bus(y) ⊃ (Agnt(x, John) ⊃ (Dest(x,Boston) ⊃ ∼ Inst(x, y))))))) Frege’s choice of operators simplified his rules of inference, but they lead to awk-ward paraphrases: It is false that for every x and y, if x is an instance of going then ifJohn is a person then if Boston is a city then if y is a bus then if the agent of x is Johnthen if the destination of x is Boston then the instrument of x is not y.Unlike Frege, who rejected Boolean algebra, Peirce developed the algebraic nota-tion for first-order logic as a generalization of the Boolean operators. Since Booletreated disjunction as logical addition and conjunction as logical multiplication,