274 14 Representing Implicit Knowledge via Hypergraphs
where the <.5> denotes the probability, which is a component of the Truth Value associated with the edge.
Next, the most basic functional edge is the Execution edge, which is ternary and denotes a relation between a Schema, its input and its output, e.g.
Execution father_of Ben_Goertzel Ted_Goertzel
or a schema father_ of that outputs the father of its argument. The ExecutionOutput (ExOut) edge denotes the output of a Schema in an implicit way, e.g.
ExOut say_hello
refers to a particular act of saying hello, whereas
ExOut add_numbers {3, 4)
refers to the Concept corresponding to 7. Note that this latter example involves a set of three entities: sets are also part of the basic SMEPH knowledge representation. A set may be thought of as a hypergraph edge that points to all its members.
In this manner we may define a set of edges and vertices modeling the habitual activity patterns of a system when in different situations. This is called the derived hypergraph of the system. Note that this hypergraph can in principle be constructed no matter what happens inside the system: whether it’s a human brain, a formal neural network, Cyc, OCP, a quantum computer, etc. Of course, constructing the hypergraph in practice is quite a different story: for instance, we currently have no accurate way of measuring the habitual activity patterns inside the human brain. {MRI and PET and other neuroimaging technologies give only a crude view, though they are continually improving.
Pattern theory enters more deeply here when one thoroughly fleshes out the Inheritance concept. Philosophers of logic have extensively debated the relationship between extensional inheritance (inheritance between sets based on their members) and intensional inheritance (in- heritance between entity-types based on their properties). A variety of formal mechanisms have been proposed to capture this conceptual distinction; see (Wang, 2006, 1995 TODO make ref) for a review along with a novel approach utilizing uncertain term logic. Pattern theory provides a novel approach to defining intension: one may associate with each Concept Vertex in a system’s derived hypergraph the set of patterns associated with the structural pattern underlying that Concept Vertex. Then, one can define the strength of the IntensionalInheritanceEdge between two ConceptVertices A and B as the percentage of A’s pattern-set that is also contained in B’s pattern-set. According to this approach, for instance, one could have
IntInhEdge whale fish <0.6>
ExtInhEdge whale fish <0.0>
since the fish and whale sets have common properties but no common members.
14.4 Implications of Patternist Philosophy for Derived Hypergraphs of Intelligent Systems
Patternist philosophy rears its head here and makes some definite hypotheses about the struc- ture of derived hypergraphs. It suggests that derived hypergraphs should have a dual network
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