Incompleteness in NEOCLASSIC

 

There are several types of incompleteness in NEOCLASSIC that need to be mentioned. For concept processing, subsumption is incomplete in several ways. Some of these incompletenesses also apply to individual processing. In addition, there are some incompletenesses in rule processing and propagations for individuals. These types of incompleteness are all discussed below.

Concept Processing

  There are a few ways in which concept subsumption is incomplete.

Test descriptions are treated as black boxes, in the same way as primitive concepts are. In fact there is really no way for NEOCLASSIC to determine the meaning of test descriptions. For example, the built-in HOST concept Integer contains the test description (testH integerp). If the concept EvenInteger is defined as (testH evenp), then it will not be classified under Integer, because NEOCLASSIC does not know that anything which satisfies the evenp test must also be an Integer. To truely define EvenInteger it must be defined using Integer (see Sections 2.1 and 2.2), as EvenInteger is:

• [](and Integer (testH evenp))

The only exception to treating test functions as black boxes is that (testH integerp) and (testH floatp) are subsumed by (testH numberp).

Also, with respect to concept subsumption, CLASSIC individuals are treated as having no properties. The reason for this is that the concept hierarchy should not change when individuals change (concept definitions cannot change). This impacts the two operators fills and oneOf.

If the individual Mary is known to be an Athlete, and concept C is the description of someone whose only child is Mary

• [](and (fills child Mary) (atMost 1 child))

then C is not subsumed by the concept of someone all of whose children are athletes

• [](all child Athlete)

This applies to CLASSIC oneOf descriptions as follows: if it is known that Joe and Mary are both Mammals, (oneOf Joe Mary) is not classified below Mammal. Thus, a CLASSIC concept containing only a oneOf description will be classified directly below either ClassicThing or another concept containing only a oneOf description, but never below a concept containing any other type of description.

Even if the properties of individuals are implied by the presence of these individuals in concept descriptions, NEOCLASSIC doesn't take these properties into account. For example, suppose that concept C1 is defined as

• []

  
  (and (fills child Sally) (all child Athlete)
  		(fills friend Sally) (atMost 1 friend))
  

and concept C2 is defined as (all friend Athlete). NEOCLASSIC does not infer that C1 is subsumed by C2.

As a more complex example, suppose that Susan is known to have Bob as a client and David is known to have Bill as a client. Let C1 be defined as

• []

  
  (and Company
  		(atLeast 1 employee)
  		(all employee (fills client Jack))
  		(all employee (oneOf Susan David))
  		(all contractor (atMost 1 client))
  		(all contractor (oneOf Susan David)))
  

and let C2 be defined as

• [](atMost 1 contractor)

NEOCLASSIC does not infer that C1 is subsumed by C2, because the establishment of the subsumption would require the use of contingent properties of Susan and David.

The above non-subsumptions are not really incompletenesses in NEOCLASSIC, as the standard definition of subsumption ignores contingent properties of individuals. However, NEOCLASSIC is incomplete with respect to this standard definition because it ignores properties of individuals that are implied by their presence in descriptions. For example, if C3 was defined as

• []

  
  (and Company
  		(atLeast 1 employee)
  		(all employee (fills client Jack))
  		(all employee (oneOf Susan David))
  		(all contractor (atMost 1 client))
  		(all contractor (oneOf Susan David))
  		(fills r Susan) (all r (fills client Bob))
  		(fills s David) (all s (fills client Bill)))
  

NEOCLASSIC would not infer that it was subsumed by C2, even though this inference does follow from the standard definition of subsumption.

To detect this subsumption, NEOCLASSIC would have to determine that either Susan or David must be an Employee; if Susan is an Employee, then she can't be a Contractor because she has to have at least 2 clients; if David is an Employee, then he can't be a Contractor because he has to have at least 2 clients. This reasoning by cases is computationally difficult, which is one reason it is not implemented in NEOCLASSIC.

As another example, suppose that the concept C4 is defined as

• []

  
  (and (fills child Sally) (all child Athlete)
  		(fills friend Sally) (atMost 1 friend))
  

and concept C5 is defined as (all friend ATHLETE). NEOCLASSIC does not infer that C4 is subsumed by C5.

HOST oneOf descriptions do not have the same types of incompleteness as CLASSIC oneOf descriptions, because the properties of HOST individuals do not change. A HOST concept containing only a oneOf description may be classified under a concept containing a test description, if all the HOST individuals in the oneOf description satisfy the test description. In addition, a HOST concept containing only a oneOf description can be classified under a concept containing an interval description, if the oneOf description contains only numbers, and they are all within the specified interval.

Individual Processing

The incompletenesses in concept subsumption can appear when determining whether or not an individual satisfies a concept description. This is because all restrictions on individuals are handled as concepts, not as descriptions of individuals. Suppose that the individual Sam is known to be a Vegetarian, and that the individual Mary is defined as someone all of whose friends have Sam as a teacher, and no one else:

• []

  
  (all friend (and (fills teacher Sam)
  		(atMost 1 teacher)))
  

Mary will not be found to satisfy the concept described by

• [](all friend (all teacher Vegetarian))

because the properties of Sam, specifically that he is a Vegetarian, are not taken into account when doing the subsumption test.

Both rules and propagations are performed only on known instances. Thus, if NEOCLASSIC knows that all Mary's sisters are Athletes, and she has at least 1 sister, it does not create a skolem individual representing the sister, in order to reason about it.

Rules in NEOCLASSIC are treated only as forward-chaining inferences, not as logical inferences. Thus, there is a rule stating that if someone is a Vegetarian, then he is known to be a HealthyThing, and NEOCLASSIC knows that Joe is an UnhealthyThing (a concept disjoint from HealthyThing), it does not infer that Joe is not a Vegetarian.