Abstract inheritance
- declarative relation among entities
Inheritance networks
- isa-trees -- partial ordering
- isa/is-not -- bipolar, is-not inference
Non-monotonic reasoning
Nixon is-a Quaker
Nixon is-a Republican
Quakers are Pacifists
Republicans are not Pacifists
Incremental system evolution is in practice non-monotonic!
slide: Knowledge representation and inheritance
One of the first formal analyses of the declarative
aspects of inheritance systems was given in
[To86].
The theoretical framework developed in [To86]
covers the inheritance formalisms found in
frame systems such as FRL, KRL, KLONE and NETL,
but also to some extent the inheritance mechanisms
of Simula, Smalltalk, Flavors and Loops.
The focus of [To86], however, is to develop a formal
theory of inheritance networks including
defaults and exceptions.
The values of
attributes play a far more important role in such networks
than in a programming context.
In particular, to determine whether the relationships expressed
in an inheritance graph are consistent, we must be able to
reason about the values of these attributes.
In contrast, the use of inheritance in programming languages
is primarily focused on sharing instance variables and
overriding (virtual) member functions, and is not so much concerned
with the actual values of instance variables.
Inheritance networks in knowledge representation systems
are often non monotonic as a result
of having is-not relations in addition to is-a
relations
and also because properties (for example can-fly)
can be deleted.
Monotonicity is basically the requirement that all
properties are preserved, which is the case for
strict inheritance satisfying the substitution principle.
It is a requirement that should be adhered to
at the risk of jeopardizing the integrity of the
system.
Nevertheless, strict inheritance may be regarded
as too inflexible to express real world
properties in a knowledge representation system.
The meaning of is-a and is-not relations in
a knowledge representation inheritance graph may
equivalently be
expressed as predicate logic statements.
For example, the statements
express the relation between, respectively,
the predicates Quaker and Republican
to the predicate Human in the graph above.
In addition, the statements
introduce the predicate Pacifist that
leads to an inconsistency when considering the
statement that Nixon is a Quaker and a Republican.
Some other examples of statements expressing
relations between entities in a taxonomic structure
are given in slide [9-logic].
