topical media & game development
Inheritance hierarchies play a role both in
knowledge representation systems and object-oriented
In effect, historically, the notions of frames
and is-a hierarchies
(that play a role in
knowledge representation) and the notions
have primarily been developed in a programming language context)
have mutually influenced each other.
programming languages, classes and inheritance are
strongly related to types and polymorphism, and directed
towards the construction of reliable programming
In contrast, the goal of knowledge representation is to
develop a semantically consistent description
of some real world domain, which allows us to reason about
the properties of the elements in that domain.
- declarative relation among entities
- isa-trees -- partial ordering
- isa/is-not -- bipolar, is-not inference
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
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
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
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
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
slide: Taxonomies and predicate logic
The latter is often used as an example of
non-monotonicity that may occur when using defaults
(in this case the assumption that all birds can fly).
The mathematical semantics for declarative taxonomic hierarchies,
as given in [To86], are based on the notion of
constructible lattices of predicates,
expressing a partial order between the predicates
involved in a taxonomy (such as, for example, Quaker and Human).
A substantial part of the analysis presented in [To86],
however, is concerned with employing the graph
representation of inheritance structures
to improve on the efficiency of reasoning
about the entities populating the graph.
In the presence of multiple inheritance and non-monotonicity
due to exceptions and defaults, care must be taken
to follow the right path through
the inheritance graph when searching for the value
of a particular attribute.
Operationally, the solution presented by [To86]
involves an ordering of inference paths (working upwards)
according to the number of intermediate nodes.
Intuitively, this corresponds to the distance
between the node using an attribute and the node defining
the value of the attribute.
In strictly monotonic situations such a measure plays
no role, however!
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