Inheritance

1
Inheritance Taxonomies and hierarchies are
natural ways of organizing knowledge because
abstraction, classification and generalization
are central tasks in knowledge representation. On
ce told that wombats are marsupials we already
know a lot about them. In other words, we know
that wombats inherit the properties of
marsupials. In description logics, inheritance
was just a way of doing certain types of logical
reasoning using a graphical metaphor if we have
a network where PianoConcerto is directly below
Concerto which is directly below MusicalWork,
then instances of PianoConcerto inherit
properties from MusicalWork because they are
logically also instances of the latter.
2
In a frame formalism, if the IS-A slot of
AdultHighSchoolStudent points to
HighSchoolStudent which points to Teenager, then
instances of AdultHighSchoolStudent may inherit
properties from HighSchoolStudent and in turn
from Teenager but there is no overt justification
for concluding that an instance of
AdultHighSchoolStudent must be an instance of
Teenager. In both cases, can instances of a
inherit properties from b? involves asking
whether we can get from a to b along a kind of
generalization path. Inheritance offers a
fundamental form of default reasoning that is a
core concept in designing intelligent systems.
3

• Inheritance networks
• We will first treat frames and descriptions as
  simple nodes in
• inheritance networks.
• Gray lt---- Elephant lt----- Clyde.
• For now, we will treat object-like concepts, like
  Elephant and properties,
• like Gray, equivalently. We could instead use a
  concept like GrayThing
• which would be a Thing whose color slot was
  filled with the individual gray.
• We will also blur over the distinction between
  classes and instances.
• We will use
• edges connecting one node to another and
  representing generalization
• relationships
• paths that are sequences of edges and
• conclusions supported by the paths in the
  example, these areClyde ? Elephant, Elephant ?
  Gray and Clyde ? Gray (the latteris true because
  the relationships are transitive.

4
Strict Inheritance is the simplest form of
inheritance and is used in description logics
and other systems based on classical logic.
Conclusions under strict inheritance are produced
by the complete transitive closures of all paths
in the network. In a tree-structured strict
inheritance network, inheritance is simple.
The supported conclusions include Ben
? Gray and Clyde ? Gray
5
The same situation obtains in a strict
inheritance network that is a DAG All
conclusions along any path are supported,
including conclusions reached by traversing
different branches from a node. Ernest
is a student and an employee. The network
supports conclusions thatErnest is an academic,
a taxpayer, and salaried. Note the negative edge.
This means that the conclusion that Ernest is
notilliterate is supported. DAGs feature
multiple inheritance, where a node inherits from
all its parents.
6
Defeasible Inheritance In the real world, the
inherited properties do not always hold they
can be defeated or overridden. This is most
obviously true in the case of default values for
slots, such as the default origin of a trip.
But virtually all properties and
characteristics can be overridden. In a
defeasible-inheritance system, one strategy of
drawing conclusions is to searching upward from
a focus node and selecting the first version of
the property under consideration. An example
7
There is an edge from Clyde to Elephant and an
edge fromElephant to Gray but a negative edge
from Clyde directlyto Gray. So, while elephants
are usually gray, Clyde is not. Intuitively, if
we were asked what the color of Clyde is,
wewould first find the negative conclusion about
Gray becauseit is directly asserted of
Clyde. The big problem is that different paths
in the network support different conclusions,
and a reasoning system must decide how to choose
the answer. In the example, there is an
argument for Clyde being gray! But there is a
better argument for concluding that he isnot,
because of the direct assertion. In many cases,
it will be impossible to say what conclusion is
right.
8
There is no formal reason to say
that Nixon is or is not a pacifist. So, this is
an ambiguous network.
9
There are two approaches to reasoning under these
conditions credulous (brave, choice) approaches
arbitrarily select a conclusion among those that
are equally well supported skeptical (cautious)
approaches often reject conclusions that are
contradicted by other paths. In the Nixon
diamond case, a credulous approach would flip a
coin. A skeptical account will refuse to draw a
conclusion.
10
Dealing with Defeasible Inheritance For DAGs
with defeasible inheritance, we need a method for
deciding what information to believe when
different conclusions can be reached through
different paths through the network. Shortest
Path and Inferential Distance
11
Shortest Path
Shows two examples ofdefeasible inheritance. In
the first DAG, it would beplausible to assume
thatClyde is not gray. The shortest path
heuristicsays that we should prefershorter
paths. And indeed, there are fewer steps
fromClyde to Gray on the pathwith the negative
edge. The same heuristic supportsthe
positive-edge path in thesecond DAG.
12
The intuition behind the shortest path heuristic
is that it makes sense to inherit from the most
specific subsuming class. If two superclasses
disagree on a property, we take the value from
the more specific one (that is, the one lower in
the inheritance hierarchy), believing that this
will be more relevant. The shortest path
heuristic is useful and capable of producing
correct conclusions in many cases but it has
flaws. It can produce intuitively incorrect
answers when there are redundant edges in the
network -- that is, edges that are already
implied by the basic network. Consider the
following
13
The edge labeled q is redundant, as from the rest
of the network it is already clear that Clyde is,
unambiguously, an elephant. But by introducing
this edge we have inadvertently changed the
polarity of the conclusion about Clydes color.
Indeed, the path from Clyde to Gray that uses
edge q is now shorter than the one with the
negative edge. Another problem with the shortest
path is that they fall prey to varying grainsize
of description in different parts of a
network. There should not be a bias against aa
long and fine-grained inheritancechain.
14
The left-hand path ends in apositive edge. The
right-handside has one more edge and ends in a
negative edge. The positive conclusion
issupported by the shortestpath. But this
network should beconsidered as ambiguous asthe
Nixon diamond network.
15
Shortest path is one of several possible
preemption strategies thatallow us to make
choices (admissibility choices) among
competingpaths. It provides us with a
specificity criterion that determines that
more specific information about an item is more
valuable than information that is true about a
more general class of items. Another
specificity criterion is inferential distance,
which is based not on linear distance but on more
general topological criteria.
16
If we start with Clyde, we want to conclude
that RoyalElephant is more specific than
Elephant, the edge q notwithstanding, because
there is a path to Elephant that passes through
RoyalElephant. Because this path is more
specific, we will preferthe negative path to
Gray over the positive one from Elephant to Gray.
In general, a node a is considered closer to
node b than to node c according to inferential
distance if and only if there is a path from a to
c through b regardless of the actual length of
any path from a to b and to c. This criterion
works for the simple examples we saw, and in the
case of the 856/857-node network is
is inconclusive, as desired.
17
But inferential distance still has
problems. What should one do if the path from a
through b to c is itself contradicted by another
path? This might require a different
formalization of inheritance.
18

• One possible formal account of inheritance
  networks
• An inheritance hierarchy ? ltV,Egt is a DAG with
  positive and negative
• edges intended to denote (normally) is-a and
  (normally) not-is-a
• relations, respectively. V are the vertices, E
  are the edges. Positive edgeswill be written as
  (a x) and negative edges will be written as (a
  ?x).
• A positive path is a sequence of one or more
  positive edges a x.
• A negative path is a sequence of zero or more
  positive edges followed bya single negative edge
  a ? x. A path is either a positive or a
  negative
• path.
• No paths have more than one negative edge.
• A path (or argument) supports a conclusion in the
  following ways
• a x supports the conclusion a ? x (a is an
  x)
• a ? x supports the conclusion a /? x (a is
  not an x).

19

• A conclusion can be supported by many arguments,
  not all equally
• believable. What makes an argument prevail, given
  other arguments
• in the network? We get this from the definition
  of admissibility
• supports a path if the corresponding set of
  edges are in E and the path
• is admissible according to the definition that
  follows. The hierarchy
• supports a conclusion a ? x (or a /? x) if it
  supports some corresponding
• path between a and x.
• A path a s1 sn (?x) is admissible if
  every edge in it is admissible
• with respect to a.
• An edge v (?x) is admissible in ? with respect
  to a if there is a positive
• path a s1 sn v (n ? 0) in E and
• each edge in a s1 sn v is admissible in ?
  wrt a (recursively)
• no edge in a s1 sn v is redundant in ?
  wrt a (see below)
• no intermediate node a, s1 sn is a preemptor
  of v (?x) wrt a (see below).

20
Thus, an edge is admissible with respect to a if
there is a nonredundant admissible path leading
to it from a that contains no preempting intermedi
aries
This is the edgein question
a
si
v
x
A node y along path a y v is a
preemptor of v x (v ?x) with respect to a
if y ?x ? E (y x ? E). For example, in the
figure, the node Whale preempts the negative edge
from Mammalto AquaticCreature with respect to
both Whale and BlueWhale.
21

• A positive edge b w is redundant in ? with
  respect to a node a if
• there is some positive path b t1 tm v ?
  E (m ? 1) for which
• each edge in b t1 tm is admissible in ?
  with respect to a(that is, none of the edges are
  themselves preempted)
• there are no c and i such that c ?ti is
  admissible in ? with respect to a
• there is no such c such that c ?w is admissible
  in ? with respect to a.

22
By this definition, the edge labeled q is
redundant with respect to BlueWhale. The
definition of redundancy for a negative edge is
analogous.
23

• Extensions
• The term extension means a possible set of
  beliefs supported by the
• network. Ambiguous networks have multiple
  extensions. Formally
• is a-connected if and only if for every node in
  ?, there is a path from
• a to x, and for every edge v (?)x there is a
  positive path from a to v.
• In other words, every node and edge is reachable
  from a.
• ? is (potentially) ambiguous with respect to
  node a at x if there is some
• node x ?V such that both a s1 sn x and a
  t1 tm ? x are paths.
• A credulous extension of an inheritance hierarchy
  ? with respect to node a
• is a maximal unambiguous a-connected subhierarchy
  of ? with respect
• to a.
• Thus, if X is a credulous extension of ?, then
  adding an edge from ? to X
• makes X either ambiguous or not a-connected.

24
An ambiguousnetwork
and two credulousextensions
If an edge is added from Mammal toMilkProducer,
this extensionwont be a-connected any more (a
is Platypus)
Adding an edge from FurryAnimalto Mammal above
and from EggLayer to Mammal below makes the
extensions ambiguous.
This shows that both extensionsare credulous.
25
Credulous extensions do not incorporate any
notion of admissibility or preemption. For
example, the network on the left has two
credulous extensions with respect to the node
Clyde, shown on the right.
26

• It would be nice to have a mechanism to rule out
  one of these extensions.
• This leads us to a definition of preferred
  extensions
• Let X and Y be credulous extensions of ? wrt a
  node a. X is preferred to Yif and only if there
  are nodes v and x such that
• X and Y agree on all edges whose endpoints
  precede v topologicallythere is an edge v x
  (or v ?x) that is inadmissible in ? and
• this edge is in Y but not in X.
• A credulous extension is a preferred extension if
  there is no other
• credulous extension that is preferred to it.
• The crucial feature of this definition is that it
  uses the notion of admissibility.

27
So, if ? is the left-hand network, the network in
the middle is a preferredextension over the one
on the right. If we use the assignments a
Clyde, v Elephant and x Gray, then the two
extensions agree up to Elephant but the edge
Elephant Gray is not admissible because it has
a preemptor, Clyde, and that edge is in the
extension on the right but not on the left.
28
Given the above formalization, several reasoning
methods are still possible credulous reasoning
choose a preferred extension and believe all
the conclusions supported by it. skeptical
reasoning Believe the conclusions supported by
any path that is present in all preferred
extensions. ideally skeptical reasoning Believe
the conclusions that are supported by all
preferred extensions. This is different from
skeptical reasoning because these conclusions
can be supported by different path in
each extension, with the consequence that this
type of reasoning cannot be computed using paths.
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