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In-Depth Information
•
X
4
A
5
: This directly points at something wrong with the literal
A
5
in the axiom. Indeed,
A
5
permits deriving the expression
≡
∃
s
:
¬
F
which yields an inconsistency in connection
∀
with
s
:
F
(derived via
A
4
).
≡
∧
∧
•
X
2
X
3
X
4
: This points at the
-operator as possible flaw. This is correct, since if
we replace the
, we can easily avoid the inconsistency. Moreover, the diagnosis
showed that both arguments of the
∧
by a
∨
are necessary for the inconsistency, which is also
correct as we have to be able to conclude
∧
A
A
A
3
to be inconsistent.
Notice the appeal of this form of refined diagnosis, as the precise location of the error is
actually marked with an "X". By then, it is up to the human engineer to do the repair properly.
4
and
5
from
4. Assurance
Suppose that an ontology upon receiving the query has presented us with a number of, say
n
1
, answers from which we can choose one. From the viewpoint of the ontology, each answer
is logically correct, and in the absence of preferences, certainty factors, or other means of
selection, we can only choose the best one subjectively. The goal of assurance is making this
decision with a provable benefit. For that matter, we briefly introduce some elements from the
theory of games, which will become handy when putting things together to a
reasoning engine
with assurance
. The reader familiar with matrix games may safely skip section 4.1, and move
on to section 4.2 directly.
4.1 Matrix-games
A(
non-cooperative n-person
)
game
}
of players being able to choose actions from their corresponding strategies within the set
of sets
PS
Γ
=(
N
,
PS
,
H
)
is a triple composed of a set
N
=
{
1, 2, . . . ,
n
=
{
}
. The
i
-th player, by taking action
s
i
∈
PS
1
,
PS
2
,...,
PS
n
PS
i
from his set
(
)
∈
PS
i
of possible strategies, receives the payoff
u
i
i
denotes the
strategies chosen by
i
's opponents. The set
H
thus comprises the set of payoff functions for
each player, i.e.
H
s
i
,
s
, where
u
i
H
and
s
−
i
−
1
PS
i
→
R
. Although we will use the general definition
here, our application use of game-theory will be with 2-player games, with player 1 being the
user of the ontology, and player 2 being the whole set of remaining entities outside the user's
scope.
A
(Nash-)equilibrium
is a choice
s
∗
=(
=
u
i
|
i
u
i
:
×
=
s
1
,...,
s
n
)
such that
s
i
,
s
∗
−
i
)
≤
s
i
,
s
∗
−
i
)
∀
u
(
u
(
s
i
∈
PS
i
N
, i.e. if any of the players solely chooses an action other than
s
i
, his revenue
will decrease. It is easy to construct examples where equilibria are not existing among the
pure strategies in
PS
i
. But, if strategies are understood as probabilities for taking certain
actions during repetitions of the game, then Glicksberg (1952) has proven that equilibria exist
for every game with continuous payoff functions. In that case, the payoff is averaged over the
repetitions of the game, i.e. we consider the
expected payoff
. Strategies which are interpreted as
probability distributions over the sets in
PS
are called
mixed strategies
, and we shall exclusively
refer to these in the sequel. The set
S
i
consists of all mixed strategies over
PS
i
. A game is
called
zero-sum
,if
and for all
i
∈
∑
i
u
i
=
0, or in the two-person case, if
u
1
=
−
u
2
. The game is called
finite
,
if the sets in
PS
are all finite. For a finite zero-sum game
Γ
0
, the average revenue under an
max
x
min
y
x
T
Ay
.
How is this related to our above reasoning problem? Player 1 will be the user of the ontology,
and player 2 will be the collection of all other agents in the system. The use of zero-sum games
is convenient because it implicitly (and perhaps pessimistically) assumes the other agents to
equilibrium strategy is the
value
of the game, denoted as
v
(
Γ
0
)=
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