Information Technology Reference
In-Depth Information
Ψ and
^
;
1. Ψ
2. Ψ
=
Th
(
Ψ
)
; and
3. for any instance "
causes
%
if
2Rsuch that 2
Ψ and " 2
^
,we
have % 2 Ψ and % 2
^
.
If Th
R
(
Ψ;
)=(
Ψ;
^
)
and F 2 Ψ , then this is denoted by
(
Ψ;
)
R
F .
It is easy to see that
R
is monotonic, that is,
Ψ
1
Ψ
2
and
1
2
implies
fF
:(
Ψ
1
;
1
)
R
F gfF
:(
Ψ
2
;
2
)
R
F g
.
Example 2.6.1.
Let
E
=
f
light
0
g
, and let
R
consist
of the causal relationships reflecting the causal dependencies between the two
switches and the light bulb in Fig. 2.3, i.e.,
up
1
;
s
1
;
s
2
g
and
F
=
f
up
(
s
1
) causes
light
if
up
(
s
2
)
:
up
(
s
1
) causes
:
light
if
>
(2.10)
up
(
s
2
) causes
light
if
up
(
s
1
)
:
up
(
s
2
) causes
:
light
if
>
Now, consider the set
Ψ
1
=
f
s
1
)
g
.
Then
Th
R
(
Ψ
1
;
)=(
Th
(
f
up
(
s
1
)
^
up
(
s
2
)
;
light
g
)
; f
up
(
s
1
)
;
light
g
); hence,
(
Ψ
1
;
)
R
light
. In contrast, suppose
Ψ
2
=
f
up
(
s
1
)
^:
light
g
and, as be-
fore,
=
f
(
s
1
)
^
(
s
2
)
g
along with
=
f
(
up
up
up
(
s
1
)
g
, then
Th
R
(
Ψ
2
;
)=(
Th
(
f
(
s
1
)
^:
g
)
; f
(
s
1
)
g
)
up
up
light
up
since no causal relationship is applicable. Hence, (
Ψ
2
;
) 1
R
:
s
2
).
The notion of performing deduction on the basis of causal relationships
shall be exploited for the construction of a xpoint characterization of suc-
cessor states which accounts for indirect eects while respecting causal in-
formation. Informally speaking, suppose an action with direct eect
E
is
performed in state
S
. Then a state
T
is considered successor i the follow-
ing holds. Set
T
includes
E
,
T
along with
E
and the underlying causal
relationships do not entail an inconsistency, and each fluent change from
S
to
T
is grounded on some causal relationship. This last condition reflects
the idea of minimizing change.
Denition 2.6.2.
Let
(
E; F; A; L
)
be a basic action domain and R a set
of causal relationships. If S is a state and a an action, then a state T is
causal minimizing-change successor
of S and a i the following holds: Set L
contains an applicable action law instance a
transforms
C
into
E such that
(
up
T
=
f `
:((
S \ T
)
[ E;E
)
R
` g
(2.11)
that is, T is xpoint of the function T: f`
:((
S \ T
)
[ E;E
)
R
`g given
S and E.
Example 2.6.2.
Let
D
be the basic action domain which models our cir-
cuit composed of two switches and a light bulb of Fig. 2.3 as in Exam-
ple 2.2.2. Furthermore, let
R
consist in the four causal relationships (2.10)
above. Suppose that the current state be
S
=
f:
up
(
s
1
)
;
up
(
s
2
)
; :
light
g
as depicted in Fig. 2.3. The only applicable action law instance for ac-
tion
a
=
toggle
(
s
1
)is
toggle
(
s
1
) transforms
f:
up
(
s
1
)
g
into
f
up
(
s
1
)
g
