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not, at least not
a priori
, respect a similar notion. Since the principle of min-
imizing change is widely considered essential for the Ramication Problem in
literature, the present section is devoted to an analysis of whether and how
this principle and the successive application of causal relationships relate.
It will turn out that causal successor states cover all states with minimal
distance to some preliminary successors in satisfying the state constraints
while respecting causality. Yet we will also show that the converse does not
hold. That is, requiring minimality is proved failing to account for all possible
causal successor states in some domains. This result challenges the common
belief in the necessity of minimizing change when addressing the Ramication
Problem.
For a general solution to the Ramication Problem, as we have seen all
along, mere state constraints contain insucient information. A suitable min-
imization strategy therefore requires additional domain knowledge. The pur-
pose of this knowledge is to help telling apart derived implications which
correspond to actual indirect eects from those which are just logical con-
sequences. For example,
up
(
s
1
)
^
up
(
s
2
)
light
, derived from state con-
straint
s
2
), indicates a causally correct implication,
whereas
up
(
s
1
)
^:
light
:
up
(
s
2
), derived from the same constraint,
does not. Though being logically equivalent, the two implications need to be
distinguished when minimizing change while accounting for indirect eects.
Causal relationships serve this purpose if taken as directed implications. Un-
like classical material implications, these directed implications must not be
applied in reverse direction. That is, while the two fluent formulas
F ^`
1
`
2
and
F ^:`
2
:`
1
are interchangeable, the semantics of the correspond-
ing relationships
`
1
causes
`
2
if
F
and
:`
2
causes
:`
1
if
F
shall dier
also in their logical reading. This reading of causal relationships as directed
implications shall be the basis for a formal denition of minimizing change
with regard to causal knowledge. This rst of all requires a notion of how
to employ causal relationships as deduction rules. Similar to the operational
reading of causal relationships, this requires to distinguish between a con-
text and a set of actually occurred eects. For the purpose of deduction, the
former is given as a set of arbitrary fluent formulas
Ψ
while the latter is,
as before, a set of fluent literals,
. In what follows, if
F
is a fluent for-
mula, then we say that a set of fluent formulas
Ψ entails F
if
F
is true
in all states satisfying
Ψ
.By
Th
(
Ψ
) we denote the set of all formulas thus
entailed by
Ψ
(that is, the
theory
of
Ψ
). The derivable consequences of a
context-eect pair given underlying causal relationships is then dened as
follows.
light
up
(
s
1
)
^
up
(
Denition 2.6.1.
Let E and F be sets of entities and fluent names, respec-
tively, and R a set of causal relationships. If Ψ is a set of fluent formulas
and a set of fluent literals, then the
theory
induced by
(
Ψ;
)
relative
to R, written Th
R
(
Ψ;
)
, is the smallest (wrt. set inclusion) pair
(
Ψ;
^
)
such that
