Information Technology Reference
In-Depth Information
typically aect only a small fraction of the environment|and in order to
concentrate on this fraction when specifying the eects of an action we need
access to it. Otherwise, i.e., if states are represented as abstract objects with-
out bearing an internal structure (as it is typical for automata theory, for
instance), the impact of an action could only be specied by a complete state
transition table. This would violate the most fundamental requirement for
adequacy.
Atomic propositions represent properties, or in general relations among
entities, which do or do not hold in a particular state. The truth-value of any
such proposition may change in the course of time as a consequence of a state
transition. Due to this dynamic characteristics the term \fluent" has been
established as a name for these propositions. Thus a state is characterized by
saying which of the various fluents are true and which are false in this state.
Denition 1.2.1.
Let E be a nite set of symbols called
entities
.Let F
denote a nite set of symbols called
fluent names
, each of which is associated
with a natural number (possibly zero) called
arity
.A
fluent
is an expression
f
(
e
1
;:::;e
n
)
where f 2F is of arity n and e
1
;:::;e
n
2E.A
fluent literal
is a fluent or its negation, denoted by :f
(
e
1
;:::;e
n
)
. A set of fluent literals
is
inconsistent
if it contains a fluent along with its negation, otherwise it is
consistent
.A
state
is a maximal consistent set of fluent literals.
Before we continue, let us illustrate these concepts with a small example.
At several places in this topic we will model the behavior of electric circuits.
Suppose a particular circuit consists of two binary switches, a light bulb,
and a battery. The various states of this system may be described using the
two entities
E
=
f
s
1
;
s
2
g
representing the two switches, along with the
unary fluent
up
denoting the position of its argument, and the nullary fluent
denoting the state of the bulb.
2
Then both
,say,
are fluent literals, and
f:
up
(
s
1
)
;
up
(
s
2
)
; :
light
g
is a maximal consistent
set of fluent literals, hence a state.
The reader may have noticed that formally any combination of truth-
values constitutes a state. Later in this topic, in Chapter 2, we will provide
means to distinguish states that cannot occur due to domain-specic depen-
dencies among fluents. For convenience, we introduce the following notational
conventions: If
`
is a fluent literal, then by
k`k
we denote its armative com-
ponent, that is,
kf
(
e
)
k
=
k:f
(
e
)
k
=
f
(
e
) where
f 2F
and
e
is a sequence
of
n
entities with
n
being the arity of
f
. This notation extends to sets of flu-
ent literals
S
as follows:
kSk
=
fk`k
:
` 2 Sg
. E.g., whenever
S
is a state,
then
kSk
is the set of all fluents. If
`
is a negative fluent literal, then
:`
should be interpreted as
k`k
. In other words,
::f
(
e
)=
f
(
e
). Finally, if
S
is a set of fluent literals, then by
:S
we denote the set
f:`
:
` 2 Sg
. E.g.,
a set
S
of fluent literals is inconsistent i
S \:S 6
=
fg
.
The second fundamental notion in any action theory are the actions them-
selves. Actions cause state transitions when being performed. As indicated
(
s
2
) and
:
light
up
light
2
The curious reader may take a peek at Fig. 2.3 on page 22.
