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,
ќ
st
is defined to be the subset of
ќ
that does not
mention any other state terms except st, does not quantify over state variables,
and does not mention Poss and <. Formally,
ќ
st
is the smallest set satisfying the
following conditions:
For any state term
st
(1) if
∈
ќ
and it does not mention any state term, then
∈
ќ
st
;
(2) for every fluent
F
(
x
1
,…,x
n
, st
)∈
ќ
s
,
F
(
x
1
,…,x
n
, st
)∈
ќ
st
;
,
ϕ
, (
∀
x
)
, (
∃
x
)
,
(
∀
a)
, and (
∃
a
)
are all in
ќ
st
, where
x
and
a
are variables of sort
o
and
sort
(3) if
,
ϕ
∈
ќ
st
, then ¬
,
∧
ϕ
,
∨
ϕ
,
ϕ
a
respectively.
ќ
st
2
is defined to denote the second-order extension of
ќ
st
by n-ary predicate
variables on domain of sort
, n≥0. Formally,
ќ
st
2
is the smallest set satisfying
o
the following conditions:
(1)
ќ
st
⊆
ќ
st
2
;
(2) if
p
is an n-ary predicate variable on domain of sort
o
, and
x
1
,…,x
n
are terms
x
1
,…,x
n
)∈
ќ
st
2
;
of sort
o
, then
p
(
(3) if
,
ϕ
∈
ќ
st
2
, then ¬
,
∧
ϕ
,
∨
ϕ
,
ϕ
↔ ϕ
, (
∀
p
)
, (
∃
p
)
,
(
∀
x
)
, (
∃
x
)
, (
∀
a
)
and (
∃
a
)
are all in
ќ
st
2
, where
x
and
a
are
variables of sort
,
o
and sort
a
respectively, and
p
is an n-ary predicate variable
on domain of sort
o
.
2.10.2
Basic action theory in LR
A basic action theory D is of the following form:
