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That is, if state
s
exists on interval
i
, the state exists at both the beginning moment
and the ending moment of this interval. State with this property is called position
state.
Definition 4.5
Motion state: if state s satisfies
∀
t
(Holds-at(
s,t
)
ŗ
∃
i
(Div(
t,i
)
∧
Holds-on(
s,i
)))
That is, if state
s
exists at moment
t
, there must be an interval containing this
moment which
exists on. State with this property is called motion state.
Then the above eight basic relations can be classified into:
Position state: EC, =, TPP, TPP
-1
Motion state: DC, PO, NTPP, NTPP
-1
The two kinds of states can be distinguished by their performance: Position
state is critical and motion state is steady.
s
2. Perturbation principle
Galton has proposed the perturbation principle according to this classification,
which is an axiom system describing how spatial states transit on the temporal
interval.
Definition 4.6
Perturbation: if RCC relation R and R' satisfies
∃
t
(Holds-at(R(
a,b
),
t
)
∧
(∃
i
(Holds-on(R'(
a,b
),
i
))
∧
(inf(
i
)=
t
)
∨
(sup(
i
)=
t
)))))
That is, if there are a state R at moment
t
and an interval
i
which begins or
ends at
. Then R and R' are the perturbations of each other.
Perturbation principle: Each RCC relation is the perturbation of itself. And
one static state can only have one motive state that they are the perturbation of
each other, vice versa (Only rigid body is involved).
Assuming
t
. R' is the state on
i
R
is a RCC state,
R
1
,R
2
,…,R
n
are the all perturbations of
R
. There
are six axioms according to perturbation principle.
(A1) Holds-on(
R
(
a,b
),
i
)
ŗ
1
n
i
=
∨
Holds-at(
R
i
(
a,b
),sup(
i
));
n
i
=
∨
(A2) Holds-on(
R
(
a,b
),
i
)
ŗ
1
Holds-at(
R
i
(
a,b
),inf(
i
));
n
i
=
∨
(A3) Holds-at(
R
(
a,b
),
t
)
ŗ
∃
t
'
Holds-on(
R
i
(
a,b
),(
t,t
'));
1
n
i
∨
Holds-on(Ri(a,b),(t',t));
(A5) Holds-on(
s
,(
t
1
,t
2
))
∧
¬Holds-at(
s,t
3
)
∧
t
2
<
t
3
ŗ
∃
t
Holds-on(
s
,(
t
1
,t
))
∧
(A4) Holds-at(
R
(
a,b
),
t
)
ŗ
∃
t
'
1
