Promise Fulfillment
Property:
Let
σ
be an arbitrary model of
ϕ
,
and
ψ
∈
Φ
ϕ
a formula that promises
r
.
If
(
σ, j
)
q
ψ
then
(
σ, k
)
q
r
for some
k
≥
j
Proof:
Follows from the semantics of temporal
formulas.
Claim:
(promise fulfillment by models)
Let
σ
be an arbitrary model of
ϕ
,
and
ψ
∈
Φ
ϕ
a formula that promises
r
.
Then
σ
contains infinitely many positions
j
≥
0
such that
(
σ, j
)
q
¬
ψ
or
(
σ, j
)
q
r
Proof:
1. Assume
σ
contains infinitely many
ψ
positions.
Then
σ
must contain infinitely many
r
positions,
since
ψ
promises
r
.
2. Assume
σ
contains finitely many
ψ
positions.
Then it contains infinitely many
¬
ψ
positions.
135
Fulfilling Atoms
Definition:
Atom
A
fulfills
ψ
∈
Φ
ϕ
(which promises
r
)
if
¬
ψ
∈
A
or
r
∈
A
.
Example:
In
T
1
p
,
Only one promising formula:
ψ
:
1
p
promises
r
:
p
A
+
1
:
{
p,
2 1
p,
1
p
}
fulfills
1
p
since
p
∈
A
1
A
+
3
:
{
p,
¬
2 1
p,
1
p
}
fulfills
1
p
since
p
∈
A
3
A
+
4
:
{¬
p,
¬
2 1
p,
¬
1
p
}
fulfills
1
p
since
¬
1
p
∈
A
4
But
A

2
:
{¬
p,
2 1
p,
1
p
}
does not fulfill
1
p
since
1
p,
¬
p
∈
A
2
136
Tableau
T
1
p
@
@
@
R
6
A
+
1
:
{
p,
2 1
p,
1
p
}

6
A

2
:
{¬
p,
2 1
p,
1
p
}
@
@
@
@
@
@
R
A
+
3
:
{
p,
¬
2 1
p,
1
p
}
?
A
+
4
:
{¬
p,
¬
2 1
p,
¬
1
p
}
137
Fulfilling Paths
Definition:
A path
π
:
A
0
, A
1
, . . .
is fulfilling
if for every
promising formula
ψ
∈
Φ
ϕ
it contains infinitely many
A
j
that fulfill
ψ
.
Example:
In
T
1
p
,
A

2
, A

2
, A

2
, A
+
3
, A
+
4
, A
+
4
, . . .
A

2
, A
+
1
, A

2
, A
+
1
, A
+
1
, A
+
1
, . . .
are fulfilling paths, but
A

2
, A

2
, A

2
, A

2
, A

2
, A

2
, A

2
, . . .
is not a fulfilling path.
138