SATISFIABILITY DEGREE THEORY FOR TEMPORAL LOGIC
∗
Jian Luo, Guiming Luo and Mo Xia
Key Laboratory for Information System Security, Ministry of Education, Tsinghua National Laboratory for Information
Science and Technology, School of Software, Tsinghua University, Beijing 100084, China
Keywords:
Satisfiability degree, Propositional logic, Temporal logic, Reasoning.
Abstract:
The truth value of propositional logic is not capable of representing the real word full of complexity and diver-
sity. The requirements of the proposition satisfiability are reviewed in this paper. Every state is labeled with
a vector, which is defined by the proposition satisfiability degree. The satisfiability degree for temporal logic
is proposed based on the vector of satisfiability degree. It is used to interpret the truth degree of the temporal
logic instead of true or false. A sound and precise reasoning system for temporal logic is established and the
computation is given. One example of a leadership election is included to show that uncertain information can
be quantized by the satisfiability degree.
1 INTRODUCTION
The idea of temporal logic (Mattolini and Nesi, 2000)
is that a formula is not statically true or false in a
time model. Instead, the models of temporal logi-
cal contain several states and a formula can be true
in some states and false in others. The formulas may
change their truth values as the system evolves from
state to state, but the truth values of the formulas are
true or false. Sometimes, a state partially satisfies
a formula, so it is not absolutely true or false, and
the semantic of the temporal logic, which is based
the classical Boolean logic, cannot interpret this case.
Thus, the world requires new ways to express uncer-
tainty. Many studies have used non-classical logic,
such as fuzzy logic (Bergmann, 2008), probabilistic
logic (Raedt and Kersting, 2003), modal logic, etc.
Satisfiability degree, a new precise logic presen-
tation, was proposed in (Luo and Yin, 2009). It de-
scribes the extent to which a proposition is true based
on the truth table by finding out the proportion of sat-
isfiable interpretations. Unlike fuzzy logic and proba-
bilistic logic, satisfiability degree does not need mem-
bership function or distribution function and it is de-
termined by the proposition itself. Moreover, satis-
fiability degree extends the concepts of satisfaction
and contradictory propositions in Boolean logic and
truth values of propositions are precisely interpreted
as their satisfiability degrees.
∗
This work is supported by the Funds NSFC 60973049,
60635020, and TNList cross-discipline foundations.
Sometimes, given the premise is true, we want
to deduce the truth degree of a considered conclu-
sion. The conditional satisfiability degree was pro-
posed in (Luo and Yin, 2009), to quantitatively repre-
sent the deductive reasoning, which is based on if the
satisfiability degree of premise is given, we deduce
the satisfiability degree of the conclusion.
There are many algorithms to compute the sat-
isfiability degree, such as the backtracking algo-
rithm (Yin et al., 2009), the satisfiability degree com-
putation based on CNF (Hu et al., 2009), the algo-
rithm based on binary decision diagrams (Luo and
Yin, 2009) and the propositional matrix search algo-
rithm (Luo and Luo, 2010). Once a propositional for-
mula is given, its satisfiability degree can be precisely
computed using those algorithms. Thus, this paper
only focuses on the performance and properties of sat-
isfiability degree based on the temporal logic.
Because the temporal logic is based on proposi-
tional logic and temporal connectives, the truth value
of a temporal formula can be precise interpreted by
satisfiability degree. Thus, if there are only several
models are available for the concerned formula, we
can choose the model with maximum satisfiability
degree. Sometimes, a model checker may not find
counterexamples but it does means the system cannot
be applied to some domains, but satisfiability degree
can provide us a quantitative analysis of model check-
ing (Kang and Park, 2005).
497
Luo J., Luo G. and Xia M..
SATISFIABILITY DEGREE THEORY FOR TEMPORAL LOGIC.
DOI: 10.5220/0003672804970500
In Proceedings of the International Conference on Evolutionary Computation Theory and Applications (FCTA-2011), pages 497-500
ISBN: 978-989-8425-83-6
Copyright
c
 2011 SCITEPRESS (Science and Technology Publications, Lda.)

2 SATISFIABILITY DEGREE
Thus, we use satisfiability degree to define the truth
values of propositions in states instead of ”true” or
false. Sometimes, it is difficult to choose some atomic
descriptions, because some properties cannot be sub-
divided into. We define a new temporal model on n
propositions set and use satisfiability degree to inter-
pret the truth values of those propositions.
2.1 Temporal Model
Definition 1. Let Σ be a set of n propositions. A
model M over Σ is a triple M = (S,R,V ) where:
• S is a set of states;
• R ⊂ S × S is a total transition relation, i.e., any
state s ∈ S, there is a state s
0
∈ S, s.t. (s,s
0
) ∈ R;
• V : Σ
n
× S → [0, 1]
n
is a function that mapping the
satisfiability degree of each proposition in s .
Thus, our model has a collection of state S, a rela-
tion R, saying how the system can move from state
to state, and, associated with each state s, one has the
n-dimension vector V (s), components of which are
satisfiability degree, denoted as f (p, s):
V (s) = ( f (p
1
,s), f (p
2
,s),... f (p
n
,s))
T
(1)
Where f (p
i
,s) is the satisfiability degree of
p
i
in state s. We can express all the in-
formation about a model M using a directed
graph and Figure 1 bellow shows us an example,
where S = {s
0
,s
1
,s
2
},Σ = {p,q,r} and V (s
0
) =
(0.5,0.6,0.2),V (s
1
) = (0.2,0.3,0.1) and V (s
2
) =
(0.8,0,1).
s
0
s
1
s
2










2.0
6.0
5.0










1.0
3.0
2.0










1
0
8.0
Figure 1: A temporal model based on satisfiability degree.
Note that, if all the propositions in set Σ are atomic
propositions, then their satisfiability degree are 0 or
1. That is the classical temporal model referred to the
paper (Huth and Ryan, 2005). Therefore, the models
we define are more expressive than that of classical
temporal models.
Definition 2. Let M = (S,R,V ) be a model. A path
ρ in M is an infinite sequence s
0
· s
1
· s
2
... of states
such that, for ∀i ≥ 0,(s
i
,s
i+1
) ∈ R.
Consider the path ρ = s
0
· s
1
· s
2
... It represents a
possible future of our system. We write ρ
i
for the
suffix starting at s
i
, e.g.ρ
3
is s
3
· s
4
...
2.2 Satisfiability Degree for CTL
∗
The syntax of CTL
∗
involves two classes of formulas,
state formula and path formula.
• state formula, which are evaluated in states:
ϕ ::= p | ¬ϕ | ϕ ∧ ϕ | A[α] | E[α] (2)
Where p ∈ Σ and α is any path formula; and
• path formula, which are evaluated along paths:
α ::= ϕ | ¬α | α∧α | α U α | G α | F α | X α (3)
The semantics (Huth and Ryan, 2005) of CTL
∗
was defined by a satisfaction relation, denoted by ,
which is characterized as the least relation on the
paths. Since the interpretation of a CTL
∗
formula
varies over states, we denote the satisfiability degree
of a CTL
∗
formula ϕ at state s according to the model
M as f
M
s
(ϕ).
Definition 3. For a model M = (S, R,V ), the satisfi-
ability degree of a CTL
∗
formula at state s is defined
inductively as:
• ∀ ϕ ∈ P, f
M
s
(ϕ) = f (ϕ,s), where P is the set of
Boolean formulas.
• f
M
s
(¬ϕ) = 1 − f
M
s
(ϕ)
• f
M
s
(ϕ
1
∧ϕ
2
) = min( f
M
s
(ϕ
1
), f
M
s
(ϕ
2
)), ϕ
1
∧ϕ
2
/∈ P
• f
M
s
(X α) = f
M
s
∗
(α), where s
∗
is the next state.
• f
M
s
(F α) = sup
∀s
∗
∈ρ
f
M
s
∗
(α), where ρ start with s and
s
∗
is any state on ρ.
• f
M
s
(G α) = inf
∀s
∗
∈ρ
f
M
s
∗
(α), where ρ start with s and
s
∗
is any state on ρ.
• f
M
s
(α
1
Uα
2
)
= sup
∀s
j
∈ρ
( f
M
s
0
(α
2
), min
0≤i≤ j
( f
M
s
i
(α
1
), f
M
s
j
(α
2
))), where
ρ = s
0
· s
1
· s
2
... , and s
0
= s, starts with s.
• f
M
s
(A[α]) = inf
∀ρ
f
M
s
(α), where ρ starts with s.
• f
M
s
(E[α]) = sup
∀ρ
f
M
s
(α), where ρ starts with s.
The truth value of each propositional formula is
determined by the state s with respect to the satisfia-
bility degree vector V (s). Suppose M = (S,R,V ) be a
model, s ∈ S, and ϕ a CTL
∗
formula. We have the fol-
low conclusions: If M, s |= ϕ if, and only if we have
f
M
s
(ϕ) = 1.
FCTA 2011 - International Conference on Fuzzy Computation Theory and Applications
498

3 A REASONING SYSTEM
A signed temporal formula can is a tuple (ϕ, f ) where
f is the satisfiability degree of ϕ . The proposed rea-
soning system is a pair ν = (A,L) where A is a set of
temporal logic axioms, and L is a collection of infer-
ence rules. An inference rule based on satisfiability
degree is a pair l = (l
op
,l
f
) where l
op
is syntactical
component that operates on temporal formulas, while
l
f
is a valuation component operating on satisfiability
degree to compute the satisfiability degree of the con-
clusion depending on the satisfiability degrees of the
premises. A rule l is usually written as:
ϕ
1
,ϕ
2
,.. .ϕ
n
l
op
(ϕ
1
,ϕ
2
... ϕ
n
)
,
f
1
, f
2
,.. . f
n
l
f
( f
1
, f
2
... f
n
)
(4)
This expression means that if the formulas
ϕ
1
,ϕ
2
... ϕ
n
have the satisfiability degree f
1
, f
2
... f
n
respectively, then l
op
(ϕ
1
,ϕ
2
,
˙
ϕ
n
) is satisfiable at least
to the satisfiability degree l
f
( f
1
, f
2
,.. . f
n
).
The inference rules for Boolean operations:
L
∧
:
p, q
p ∧ q
,
f
1
, f
2
f
1
f
2
(5)
L
∨
:
p, q
p ∨ q
,
f
1
, f
2
f
1
+ f
2
− f
1
f
2
(6)
L
→
:
p, q
p → q
,
f
1
, f
2
1 − f
1
+ f
1
f
2
(7)
And for any CTL
∗
formula ϕ, we have
L
¬
:
ϕ
¬ϕ
,
f
1 − f
(8)
The theorems proved by the natural deduction can
also be proved by our reasoning system, because we
have the following theorem. Its proof is omitted.
Theorem 1. For propositional formulas ϕ and φ :
(1) ϕ  φ if and only if f (φ | ϕ) = 1;
(2) ϕ ` φ if and only if f (φ | ϕ) = 1.
In addition, there are four kinds of generalization
rules for temporal logic.
L
AG
:
ϕ
AG ϕ
,
f
f
(9)
L
GF
:
G ϕ
F ϕ
,
f
f
(10)
L
X
:
ϕ
F ϕ
,
f
f
(11)
L
AE
:
A α
E α
,
f
f
(12)
Theorem 2. Inference rules (5)-(12) are sound.
Proof: the rule L
¬
is trivial. Let ω
1
,ω
2
∈ Ω be in-
terpretations such that p(ω
1
) = 1, and q(ω
2
) = 1, then
ω = (ω
1
,ω
2
) ∈ Ω × Ω, p ∧ q(ω) = 1. By the Defini-
tion of satisfiability degree, we have
f (p ∧ q) =
|Ω
p
× Ω
q
|
|Ω × Ω|
= f (p) f (q) = f
1
f
2
(13)
The rules such as L
∨
and L
→
can be derive from
L
∧
and L
¬
, since p ∧ q = ¬(¬p ∧ ¬q) and p → q =
¬p ∨q. If a CTL formula ϕ has the minimum satisfia-
bility degree f regardless of models, then G ϕ has the
same satisfiability degree f , so L
G
is sound. Since for
any state formula, we have:
f
M
s
(M ϕ) ≥ f
M
s
(G ϕ) (14)
Thus L
GF
is sound. L
X
and L
AE
are trivial sound.
It is very difficult to establish a complete one for
CTL
∗
, the completeness of our deductive system for
CTL
∗
need to be proved in our future works.
4 APPLICATION
This section presents another application, called
Leader Election. This problem provide a protocol to
select a leader out of their midst. The winning can-
didate needs more than half the available votes cast.
Assume everyone is a candidate as well as a voter,
he has a unique identity, id for short, selected from
{1, 2,.., N}, we have two methods: one, the candidate
can randomly vote his guy, including himself, then we
check the result, if someone has more than half votes,
he wins, otherwise we enter the next round and do it
again until someone is elected as the leader. Second
method, we do the same thing on the first round, but if
nobody is selected, we change the rule on next round,
we choose only two candidates with the most votes,
of course, if there exist equal votes, select the one
with bigger id. Then, vote again. We cannot make
the election round and round because the expense of
election is high. We should emphasize that a voter
may change his opinion or position such that the re-
sults of each round are different, thus the extent of
successful election varies with states or time. We use
two propositions to describe the selection process of
the two methods:
• p : a leader is elected now.
• r : it needs the next election.
Let M
1
be the model of the first method described as
Figure 2. The meanings of each state are:
• s
10
: the initial sate that no leader is elected and it
does need the next election.
SATISFIABILITY DEGREE THEORY FOR TEMPORAL LOGIC
499

• s
11
: the last election can come to nothing such
that it will need the next election.
• s
12
: The last election can be successful, so it
needs no election any more.
s
10
s
11
s
12








1
0








1
)(k 








0
)(k 
Figure 2: The model of the first method.
Note that α(k) is the satisfiability degree of a suc-
cessful kth election, β(k) for unsuccessful kth elec-
tion, where
α(k) + β(k) = 1 (15)
Because, the second method can change its rules
if the first election fails, so it is different from M
1
.
We use M
2
to model the second method, please see
Figure 3. The meanings of each state are:
• s
20
: the initial state that no leader is elected.
• s
21
: the last election can come to nothing such
that it will need the next election. Once the first
election fails, it will go to state s
21
and change its
rules. Thus, it will go to itself rather than s
20
like
the model M
1
.
• s
22
: the last election can be successful, so it needs
no election any more.
s
20
s
21








1
0








1
)(k 








0
)(k 
s
22
Figure 3: The model of the second method.
Note that λ(k) is the satisfiability degree of a suc-
cessful kth election, µ(k) for unsuccessful kth elec-
tion, where
λ(k) + µ(k) = 1 (16)
Then, we concern the property that a leader is
elected in only two rounds, which is written as:
ϕ = X (p ∧ r = 0) ∨ X X(p ∧ r = 0) (17)
If f
M
1
s
10
(ϕ) < f
M
2
s
20
(ϕ), M
2
is better than M
1
, vice
versa. However, if we use model checker, both M
1
and M
2
may not satisfy the formula, thus satisfiability
degree can better differentiate the two protocols.
5 CONCLUSIONS
Satisfiability degree is a new method to precisely ex-
press the satisfiable extent of a formula. It is an in-
herent attribute of the proposition not depending on
the interpretations. Since the propositional logic is
the basis of temporal logic, satisfiability degree can
be extended to the temporal logic such that the new
temporal model can be constructed by satisfiability
degree. That model is more expressive than the clas-
sical temporal model. On a model, the truth value
of a temporal formula is precisely interpreted as its
satisfiability degree. Based on satisfiability degree, a
deductive system for CTL
∗
is established; some in-
terference rules and axiom schemes are given, and its
soundness is proved. The example, Leader Election,
shows that satisfiability degree can be used to do a
quantitative analysis for uncertain model checking.
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