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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.)