Table 2: The Run Time of each algorithm.
formula size EA XA BA MA A2
172× 164 > 24 h > 24 h > 24 h 0.1 s
a
0.04 s
169× 169
b
> 48 h
c
> 48 h > 48 h 71.8 s 1.2 s
2437× 852 > 48 h > 48 h > 48 h 11412 s 1080.4 s
2281× 1763 > 48 h > 48 h > 48 h 12000 s 932.4 s
a
h means hours and s for seconds.
b
The term ”169× 169” means 169 clauses and 169 atoms.
c
The term ”> 48 h” means that it takes at least 48 hours to solve the
problem, and so on.
in time less than 2
n
, it is an unsuitable approach; be-
cause sometimes those optimizations cannot reduce
computation times, not to mention its space explo-
sion problem. The backtracking depth will increase
for lager formulas, so it slows the backtracking algo-
rithm down. The backtracking search algorithm ver-
sion is marked as BA. The propositional matrix algo-
rithm, marked as MA, is more effective than the three
algorithm just mentioned before, since it analyzes the
structure of the formula to divide large formulas into
smaller ones such that their satisfiability degrees can
be easily obtained; but it needs more space and its rum
time is higher than Algorithm 2, because the number
of smaller formulas divided by Algorithm 2 is less.
Besides, Algorithm 2 needs less space than other four
alogrithms.
In all, the experimental results verify and support
the theoretical analysis in section 4.
6 CONCLUSIONS
Satisfiability degree is a new method to precisely in-
terpret the truth value of propositional logic. An algo-
rithm to computed satisfiability degree was proposed.
It divides a large formula into two smaller formu-
las; and they can be further simplified by unit clauses
such that their satisfiability degrees can be easily ob-
tained once they contain only a clause or only unit
clauses. The satisfiability degree of the large formula
equals the difference of the two formulas. The cor-
rectness of the proposed algorithm was proved as well
as α
min(m,n)
time complexity, where α is greater than 1
but smaller than 2. Thus, Algorithm 2 has a less time
complexity than the enumeration algorithm, XOBBD
algorithm and the search algorithms. In addition, Al-
gorithm use a set to represent a CNF formula such
thats pace consumed is greatly reduced. Because it
only need to store the atoms in the considered formula
not the interpretations. Experimental results further
demonstrates that conclusion.
Further work can apply satisfiability degree to pre-
diction, model checking and reasoning. We also in-
tend to use satisfiability degree to analyze the satis-
fiable extent of predicate logic, higher order logic as
well as temporal logic. By experimental results, sat-
isfiability degree can be used for circuit test.
REFERENCES
Gerla, G. (1994). Inferences in probability logic. In Artifi-
cial Intelligence. vol. 70, no. 1-2, 1994, pp. 33-52.
Huth, M. and Ryan, M. (2005). Logic in Computer Science.
China Machine Press, Beijing, 2nd edition.
Luo, J. and Luo, G. M. (2010). Propositional matrix search
algorithm for satisfiability degree computation. In the
9th IEEE International Conference on Cognitive In-
formatics. Beijing, China, pp.974-977.
Luo, G. M., Y. C. Y. and Hu, P. (2009). An algorithm for
calculating the satisfiability degee. In proceedings of
the 2009 sixth international Conference on Fuzzy Sys-
tem and Knowledge Discovery. Tianjin, China, 2009,
pp.322-326.
Malik, S., M. Y. and Fu, Z. (2005). Zchaff2004: An efficient
sat solver. In in LNCS 3542, Theory and Applications
of Satisfiability Testing. pp. 360C375.
McMillan, K. L. (2002). Applying sat methods in un-
bounded symbolic model checking. In in Proc. Int.
Conf. Computer-Aided Verification. vol.2404.
Yin, C. Y., L. G. M. and Hu, P. (2009). Backtracking
search algorithm for satisfiability degree calculation.
In proceedings of the 2009 sixth international Con-
ference on Fuzzy System and Knowledge Discovery.
Tianjin,China,2009,pp.3-7.
Zadeh, L. A. (1965). Fuzzy sets. In Information & Control.
vol. 8, no. 3, 1965, pp. 338-353.
Zhou, H. J. and Wang, G. J. (2006). A new theory
consistency index based on deduction theorems in
several logic systems. In Fuzzy Sets and Systems.
vol.157, no.3, pp.427-443.
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