Download Truth Value Solver: A Software for Calculating Truth Value with

Truth Value Solver: A Software for Calculating
Truth Value with Uncertain Logic
Yang Zuo
Uncertainty Theory Laboratory, Department of Mathematical Sciences
Tsinghua University, Beijing 100084, China
[email protected]
Abstract
Uncertain logic based on uncertainty theory is a generalization of classical logic for dealing with
uncertain knowledge. One advantage of uncertain logic is that it is consistent with the law of excluded
middle and the law of contraction. The truth value is defined as the uncertain measure that a
proposition is true. Truth value theorem gives a formula for computing the truth value of independent
uncertain propositions. In this paper, a software based on truth value theorem is given, which is used
to computing truth value.
Keywords: Uncertain measure, uncertain variable, uncertain logic, truth value
1
Introduction
Classical logic is used to deal with certain information, but on the other hand multi-valued logic is used to
deal with uncertain information. Multi-valued logic was proposed by Lukasiewicz and extended by many
researchers such as Adams [1], Shafer [9], Zadeh [10], etc. Multi-valued logic has been well developed,
but the interpretation of the truth value is controversial. Although multi-valued logic theory was well
developed and widely applied, there is a fatal defect that it disobeys the law of excluded middle, which is
one of the most basic laws in classical logic. In 1976, Nilsson [8] considered the truth value as a probability
and initiated probability logic. Recently, Li and Liu [3] introduced credibilistic logic to deal with fuzzy
knowledge. Li and Liu [4] proposed uncertain logic, which can be seen as a generalization of all these
multi-valued logics, in which the truth value is defined as the uncertain measure that the proposition
is true. One advantage of uncertain logic is its consistency with classical logic. Chen and Ralescu [2]
proved truth value theorem that provides a numerical method for calculating the truth value of uncertain
formulas.
1
Although truth value theorem was given, it is still a complexity of the calculations to compute the
truth value of uncertain formula. In this paper, a software to calculate truth value is introduced, which
is based on truth value theorem. The rest of this paper is organized as follows: some basic concepts of
uncertainty theory are recalled in Section 2. Several basic definitions of uncertain logic are contained in
Section 3. A software to calculate truth value is introduced and some examples are calculated in Section
4. Finally, ta brief summary is given in Section 5.
2
Preliminaries
Let Γ be a nonempty set, and L a σ-algebra over Γ. Each element Λ ∈ L is called an event.
Definition 1. (Liu[5]) The set function M is called an uncertain measure if it satisfies the following four
axioms:
Axiom 1. (Normality) M{Γ} = 1;
Axiom 2. (Monotonicity) M{Λ1 } ≤ M{Λ2 } whenever Λ1 ⊂ Λ2 ;
Axiom 3. (Self-Duality) M{Λ} + M{Λc } = 1 for any event Λ;
Axiom 4. (Countable Subadditivity) For every countable sequence of events {Λi }, we have
(∞ )
∞
X
[
M{Λi }.
M
Λi ≤
i=1
i=1
.
An uncertain variable is a measurable function from an uncertainty space (Γ, L, M) to the set of real
numbers.
Definition 2. (Liu [6]) The uncertain variables X1 , X2 , · · · , Xm are said to be independent if
(m
)
\
M
Xi ∈ Bi = min M{Xi ∈ Bi }
1≤i≤m
i=1
(1)
for any Borel sets B1 , B2 , · · · , Bm of real numbers.
Theorem 1. (Liu [7], Operational Law) Let ξ1 , ξ2 , · · · , ξn be independent uncertain variables, and f :
<n → < a measurable function. Then ξ = f (ξ1 , ξ2 , · · · , ξn ) is an uncertain variable such that







M{ξ ∈ B} =






sup
min Mk {Bk },
f (B1 ,B2 ,··· ,Bn )⊂B 1≤k≤n
1−
sup
if
min Mk {Bk }, if
f (B1 ,B2 ,··· ,Bn )⊂B c 1≤k≤n
0.5,
sup
sup
min Mk {Bk } > 0.5
f (B1 ,B2 ,··· ,Bn )⊂B c 1≤k≤n
otherwise
2
min Mk {Bk } > 0.5
f (B1 ,B2 ,··· ,Bn )⊂B 1≤k≤n
3
Uncertain Logic
Uncertain logic was designed by Li and Liu [4] as a generalization of classical logic. Li and Liu gave the
definition of uncertain proposition which is a statement with truth value in [0, 1]. In fact, the uncertain
proposition X is essentially an uncertain variable taking values 0 or 1, where X = 1 means X is true and
X = 0 means X is false. Uncertain propositions are called independent if they are independent uncertain
variables.
An uncertain formula is a finite sequence of uncertain propositions and connective symbols that are
well defined. If ξ, η, τ are uncertain propositions, then X = ¬ξ,X = ξ∧η,X = (ξ∨η) → τ are all uncertain
formulas. Uncertain formulas are called independent if they are independent uncertain variables.
Truth value is a key concept in uncertain logic and is defined as the uncertain measure that the
uncertain formula is true.
Definition 3. (Li and Liu [4]) Let X be an uncertain formula. Then the truth value of X is defined as
the uncertain measure that the uncertain formula X is true, i.e.,
T (X) = M{X = 1}.
Theorem 2. (Truth Value Theorem [2]) Let X be an uncertain formula containing independent uncertain
propositions ξ1 , ξ2 , · · · , ξn whose truth function is f . Then the truth value of X is



sup
min vi (xi ),
if
sup
min vi (xi ) < 0.5

f (x1 ,x2 ,··· ,xn )=1 1≤i≤n
f (x1 ,x2 ,··· ,xn )=1 1≤i≤n
T (X) =


sup
min vi (xi ), if
sup
min vi (xi ) ≥ 0.5
 1−
f (x1 ,x2 ,··· ,xn )=0 1≤i≤n
f (x1 ,x2 ,··· ,xn )=1 1≤i≤n
where xi take values either 0 or 1, and vi are uncertainty mass functions of ξi with
vi (1) = M{ξi = 1}, vi (0) = M{ξi = 0}
for i = 1, 2, · · · , n, respectively.
4
Truth Value Solver
Truth Value Solver is a free software for computing the truth values of uncertain formula based on truth
functions. This software and its source code may be downloaded from http://orsc.edu.cn/liu/resources.htm.
This software can compute truth values of uncertain formula when they are composed of no more than
eight propositions. The usage of this software is very simple. All you have to do is inputting truth values
of propositions and the truth function of the uncertain formula. Then it shows you the truth value of the
formula in the result window. Now let us perform it via some numerical examples.
3
Example 1. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1,
0.3, 0.5, 0.7, 0.9, respectively. Let
X = only 1 propositions of ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are true.
(2)
It is clear that the truth function is

 1, if x1 + x2 + x3 + x4 + x5 = 1
f (x1 , x2 , x3 , x4 , x5 ) =
 0, if x + x + x + x + x 6= 1.
1
2
3
4
5
A run of the truth value solver shows that T (X) = 0.3.
Example 2. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1,
0.3, 0.5, 0.7, 0.9, respectively. Let
X = only 3 propositions of ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are true.
(3)
It is clear that the truth function is

 1, if x1 + x2 + x3 + x4 + x5 = 3
f (x1 , x2 , x3 , x4 , x5 ) =
 0, if x + x + x + x + x 6= 3.
1
2
3
4
5
A run of the truth value solver shows that T (X) = 0.5.
Example 3. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1,
0.3, 0.5, 0.7, 0.9, respectively. Let
X = 5 of propositions of ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are true.
(4)
It is clear that the truth function is

 1, if x1 + x2 + x3 + x4 + x5 = 5
f (x1 , x2 , x3 , x4 , x5 ) =
 0, if x + x + x + x + x 6= 5.
1
2
3
4
5
A run of the truth value solver shows that T (X) = 0.1.
Example 4. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1,
0.3, 0.5, 0.7, 0.9, respectively. Let
X = odd number of propositions of ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are true.
It is clear that the truth function is

 1, if x1 + x2 + x3 + x4 + x5 ∈ {1, 3, 5}
f (x1 , x2 , x3 , x4 , x5 ) =
 0, if x + x + x + x + x ∈ {0, 2, 4}.
1
2
3
4
5
A run of the truth value solver shows that T (X) = 0.5.
4
(5)
Example 5. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1,
0.3, 0.5, 0.7, 0.9, respectively. Let
X = even number of propositions of ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are true.
(6)
It is clear that the truth function is

 1, if x1 + x2 + x3 + x4 + x5 ∈ {0, 2, 4}
f (x1 , x2 , x3 , x4 , x5 ) =
 0, if x + x + x + x + x ∈ {1, 3, 5}.
1
2
3
4
5
A run of the truth value solver shows that T (X) = 0.5.
Example 6. Assume that ξ1 , ξ2 , ξ3 , ξ4 , ξ5 are independent uncertain propositions with truth values 0.1,
0.3, 0.5, 0.7, 0.9, respectively. Let
X = (ξ1 ∧ ξ2 ) ∨ (ξ2 ∧ ξ3 ) ∨ (ξ3 ∧ ξ4 ) ∨ (ξ4 ∧ ξ5 ).
(7)
It is clear that the truth function is



1, if x1 + x2 = 2






1, if x2 + x3 = 2



f (x1 , x2 , x3 , x4 , x5 ) =
1, if x3 + x4 = 2





1, if x4 + x5 = 2





 0, otherwise.
A run of the truth value solver shows that T (X) = 0.7.
5
Conclusion
Uncertain logic was defined within the framework of uncertainty theory as a generalization of classical
logic. One advantage of uncertain logic is its consistency with classical logic. We gave a software to deal
with the computing of truth value of uncertain formula. Finally, some examples were given.
Acknowledgments
This work was supported by National Natural Science Foundation of China Grant No.60874067.
References
[1] Adams, E., and Levine, H., On the uncertainties transmitted from premises to conclusions in deductive
inferences, Synthese, vol.30, 429-460, 1975.
5
[2] Chen XW, and Ralescu DA, A note on truth value in uncertain logic,Proceedings of the Eighth International
Conference on Information and Management Sciences, Kunming, China, July 20-28, 2009, pp.739-741.
[3] Li, X., Liu, B., Foundation of credibilistic logic, Fuzzy Optimization and Decision Making, vol.8, no.1, 91-102,
2009.
[4] Li, X., Liu, B., Hybrid logic and uncertain logic, Journal of Uncertain Systems, vol.3, no.2, 83-94, 2009.
[5] Liu, B., Uncertainty Theory, 2nd ed., Springer-Verlag, Berlin, 2007.
[6] Liu, B., Uncertainty Theory, 3nd ed., http://orsc.edu.cn/liu/ut.pdf.
[7] Liu, B., Some research problems in uncertainty theory, Journal of Uncertain Systems, vol.3, no.1, 3-10, 2009.
[8] Nilsson, N., Probability logic, Artificial Intelligence, vol.28, 71-78, 1986.
[9] Shafer, G., Mathematical Theory of Evidence, Princeton, New Jersey, 1979.
[10] Zadeh, L., Fuzzy logic and approximate reasoning, Synthese, vol.30, 407-428, 1975.
6