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Uncertainty measures in rough algebra with applications to rough logic

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Abstract

The present paper is devoted to the measurement of uncertainty in rough algebra. Specifically, we employ the probability measure on the set of homomorphism of pre-rough algebra into \({\{0,\frac{1}{2},1\}}\) to present the graded version of rough truth value for elements in pre-rough algebra, which leads to the definition of rough (upper, lower) truth degree. These notions are subsequently used to introduce some other types of uncertainty measures including roughness degree, accuracy degree, rough inclusion degree, etc. A comparative study is conducted between these proposed uncertainty measures and the existing notions in rough logic and it is shown that the obtained results in She et al. (Fundam Inform 107:1–17, 2011) can be regarded as a special case of the present paper.

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Abbreviations

Acc(a):

Accuracy degree of a

Acc′(a):

An equivalent form of Acc(a), shown in Definition 7

\(\bar{a}\) :

A mapping induced by the element a in pre-rough algebra, which is defined by \(\forall v\in\Upomega, \bar{a}(v)=v(a)\)

B :

Boolean algebra

D(a, b):

Inclusion degree between a and b in Xu et al. [30]

(E,  ≤):

Partially ordered set

Inc(a, b):

Rough inclusion degree between a and b

\(\overline{Inc}(a,b)\) :

Rough upper inclusion degree between a and b

\(\underline{Inc}(a,b)\) :

Rough lower inclusion degree between a and b

L :

An operator on pre-rough algebra, \(\forall a\in P,\, La\) is understood as rough lower approximation of a

M :

An operator on pre-rough algebra, \(\forall a\in P, \,Ma\) is understood as rough upper approximation of a

\(\mathcal{P}\) :

Pre-rough algebra

P :

The underlying lattice of \(\mathcal{P}\)

PRL :

Pre-rough logic

R :

Equivalence relation on U

R 3 :

The standard pre-rough algebra

\({\{0,\frac{1}{2},1\}}\) :

\(\bar{R}(X)\)Rough upper approximation of X

\(\underline{R}(X)\) :

Rough lower approximation of X

Rou(a):

Roughness degree of a

Rou′(a):

An equivalent form of Rou(a), shown in Definition 7

U :

Universe of discourse

v :

A valuation on \(\mathcal{P}\)

[x]:

Equivalence class containing x

\(\Upomega\) :

The set of all valuations on pre-rough algebra \(\mathcal{P}\)

μ :

Probability measure on \(\Upomega\)

\(\bar{\tau}(a)\) :

Rough upper truth degree of a

τ(a):

Rough truth degree of a

\(\underline{\tau}(a)\) :

Rough lower truth degree of a

ξ(a, b):

Rough similarity degree between a and b

\(\bar{\xi}(a,b)\) :

Rough upper similarity degree between a and b

\(\underline{\xi}(a,b)\) :

Rough lower similarity degree between a and b

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Acknowledgments

This work was supported by the National Nature Science Fund of China under Grant 61103133, The Innovation Foundation of Science and Technology for Young Scholars, Xi’an Shiyou University (No. 2012QN011).

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Authors and Affiliations

  1. College of Science, Xi’an Shiyou University, Xi’an, China

    Yanhong She & Xiaoli He

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  1. Yanhong She
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  2. Xiaoli He
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Correspondence to Yanhong She.

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She, Y., He, X. Uncertainty measures in rough algebra with applications to rough logic. Int. J. Mach. Learn. & Cyber. 5, 671–681 (2014). https://doi.org/10.1007/s13042-013-0206-0

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