Uncertainty measures in rough algebra with applications to rough logic
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ORIGINAL ARTICLE
Uncertainty measures in rough algebra with applications to rough logic Yanhong She • Xiaoli He
Received: 3 March 2013 / Accepted: 5 October 2013 Ó Springer-Verlag Berlin Heidelberg 2013
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 f0; 12 ; 1g to present the graded version of rough truth value for elements in prerough 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. Keywords Rough algebra Rough truth degree Rough similarity degree Accuracy degree Roughness degree Rough inclusion degree Abbreviations Acc(a) Accuracy degree of a Acc0 (a) An equivalent form of Acc(a), shown in Definition 7 a A mapping induced by the element a in prerough algebra, which is defined by 8v 2 X; aðvÞ ¼ vðaÞ B Boolean algebra D(a, b) Inclusion degree between a and b in Xu et al. [30] (E, B) Partially ordered set
Inc(a, b) Incða; bÞ Incða; bÞ L M P P PRL R R3 f0; 12 ; 1g RðXÞ Rou(a) Rou0 (a) U v [x] X l sðaÞ s(a) sðaÞ n(a, b) bÞ nða; nða; bÞ
Rough inclusion degree between a and b Rough upper inclusion degree between a and b Rough lower inclusion degree between a and b An operator on pre-rough algebra, 8a 2 P; La is understood as rough lower approximation of a An operator on pre-rough algebra, 8a 2 P; Ma is understood as rough upper approximation of a Pre-rough algebra The underlying lattice of P Pre-rough logic Equivalence relation on U The standard pre-rough algebra RðXÞRough upper approximation of X Rough lower approximation of X Roughness degree of a An equivalent form of Rou(a), shown in Definition 7 Universe of discourse A valuation on P Equivalence class containing x The set of all valuations on pre-rough algebra P Probability measure on X Rough upper truth degree of a Rough truth degree of a Rough lower truth degree of a Rough similarity degree between a and b Rough upper similarity degree between a and b Rough lower similarity degree between a and b
1 Introduction Y. She (&) X. He College of Science, Xi’an Shiyou University, Xi’an, China e-mail: [email protected]
Rough set theory, originated by Pawlak [22], is a mathematical tool in dealing with uncertain and imprecise
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Int. J. Mach. Learn. & Cyber.
information. By employing the concepts of upper and lower approximations in rough set theory, knowledge hidden in information tables may be unraveled and expressed in the forms of decision rules [25, 32]. Since its inception, algebraic aspects have emerged as one of the important issues of rough set t
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