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Generalized Rough Sets via Quantum Implications on Quantum Logic
Generalized Rough Sets via Quantum Implications on Quantum Logic
axioms Article Generalized Rough Sets via Quantum Implications on Quantum Logic Songsong Dai School of Electronics and Information Engineering, Taizhou University, Taizhou 318000, China; firstname.lastname@example.org Abstract: This paper introduces some new concepts of rough approximations via ﬁve quantum implications satisfying Birkhoff–von Neumann condition. We ﬁrst establish rough approximations via Sasaki implication and show the equivalence between distributivity of multiplication over join and some properties of rough approximations. We further establish rough approximations via other four quantum implication and examine their properties. Keywords: rough set; quantum logic; orthomodular lattice; quantum implication; Sasaki implication MSC: 03G12; 06D72 1. Introduction In 1982, aiming to give a mathematical tool for incomplete information processing, Pawlak  introduced the theory of rough sets. This theory has been widely used in many ﬁelds. The key of rough set is a pair of operators called lower and upper approximation operators. Many scholars have generalized the notion of rough approximation operators in different way. One way is to deﬁne these operators in different mathematical structures, such as modal logics [2–4], topological structures [5,6], Boolean algebra [7,8], lattice effect Citation: Dai, S. Generalized Rough Sets via Quantum Implications on algebra , residuated lattices [10–15], and others [16–18]. Quantum Logic. Axioms 2022, 11, 2. Quantum computers were ﬁrst introduced by Feynman [19,20] and formalized by https://doi.org/10.3390/axioms Deutsch . Shor  gave a polynomial-time quantum algorithm for factoring integers in 1994 and Grover  introduced a quantum algorithm for unstructured searching in 1996. Their works greatly stimulated the research of quantum computation. With the Academic Editors: Alexander Šostak, advent of quantum computation, it is natural to ask the question: how to use the rough Michal Holcapek, Antonin Dvorak sets method in quantum computation and vice versa. Our method is from a logical point and Amit K. Shukla of view. Since quantum computation is a beautiful combination of quantum theory and Received: 6 November 2021 computer science. As early as in 1936, in order to give a logic of quantum mechanics, Accepted: 21 December 2021 Birkhoff and von Neumann  introduced quantum logic, whose algebraic model is Published: 22 December 2021 an orthomodular lattice. Then, the issue is how to apply quantum logic in the analysis Publisher’s Note: MDPI stays neutral and design of rough sets. In the recent years, some scholars studied rough sets based on with regard to jurisdictional claims in quantum logic. In 2017, Hassan  showed that rough set model with quantum logic published maps and institutional afﬁl- can be used for recognition and classiﬁcation systems. In our previous work [26,27], we iations. proposed a rough set model based on quantum logic. We deﬁned rough approximation operators via join and meet on a complete orthomodular lattice (COL). Some properties in our previous work are based on the distributivity of meet over join. However, any orthomodular lattice satisfying distributivity of meet over join reduces to a Boolean algebra. Copyright: © 2021 by the author. Moreover, some straightforward equivalences between distributivity and properties are Licensee MDPI, Basel, Switzerland. proved. This means that these properties of rough sets theory hold if, and only if, the This article is an open access article orthomodular lattice is a Boolean algebra. So these properties of rough sets theory hold in distributed under the terms and the frame of classical logic and may not hold in the frame of quantum logic. It is necessary to conditions of the Creative Commons consider other rough sets model based on quantum logic, making more properties of rough Attribution (CC BY) license (https:// sets hold in the frame of quantum logic. Quantum implication operators are important in creativecommons.org/licenses/by/ the study of quantum logic. For example, they can be used to deﬁne deduction rules in 4.0/). Axioms 2022, 11, 2. https://doi.org/10.3390/axioms11010002 https://www.mdpi.com/journal/axioms Axioms 2022, 11, 2 2 of 15 quantum reasoning. This paper, therefore, discusses the quantum rough approximation operators based on quantum implications. The paper is organized as follows: Section 2, we recall the concepts of orthomodular lattices. Since there are ﬁve quantum implications satisfying Birkhoff–von Neumann condition. Section 3, we redeﬁne the rough approximation operators via the multiplication and Sasaki implication. In Section 4, we introduce the rough approximation operators via other four quantum implications. The conclusion is given in the ﬁnal section. 2. Preliminaries 2.1. Quantum Implicator First, we recall the concept of COL and its implicators [28–34]. A COL L =< L,,^,_,?, 0, 1 > is a complete bounded lattice with a unary operator ? which has the following properties: for all u, v 2 L ? ? (C1) u _ u = 1, u ^ u = 0; ?? (C2) u = u; ? ? (C3) u v ) v u ; (C4) u v ) u^ (u _ v) = v. where 0 and 1 are the least and greatest elements of L, is the partial ordering in L, u^ v and u_ v stand for the greatest lower bound and the least upper bound of u and v. Quantum logic is a COL-valued logic and classical logic is treated as a Boolean algebra. The former is weaker than the latter. For example, the distributivity of meet over join holds in Boolean algebra, i.e., for all u, v, w 2 L, (v^ u)_ (w^ u) = (v_ w)^ u. (1) However, it is not valid in a COL. Implication operators in quantum logic can be deﬁned in terms of ?, _, and ^. They are required to satisfy the Birkhoff–von Neumann condition : for any u, v 2 L, u ! v = 1 if, and only if, u v. There are only ﬁve implication operators satisfying this condition [35,36]: Sasaki implication: u ! v = u _ (u^ v) (2) Dishkant implication: ? ? u ! v = v_ (u ^ v ) (3) Kalmbach implication: ? ? ? u ! v = (u ^ v)_ (u^ v)_ (u ^ v ) (4) Non-tollens implication: ? ? ? u ! v = (u ^ v)_ (u^ v)_ ((u _ v)^ v ) (5) Relevance implication: ? ? ? ? u ! v = (u ^ v)_ (u ^ v )_ (u^ (u _ v)). (6) Moreover, the multiplication operator is deﬁned as follows: for all u, v 2 L, u&v = (u_ v )^ v. (7) de f ? ? ? ? Remark 1. For any u, v 2 L, u ! v = v ! u , u ! v = v ! u . 2 1 4 3 If L is a Boolean algebra, u ! v (i = 1, 2, ..., 5) is equivalent to u ! v = u _ v Remark 2. i 0 which is named “material implication”. Axioms 2022, 11, 2 3 of 15 Remark 3. For any orthomodular lattice, among ! (i = 1, 2, 3, 4, 5), Sasaki implication ! is i 1 unique one satisfying the following condition [37,38]: there exists binary operation and such that for any u, v, w 2 L, u&v w if, and only if, u v ! w. The following are some properties of the Sasaki implication and the multiplication: (C5) u&v w iff u v ! w; (C6) 0&u = 0, u&1 = u, 1 ! u = u and u ! 1 = 1; (C7) u^ v u&v; (C8) u v if, and only if, u ! v = 1 Let l =< L,,^,_,?, 0, 1 > be a COL, then L is a Boolean algebra, if, and only if, any one of the following condition holds: (C9) & is commutative, i.e., u&v = v&u for any u, v 2 L; (C10) v w ) u&v u&w for any u, v, w 2 L. 2.2. Dual Operator of Quantum Implicator Based on?, a dual operator ,! of quantum implicator! is deﬁned as follow: for all i i u, v 2 L, ? ? u ,! v = u ! v . (8) i i Proposition 1. All ﬁve operator J (i=1,2,...,5) satisfy the condition: for any u, v 2 L, v u if, and only if, u ,! v = 0. Proof. It can be deduced from the following, for any u, v 2 L, ? ? v u , u v ? ? , u ! v = 1 ? ? ? , (u ! v ) = 0 , u ,! v = 0. Proposition 2. For any u, v 2 L, v ,! u = u&v Proof. It can be deduced from the following, ? ? ? v ,! u = (v ! u ) 1 1 ? ? ? = (v _ (v^ u )) = v^ (v _ u) = u&v. The bi-implication operator corresponding to the Sasaki implication is deﬁned as follows: for any u, v 2 L, u $ v = (u ! v)^ (v ! u). (9) de f Clearly, u = v if, and only if, u $ v = 1. Let X be a ﬁnite set, L a COL, E a binary relation on X relative to L. Then, E is serial if for all u 2 X, _ E(u, v) = 1. v2X E is reﬂexive if E(u, u) = 1 holds for all u 2 X. E is symmetric if E(u, v) = E(v, u) holds for all u, v 2 X. E is &