A temporal semantics for Nilpotent Minimum logic

In [Ban97] a connection among rough sets (in particular, pre-rough algebras) and three-valued {\L}ukasiewicz logic {\L}3 is pointed out. In this paper we present a temporal like semantics for Nilpotent Minimum logic NM ([Fod95, EG01]), in which the logic of every instant is given by {\L}3: a completeness theorem will be shown. This is the prosecution of the work initiated in [AGM08] and [ABM09], in which the authors construct a temporal semantics for the many-valued logics of G\"odel ([G\"od32], [Dum59]) and Basic Logic ([H\'aj98]).Comment: 19 pages, 2 table

Representation of Nelson Algebras by Rough Sets Determined by Quasiorders

In this paper, we show that every quasiorder $R$ induces a Nelson algebra $\mathbb{RS}$ such that the underlying rough set lattice $RS$ is algebraic. We note that $\mathbb{RS}$ is a three-valued {\L}ukasiewicz algebra if and only if $R$ is an equivalence. Our main result says that if $\mathbb{A}$ is a Nelson algebra defined on an algebraic lattice, then there exists a set $U$ and a quasiorder $R$ on $U$ such that $\mathbb{A} \cong \mathbb{RS}$.Comment: 16 page

Matroidal structure of generalized rough sets based on symmetric and transitive relations

Rough sets are efficient for data pre-process in data mining. Lower and upper approximations are two core concepts of rough sets. This paper studies generalized rough sets based on symmetric and transitive relations from the operator-oriented view by matroidal approaches. We firstly construct a matroidal structure of generalized rough sets based on symmetric and transitive relations, and provide an approach to study the matroid induced by a symmetric and transitive relation. Secondly, this paper establishes a close relationship between matroids and generalized rough sets. Approximation quality and roughness of generalized rough sets can be computed by the circuit of matroid theory. At last, a symmetric and transitive relation can be constructed by a matroid with some special properties.Comment: 5 page

Rough matroids based on coverings

The introduction of covering-based rough sets has made a substantial contribution to the classical rough sets. However, many vital problems in rough sets, including attribution reduction, are NP-hard and therefore the algorithms for solving them are usually greedy. Matroid, as a generalization of linear independence in vector spaces, it has a variety of applications in many fields such as algorithm design and combinatorial optimization. An excellent introduction to the topic of rough matroids is due to Zhu and Wang. On the basis of their work, we study the rough matroids based on coverings in this paper. First, we investigate some properties of the definable sets with respect to a covering. Specifically, it is interesting that the set of all definable sets with respect to a covering, equipped with the binary relation of inclusion $\subseteq$, constructs a lattice. Second, we propose the rough matroids based on coverings, which are a generalization of the rough matroids based on relations. Finally, some properties of rough matroids based on coverings are explored. Moreover, an equivalent formulation of rough matroids based on coverings is presented. These interesting and important results exhibit many potential connections between rough sets and matroids.Comment: 15page

Black Box White Arrow

The present paper proposes a new and systematic approach to the so-called black box group methods in computational group theory. Instead of a single black box, we consider categories of black boxes and their morphisms. This makes new classes of black box problems accessible. For example, we can enrich black box groups by actions of outer automorphisms. As an example of application of this technique, we construct Frobenius maps on black box groups of untwisted Lie type in odd characteristic (Section 6) and inverse-transpose automorphisms on black box groups encrypting ${\rm (P)SL}_n(\mathbb{F}_q)$. One of the advantages of our approach is that it allows us to work in black box groups over finite fields of big characteristic. Another advantage is explanatory power of our methods; as an example, we explain Kantor's and Kassabov's construction of an involution in black box groups encrypting ${\rm SL}_2(2^n)$. Due to the nature of our work we also have to discuss a few methodological issues of the black box group theory. The paper is further development of our text "Fifty shades of black" [arXiv:1308.2487], and repeats parts of it, but under a weaker axioms for black box groups.Comment: arXiv admin note: substantial text overlap with arXiv:1308.248