Sequences of Refinements of Rough Sets: Logical and Algebraic Aspects

In this thesis, a generalization of the classical Rough set theory [83] is developed considering the so-called sequences of orthopairs that we define in [20] as special sequences of rough sets. Mainly, our aim is to introduce some operations between sequences of orthopairs, and to discover how to generate them starting from the operations concerning standard rough sets (defined in [32]). Also, we prove several representation theorems representing the class of finite centered Kleene algebras with the interpolation property [31], and some classes of finite residuated lattices (more precisely, we consider Nelson algebras [87], Nelson lattices [23], IUML-algebras [73] and Kleene lattice with implication [27]) as sequences of orthopairs. Moreover, as an application, we show that a sequence of orthopairs can be used to represent an examiner’s opinion on a number of candidates applying for a job, and we show that opinions of two or more examiners can be combined using operations between sequences of orthopairs in order to get a final decision on each candidate. Finally, we provide the original modal logic SOn with semantics based on sequences of orthopairs, and we employ it to describe the knowledge of an agent that increases over time, as new information is provided. Modal logic SOn is characterized by the sequences (□ 1, …, □ n) and (◯ 1, …, ◯ n) of n modal operators corresponding to a sequence (t1, …, tn) of consecutive times. Furthermore, the operator □ i of (□ 1, …, □ n) represents the knowledge of an agent at time ti, and it coincides with the necessity modal operator of S5 logic [29]. On the other hand, the main innovative aspect of modal logic SOn is the presence of the sequence (◯ 1, …, ◯ n), since ◯ i establishes whether an agent is interested in knowing a given fact at time ti.