1,720,980 research outputs found
Sequences of Refinements of Rough Sets: Logical and Algebraic Aspects
No full textIn 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
How to merge opinions by using operations between sequences of orthopairs
No full textOrthopairs are, analogously to rough sets, a mathematical tool for dealing with uncertainty. Sequences of orthopairs also take into account the possibility to deal with a refinement process of information and with missing information. After recalling the main definitions, we present different operations of conjunction between sequences of orthopairs. We further present an example in which, having available non complete information about applicants for a job, two examiners evaluate them in order to find the better candidates and then their opinions are merged into an individual result
Invertible substitutions in logics with algebraic semantics equivalent to Product algebras
No full textProduct logic is considered one of the major truth-functional fuzzy propositional logics. Its semantics is given by the variety of Product algebras {mathbb{P}}. In the hierarchy of fuzzy logics based on left-continuous t-norms there are a few logics whose algebraic semantics are varieties categorically equivalent with {mathbb{P}}. For these logics we shall describe finitely generated free algebras and their group of automorphisms, that is, invertible substitutions
Averaging the Truth Value of Formulas in Gödel Logic
No full textLet L be a propositional mathematical fuzzy logic with the real unit interval [0, 1] as its set of truth values. Assume that L has an algebraic semantics given by a variety V of algebras. A finitely additive probability measure (or, state) over the free n-generated V-algebra provides an average value over all assignments of a formula with n many variables in L only if this measure is invariant with respect to automorphisms of the free algebra. In this paper we characterise the states that are invariant with respect to automorphisms of the free n-generated Gödel algebra
Tropicalization through the lens of Łukasiewicz logic, with a topos theoretic perspective
No full text
- …
