https://orcid.org/0000-0002-7517-0923
Rough Sets were introduced by Z. Pawlak in the year 1982 with the intention to address knowledge representation and data processing from the angle of computation and decision making. The main idea behind rough sets is to approximate the extension of a concept or a relation by two subsets of the universe of discourse, namely, lower and upper approximations. Various types of algebras on the power set of the universe with lower/upper approximation operators were soon proposed, and algebraic studies have since formed an important branch of research for rough set theory.
In parallel, over time, logics inspired by rough sets have evolved in a variety of ways. Such logics may be divided broadly into two groups: algebraic and modal. In the first, logics are rendered with algebraic semantics where the algebras are abstractions from rough set models. Among these algebras, a notable one that has seen a lot of investigation is the topological quasi-Boolean algebra and related algebras. This direction of research includes a study of representation theorems for the abstract algebraic structures in terms of rough set structures and their subsequent use in formulating a rough set semantics for the corresponding logics.
In the second group, various modal logic systems were developed depending on the properties of the lower/upper approximation operators. Several new types of modal logic systems, including non-standard ones, have been proposed to reflect rough set models. Besides traditional modal logics that were equipped with rough set-based semantics, modal and multimodal systems, generated from generalisations of rough set models, have been developed. An interplay between modal systems and rough set theory has thus become a significant area of research.
Yet, other types of development took place that include generalising the consequence relation to the rough consequence relation based on rough modus ponens rules. Another research direction was based on studying many-valued logics, where the boundary region is represented by non-classical truth values unknown and/or inconsistent.
To exchange ideas related to the aforementioned areas, a workshop on Logics from Rough Sets was organised by M.K. Chakraborty as a part of UNILOG 2022: the 7th World Congress and School on Universal Logic.Footnote1 The current special issue mainly contains ideas in the papers that were accepted for presentation in the workshop. The papers submitted to the special issue were peer reviewed by two referees each. The eight you find in the issue are the result of a selection based on their recommendations, giving a snapshot of the current state of the art in the intersection of logic and rough sets:
W. Conradie, K. Manoorkar, A. Palmigiano and M. Panettiere: Modal reduction principles: a parametric shift to graphs, which addresses graph-based frames motivated by information deficiencies resulting from perceptual, evidential or linguistic limits. It investigates graph-based frames as a framework generalising rough set theory and shows connections between the first-order correspondents of Sahlqvist modal reduction principles on Kripke frames, and on graph-based and polarity-based frames.
Md. A. Khan: A Study of Modal Logic With Semantics Based on Rough Set Theory, which presents a survey of modal logic focusing on semantics reflecting properties of structures generalising rough set theory. In particular, the paper addresses multi-granulation rough set model, covering rough set model and boundary operators. Multi-modal languages, languages with modal operators indexed with numbers and sets of parameters, are also considered.
A. Kumar and M. Banerjee: Some Algebras and Logics from Quasiorder-generated Covering-based Approximation Spaces, which characterises the quasiorder-generated covering-based approximation spaces whose corresponding collections of definable and rough sets form Stone algebras. Rough Stone algebras and other rough lattices are introduced, and representations of these algebras in terms of rough sets are obtained. Logics for these algebras are formulated and shown to be sound and complete with respect to rough set semantics.
Z. Lin, Y. Wang and M. Ma: Decidability of Topological Quasi-Boolean Algebras, where sequent calculus for topological quasi-Boolean algebras are investigated, and its decidability is shown using the finite model property. Non-distributive variants of topological quasi-Boolean algebras are also considered, related sequent calculi are developed and the decidability is demonstrated.
A. Mani: Granular Knowledge and Rational Approximation in General Rough Sets-I, where rough sets are associated with rationality. Granular generalisations of graded and variable precision rough sets are surveyed and investigated from the perspective of rationality of approximations. The approach is also extended to ideal-based approximations together with co-granular or point-wise approximations.
P. Pagliani: Crypto-preorders, topological relations, information and logic, which addresses topological properties related to rough set models. In particular, using the fact that any preorder on a set induces an Alexandrov topology on the set, it is shown that an Alexandrov topology can be transformed into different types of logic-algebraic models. The paper studies the conditions under which a relation is a crypto-preorder and how to transform it into a suitable preorder.
U. Rivieccio, J. Järvinen and S. Radeleczki: Nelson Algebras, Residuated Lattices and Rough Sets: A Survey, where a comprehensive survey of work related to Nelson algebras, the algebraic counterpart of Nelson's constructive logic with strong negation, is provided. Since each Nelson algebra defined on an algebraic lattice is isomorphic to the rough set algebra based on a quasiorder, a strong link between Nelson algebras and rough sets is also emphasised.
A. Saha and J. Sen: Modality free pre-rough logic, where an alternative but equivalent definition of a pre-rough algebra is proposed, without a unary operator corresponding to modality. Modality-free versions for some algebras weaker than pre-rough algebra are also obtained, while for some other structures weaker than pre-rough algebra it is shown that modality-free versions do not exist. Hilbert-type axiomatisation as well as sequent calculi of all proposed algebras are provided.
We would like to thank all the authors who submitted contributions, the referees for this special issue as well as Andreas Herzig, the JANCL Editor in Chief, for his great help.