Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2504.04218

A Categorical Foundation of Rough Sets

Yoshihiko Kakutani
View original source ↗Read source document ↗Export metadata ↓

Abstract

Rough sets are approximations of concrete sets. The theory of rough sets has been used widely for data-mining. While it is well-known that adjunctions are underlying in rough approximations, such adjunctions are not enough for characterization of rough sets. This paper provides a way to characterize rough sets in terms of category theory. We reformulate rough sets as adjunctions between preordered sets in a general way. Our formulation of rough sets can enjoy benefits of adjunctions and category theory. Especially, our characterization is closed under composition. We can also explain the notions of attribute reduction and data insertion in our theory. It is novel that our theory enables us to guess decision rules for unknown data. If we change the answer set, we can get a refinement of rough sets without any difficulty. Our refined rough sets lead rough fuzzy sets or more general approximations of functions. Moreover, our theory of rough sets can be generalized in the manner of enriched category theory. The derived enriched theory covers the usual theory of fuzzy rough sets.

Explore related subjects

cs.LO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yoshihiko Kakutani. 2025-04-05. A Categorical Foundation of Rough Sets. https://arxiv.org/abs/2504.04218

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

PICID: Proof-Driven Clause Learning in Neural Network Verification

Current Deep Neural Network (DNN) verifiers are typically designed to prioritize scalability over reliability. Reliability can be reinforced through the generation of proofs that are checkable by trusted, external proof checkers. To date, only a handful of verifiers support proof production; and these rely on verifier-specific formats, and balance between scalability, proof detail, and the trustworthiness of their proof checker. In this tool paper, we introduce PICID, a DNN verifier that produces proofs in the standard Alethe format for SMT solving, checkable by an independent checker. PICID implements a parallel CDCL(T) architecture that integrates the state-of-the-art, proof-producing CaDiCaL SAT solver with the Marabou DNN verifier. Furthermore, PICID leverages UNSAT proofs to derive conflict clauses. Our evaluation shows that PICID generates valid proofs in the vast majority of cases and significantly outperforms existing tools that produce comparable proofs.

cs.LO↗
arXiv2026

Confluence of conditional rewriting modulo

Sets of equations E play an important computational role in rewriting-based systems R. The equivalence relation =E induced by E introduces a partition of terms into E-equivalence classes on which rewriting computations, denoted ->R/E and called rewriting modulo E, are issued. This paper investigates confluence of ->R/E, usually called E-confluence, for conditional rewriting-based systems, where rewriting steps are determined by conditional rules. We rely on Jouannaud and Kirchner's framework to investigate confluence of an abstract relation R modulo an abstract equivalence relation E on a set A. We show how to particularize such a framework to be used with conditional systems. Then, we show how to define appropriate finite sets of conditional pairs to prove and disprove E-confluence. We introduce (i) Logic-based Conditional Critical Pairs, which do not require the use of (often infinitely many) E-unifiers to provide a finite representation of the local peaks considered in the abstract framework. We also introduce (ii) parametric Conditional Variable Pairs which are essential to deal with conditional rules in the analysis of E-confluence. Finally, we introduce (iii) Down Conditional Pairs which are often necessary to disprove E-confluence. Our results apply to well-known classes of rewriting-based systems, improving on previous results. As for unconditional systems, our results apply to Equational Term Rewriting Systems, first investigated by Huet and then by Jouannaud, and Jouannaud and Kirchner, among others. As for conditional systems, our results also apply to conditional rewrite theories and Maude.

cs.LO↗
arXiv2026

Verifying Numerical Methods with Isabelle/HOL

Modern machine learning pipelines are built on numerical algorithms. Reliable numerical methods are thus a prerequisite for trustworthy machine learning and cyber-physical systems. Therefore, we contribute a framework for verified numerical methods in Isabelle/HOL based on ITrees. Our user-friendly specification language enables the direct declaration of numerical programs that can be annotated with variants and invariants for reasoning about correctness specifications. The generated verification conditions can be discharged via automated proof methods and lemmas from the HOL-Analysis library. The ITrees foundation interacts with Isabelle's code generator to export source code. This provides an end-to-end path from formal specifications with machine-checked guarantees to executable sources. We illustrate the process of modelling numerical methods and demonstrate the effectiveness of the verification by focusing on two well-known methods, the bisection method and the fixed-point iteration method. We also contribute crucial extensions to the libraries of formalised mathematics required for this objective: higher-order derivatives and Taylor's theorem in Peano form. Finally, we qualitatively evaluate the use of the framework for verifying numerical methods.

cs.LO↗