[Paper Review] An improved axiomatic definition of information granulation

Ping Zhu|ArXiv.org|Aug 27, 2009

Rough Sets and Fuzzy Logic5 references4 citations

TL;DR

This paper proposes an improved axiomatic definition of information granulation by linking it to monotonic functions of multiple variables, providing a universal construction method. It demonstrates that key existing granulation measures—such as those in [2, 6, 7, 9, 10]—are instances of this framework, enhancing mathematical rigor and unifying prior approaches.

ABSTRACT

To capture the uncertainty of information or knowledge in information systems, various information granulations, also known as knowledge granulations, have been proposed. Recently, several axiomatic definitions of information granulation have been introduced. In this paper, we try to improve these axiomatic definitions and give a universal construction of information granulation by relating information granulations with a class of functions of multiple variables. We show that the improved axiomatic definition has some concrete information granulations in the literature as instances.

Motivation & Objective

• To address formal and mathematical shortcomings in prior axiomatic definitions of information granulation.
• To unify diverse information granulation measures from the literature under a single, robust axiomatic framework.
• To establish a universal construction method for information granulation based on multivariable monotonic functions.
• To ensure that known granulation measures satisfy the new axiomatic definition, thereby validating its generality and applicability.

Proposed method

• Define a mapping α(P) that associates each attribute subset P ⊆ A with a vector of tolerance class sizes for all objects in the universe.
• Introduce a function f: ℝⁿ → ℝ that is monotonic with respect to the component-wise partial order on sorted vectors of class sizes.
• Construct a granulation measure G_f(P) = f(βα(P)) + c, where βα(P) sorts the vector of class sizes and c ≥ −min(f) ensures non-negativity.
• Prove that any such G_f satisfies the three axioms: non-negativity, invariance under equivalence of attribute sets, and monotonicity under refinement.
• Demonstrate that the construction is universal by showing that existing granulations (e.g., in [2, 6, 7, 9, 10]) arise as special cases of G_f for specific f and c.
• Use the inverse construction: given a granulation G, define f_G via g, ˆg, and π to recover f such that f∘βα = G.

Experimental results

Research questions

• RQ1 Can a more mathematically rigorous axiomatic definition of information granulation be developed to correct flaws in prior formulations?
• RQ2 How can a universal construction method be established that generates all known information granulation measures?
• RQ3 Do prominent granulation measures from the literature—such as those in [2, 6, 7, 9, 10]—satisfy the new axiomatic definition?
• RQ4 What functional class of functions on multivariable vectors can generate all valid information granulation measures?
• RQ5 Is there a one-to-one correspondence between monotonic functions f and information granulation measures G_f?

Key findings

• The improved axiomatic definition satisfies non-negativity, invariance under equivalent attribute sets, and monotonicity under refinement, correcting prior formal weaknesses.
• The construction G_f(P) = f(βα(P)) + c with monotonic f and c ≥ −min(f) yields a valid information granulation for any information system.
• The granulation in [2] is recovered as G_f(P) = (1/n²)∑|S_P(xi)| with f(r) = (1/n²)∑r_i and c=0.
• The combination granulation in [9] is obtained with f(r) = (1/n)∑r_i(r_i−1)/(n(n−1)) and c=0.
• The rough entropy in [7] is realized via f(r) = (1/n)∑log₂r_i and c=0, showing consistency with existing entropy-based measures.
• The framework unifies multiple granulation measures under a single, general construction, proving their compatibility with the improved axiomatic definition.