arxiv: 2605.14560 · v1 · submitted 2026-05-14 · 🧮 math.GM

Recognition: 2 theorem links

· Lean Theorem

Roughness and entropy measures of a soft set

Santanu Acharjee , Sankar K. Pal

Authors on Pith no claims yet

Pith reviewed 2026-05-15 00:53 UTC · model grok-4.3

classification 🧮 math.GM

keywords soft setsroughness measuresentropy measuresuncertaintysoft computingrough sets comparison

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The pith

Soft sets receive two roughness measures and six entropy measures defined within their native framework.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops two roughness measures and six entropy measures specifically for soft sets. The development follows Molodtsov's original soft set principles without modification. Properties of these measures are examined through both mathematical proofs and numerical examples. The result is a roughness characterization shown to be different from that in classical rough set theory. This provides tools for handling uncertainty in soft set applications across various fields.

Core claim

The authors establish two distinct roughness measures for soft sets based on separate conceptual frameworks and introduce six entropy measures, then prove their key properties. These constructions maintain the attribute-oriented structure of soft sets and yield a roughness analysis that the paper demonstrates is not equivalent to classical rough set approaches.

What carries the argument

Roughness measures defined in two conceptual frameworks for soft sets, which quantify the degree of uncertainty or incompleteness while respecting the parametrization of attributes.

Load-bearing premise

That the proposed roughness measures capture a form of uncertainty in soft sets that cannot be reduced to the standard rough set definitions.

What would settle it

A counterexample consisting of a specific soft set and its approximations where the calculated roughness value matches exactly what classical rough set theory would predict for the corresponding set.

read the original abstract

Soft set theory is an important and emerging area within soft computing, owing to its attribute-oriented mathematical framework and its wide applicability in diverse domains, including science and social sciences. The theoretical constraints associated with the selection of subsets of the sets of attributes in soft set theory have further motivated the development of hybrid and extended theoretical models. In this paper, we introduce two distinct roughness measures and six entropy measures for soft sets and systematically investigate their properties using both theoretical analysis and computational techniques. The proposed roughness measures are defined within two distinct conceptual frameworks. Throughout the development of these measures and the corresponding results, the foundational principles of soft set theory, as established by Molodtsov, are strictly preserved. Furthermore, the proposed framework is shown to be novel with respect to roughness characterization, and a comparative analysis with classical rough set theory is presented to highlight the theoretical distinctions and contributions of this work.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper claims to define two roughness measures and six entropy measures for soft sets that stay faithful to Molodtsov's axioms and differ from rough sets, though only the abstract is here to judge.

read the letter

The main takeaway is that this paper introduces two distinct roughness measures and six entropy measures for soft sets. The authors state that these preserve the core axioms from Molodtsov and provide a roughness framework that is different from what classical rough set theory offers. They also mention doing both theoretical analysis and computational checks on the properties. What stands out as positive is the effort to build these measures directly on the soft set structure without altering its basic rules. The abstract points to a comparative analysis with rough sets to show the distinctions, which could help clarify how soft sets handle uncertainty in ways that rough sets do not. This kind of extension fits into the broader use of soft sets in decision making and soft computing applications. The clear limitation comes from having only the abstract. There are no actual definitions, proofs, or examples provided here, so it's impossible to confirm whether the new measures truly satisfy the expected properties or if they avoid overlap with existing concepts. The claim of novelty depends entirely on the authors' assertion in the abstract, and without the details, any potential issues with how the measures are constructed remain hidden. Readers who work in soft set theory or related areas of uncertainty modeling would be the ones to benefit most. Someone looking for quantitative tools to apply in hybrid models might pick up ideas from the framework if the full development holds together. I would recommend sending the full paper to peer review. The topic is focused enough that referees familiar with the area can evaluate the derivations and the computational results to see if they add real value.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces two distinct roughness measures and six entropy measures for soft sets. These are defined within two conceptual frameworks and their properties are investigated via theoretical analysis and computational techniques. The work claims to strictly preserve Molodtsov's foundational axioms for soft sets while establishing novelty relative to classical rough-set roughness measures, supported by a comparative analysis.

Significance. If the definitions prove to be non-redundant with existing measures and the claimed properties hold under Molodtsov's axioms, the contribution would supply additional quantitative tools for uncertainty in soft-set models, potentially useful in applications involving attribute-based approximations. The dual-framework approach and entropy extensions could broaden the toolkit beyond standard rough-set entropy, but the abstract alone does not establish whether the novelty is substantive or merely notational.

major comments (1)

[Abstract] Abstract: the central claim that the roughness measures constitute a 'genuinely novel roughness characterization' distinct from classical rough sets cannot be evaluated, because the explicit definitions of the two measures, the six entropy measures, and any supporting theorems or computational examples are absent from the provided text.

minor comments (1)

[Abstract] The abstract refers to 'two distinct conceptual frameworks' without naming or briefly characterizing them, which obscures how the measures differ from each other.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed review and for highlighting the need for clarity in evaluating our claims. We address the major comment point by point below.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the roughness measures constitute a 'genuinely novel roughness characterization' distinct from classical rough sets cannot be evaluated, because the explicit definitions of the two measures, the six entropy measures, and any supporting theorems or computational examples are absent from the provided text.

Authors: We agree that the abstract, being a concise summary, does not include the explicit mathematical definitions, theorems, or examples. These appear in the full manuscript: the two roughness measures are defined in Section 3 within two distinct frameworks, the six entropy measures in Section 4, with properties, proofs preserving Molodtsov's axioms, and comparative analysis versus classical rough sets in Section 5, supported by computational examples. The abstract accurately summarizes the novelty established in the body. If the full text was not provided to the referee, we are happy to supply it. No changes to the abstract are required, as expanding it would violate standard length and style constraints. revision: no

Circularity Check

0 steps flagged

No significant circularity in available abstract

full rationale

The provided abstract introduces two new roughness measures and six entropy measures as fresh definitions that preserve Molodtsov's soft-set axioms while claiming novelty relative to classical rough sets. No equations, derivation steps, fitted parameters, or self-citations are visible that would reduce any claimed result to its own inputs by construction. The text describes definitional extensions and comparative analysis rather than any prediction or uniqueness theorem that loops back to the paper's own content. With only the abstract available and no load-bearing technical steps exhibited, the derivation chain cannot be shown to contain circularity under the enumerated patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the existing axioms of soft set theory; no free parameters, fitted constants, or new invented entities are mentioned.

axioms (1)

domain assumption Foundational principles of soft set theory as established by Molodtsov
Strictly preserved throughout the development of the measures.

pith-pipeline@v0.9.0 · 5416 in / 1137 out tokens · 34853 ms · 2026-05-15T00:53:43.302948+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce two distinct roughness measures and six entropy measures for soft sets... foundational principles of soft set theory, as established by Molodtsov, are strictly preserved.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 3.1... lower soft approximation... Definition 3.2... upper soft approximation... accuracy measure ρ_P_R(S,A) = sum |lower| / sum |upper|

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.