模糊软集λ-三I方法的鲁棒性分析

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本文主要探讨了模糊软集领域内一般三I方法的鲁棒性，针对模糊软集推理中的稳定性问题，作者Lu Wang和Keyun Qin提出了基于模糊软模因（Fuzzy Soft Modus Ponens, FSMP）和模糊软模反驳（Fuzzy Soft Modus Tollens, FSMT）的λ-三元组推理方法。在这一框架下，他们着重分析了与残差对相关的推理结论的计算公式，这对于理解和评估模糊推理过程中的精确性和抗干扰能力至关重要。在模糊推理过程中，尤其是当面对不确定性和环境噪声时，鲁棒性是一个核心考量因素。鲁棒性研究的是在输入数据存在误差或不精确的情况下，推理结果的稳定性和一致性。作者通过引入λ参数，对λ-三元组推理方法在FSMP和FSMT中的表现进行了深入的剖析，旨在揭示这些方法在处理模糊和软信息时的稳健特性。特别地，文章进一步扩展到多模糊规则的FSMP模型中的通用三元组推理方法的鲁棒性分析。这涉及到如何确保在多个模糊规则的组合下，推理结论依然能在一定程度上抵抗外部干扰，保持一定的决策可靠性。为了实现这一点，研究者可能采用了左连续范数等数学工具来量化和衡量这种鲁棒性。这篇研究论文提供了对模糊软集理论中λ-三元组推理方法的稳健性分析，为理解模糊推理系统在实际应用中的鲁棒性提供了新的视角，并为设计更可靠、适应复杂环境的模糊推理算法提供了理论支持。对于从事模糊逻辑、人工智能以及软计算领域的研究人员来说，理解和掌握这一领域的研究成果是提升推理系统性能的关键一步。

Biresidual p ↔ q can be used to characterize the

similarity between p and q. Now, we are going to

recall further properties in terms of the biresidual re-

lated to the left-continuous triangular norm.

Lemma 2.

29,30,31

Assume that T is a left-continuous

triangular norm and ρ is the biresidual associated

with T . Then, for any x, y, z, o ∈ [0, 1]:

(1) x ↔ 1 = x;

(2) x = y ⇔ x ↔ y = 1;

(3) x ↔ y = y ↔ x;

(4) x ↔ y 6 R(x, 0) ↔ R(y, 0)

(5) T (x ↔ y, y ↔ z) 6 x ↔ z;

(6) (x ↔ y) ∧ (z ↔ o) 6 (x ∨z) ↔ (y ∨ o);

(7) T (x ↔ y, z ↔ o) 6 T (x, z) ↔ T(y, o);

(8) T (x ↔ y, z ↔ o) 6 R(x, z) ↔ R(y, o).

Lemma 3.

Assume that two arbitrary functions

are h

, h

: O → [0, 1], ρ is the biresidual and a non-

empty set is O. Then

(1)

∧

o∈O



(o) ↔ h

(o)



∧

o∈O

(o) ↔

∧

o∈O

(o);

(2)

∧

o∈O



(o) ↔ h

(o)



∨

o∈O

(o) ↔

∨

o∈O

(o).

Molodtsov presented the theory of soft set for hand-

ing uncertainties

. Assume that U is an initial uni-

verse set and E is parameters on the universal set.

(U, E) is called a soft space.

Deﬁnition 3.

The pair ( f , S) is said to be a soft set

over U if S ⊆ E and f : S → P(U).

That is to say, soft set in Deﬁnition 3 is a parame-

terized family of subsets of universe set. In general,

the parameters of a soft space are fuzzy words or

sentences associated with fuzzy words. With this in

mind, Maji et al. gave the deﬁnition of fuzzy soft

set

Deﬁnition 4.

Assume that S is subset of E, a pair

( f , S) is a fuzzy soft set over U if f : S → F(U),

where F(U) denote the set of all fuzzy subsets over

In order to investigate fuzzy soft sets based ap-

proximate reasoning, Qin extended the deﬁnition of

fuzzy implication operator to fuzzy soft implication

operator

Deﬁnition 5.

Assume that ( f , S) and (g, T ) are

two fuzzy soft sets over U and V respectively.

A fuzzy soft implication operator from ( f , S) to

(g, T ) is a fuzzy soft set (h, K) over U × V , where

K ⊆ S × T , h : K → F(U ×V ) is given by

H(s, t)(u, v) = F(s)(u) → G(t)(v)

for all (s,t) ∈ K, (u, v) ∈ U × V , and → is a fuzzy

implication operator.

Based on fuzzy soft implication operators, Qin

presented fuzzy soft inference method

. Triple I in-

ference methods for FSMP and FSMT are present-

ed. The related calculation formulas for FSMP and

FSMT are studied. FSMP and FSMT can be ex-

pressed as follows

FSMP: Major premise: ( f , S) → (g, T ) and Minor

premise: ( f

∗

, S). Conclusion:(g

∗

, T );

FSMT: Major premise: ( f , S) → (g, T ) and Minor

premise: (g

∗

, T ). Conclusion: ( f

∗

, S).

Where ( f , S), ( f

∗

, S) and (g, T ), (g

∗

, T ) are fuzzy

soft sets over U and V respectively. Based on the F-

SMP and FSMT, Qin presented Principle of triple I

in the following:

Principle of triple I for FSMP

: The conclusion

∗

, T ) of FSMP is the smallest fuzzy soft set over

V satisfying





f (s)(u), g(t)(v)



, R



∗

(s)(u), g

∗

(t)(v)





= 1;

Principle of triple I for FSMT

: The conclusion

( f

∗

, S) of FSMT is the largest fuzzy soft set over U

satisfying





f (s)(u), g(t)(v)



, R



∗

(s)(u), g

∗

(t)(v)





= 1;

Theorem 4.

Assume that the implication operator

in FSMP and FSMT is the residual implication R of

left-continuous t-norm T .

1113

___________________________________________________________________________________________________________

International Journal of Computational Intelligence Systems, Vol. 11 (2018) 1111-1122

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