《1 引言》

1 引言

《2 统一集以及相关的概念》

2 统一集以及相关的概念

《2.1统一集及其表示》

2.1统一集及其表示

《图1》

《图2》

$\begin{array}{l}S=\left\{\left(a,F\left(a\right)\right)|a\in A,F\left(a\right)\in B,\\ F\left(a\right)\text{满}\text{足}\text{约}\text{束}J\right\}。\text{ }\text{ }\text{ }\left(3\right)\end{array}$

《2.2统一集的扩展》

2.2统一集的扩展

$S{}_{1}=\left\{a,F{}_{1}\left(a\right)|a\in A{}_{1},F{}_{1}\left(a\right)\in B{}_{1}\right\},\text{ }\text{ }\text{ }\left(4\right)$

《2.3统一集与各种集合论》

2.3统一集与各种集合论

《2.3.1 普通统一集》

2.3.1 普通统一集

0阶扩展统一集称为普通统一集。在普通统一集中, 集合中的元素是基本的, 经典集合、模糊集合、可拓集合、Vague集合、FHW、FEEC等都可以认为是普通的统一集。

1) 经典集合[2]

$S{}_{\text{c}\text{r}\text{i}\text{s}\text{p}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\left(U,\phantom{\rule{0.25em}{0ex}}\left\{0,1\right\},\phantom{\rule{0.25em}{0ex}}C\left(x\right),\varnothing \right),\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\left(5\right)$

2) 模糊集合[3]

$S{}_{\text{f}\text{u}\text{z}\text{z}\text{y}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\left(U,\phantom{\rule{0.25em}{0ex}}\left\{0,1\right\},\phantom{\rule{0.25em}{0ex}}\mu \left(x\right),\phantom{\rule{0.25em}{0ex}}\varnothing \right),\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(6\right)$

3) 可拓集合[4]

$S{}_{\text{e}\text{x}\text{t}\text{e}\text{n}\text{s}\text{i}\text{o}\text{n}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\left(U,\phantom{\rule{0.25em}{0ex}}\left(-\infty ,+\infty \right),k\left(x\right),\varnothing \right)‚\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(7\right)$

4) Vague集合[5]

Vague集合是从Fuzzy集合中直接扩展出来的, 它弥补了传统Fuzzy集合的只能描述事物正隶属度的不足, 而是从隶属于一个概念的真假两个方面来考虑。用统一集可以把Vague集合表示为

$\begin{array}{l}S{}_{\text{v}\text{a}\text{g}\text{u}\text{e}}\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\left(U,\phantom{\rule{0.25em}{0ex}}\left[0,1\right]{}^{2},\phantom{\rule{0.25em}{0ex}}\left(t\left(x\right),\phantom{\rule{0.25em}{0ex}}f\left(x\right)\right),\phantom{\rule{0.25em}{0ex}}t\left(x\right)\\ +f\left(x\right)\le 1\right)‚\text{ }\text{ }\text{ }\left(8\right)\end{array}$

5) 模糊灰色物元空间 (FHW) [6]

FHW是一套用于宏观复杂大系统的决策支持系统, 它融合了模糊数学、可拓学、灰色系统、思维科学等多种学科, 曾在长江三峡的决策中产生了很好的实际应用效果。在FHW对方案进行评比选优时, 每一个方案都可以用一组模糊集合来对这个方案从当前、潜在、优劣度等多个方面进行评价。一个FHW空间用统一集的形式表示为

《图3》

6) 模糊可拓经济控制 (FEEC) [7]

FEEC是一种模糊可拓经济控制的方法, 它结合了可拓学、模糊控制技术、集对分析和其他一些最新系统科学理论, 将经济系统看作一个开放的宏观复杂大系统, 运用从定性到定量综合集成法进行经济分析和控制研究。进行系统分析时, 首先要建立一个模糊可拓经济空间 (FEES) , 它用统一集的形式表示为

$S{}_{\text{F}\text{E}\text{E}\text{C}}=\left(W×V,W×V,Τ,J\right)‚\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(10\right)$

《2.3.2 一阶统一集》

2.3.2 一阶统一集

1) 集对分析[8]。

《图4》

SSPA中的任意一个元素可以表示为:r= ( (S1, S2) , (a, b, c) ) , 其中 (S1, S2) 是一个集对, (a, b, c) 是这个集对的同一度、差异度、对立度。S1S2都是经典集合, 用统一集表示为

《图5》

《图6》

2) 粗糙集合类[12]。

$S{}_{\text{r}\text{o}\text{u}\text{g}\text{h}}=\left(Ρ,Ρ×Ρ,\left(F{}_{1}:Ρ\to Ρ,F{}_{2}:Ρ\to Ρ\right),R\right),\text{ }\text{ }\text{ }\left(13\right)$

$r=\left(x,\left(R{}^{-}\phantom{\rule{0.25em}{0ex}}\left(X\right),R{}_{-}\phantom{\rule{0.25em}{0ex}}\left(X\right)\right),\text{ }\text{ }\text{ }\left(14\right)$

$\begin{array}{l}X\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\left(U,\phantom{\rule{0.25em}{0ex}}\left\{0,1\right\},\phantom{\rule{0.25em}{0ex}}C{}_{1}\phantom{\rule{0.25em}{0ex}}\left(x\right),\phantom{\rule{0.25em}{0ex}}\varnothing \right),\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\\ R{}^{-}\phantom{\rule{0.25em}{0ex}}\left(X\right)\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\left(U,\phantom{\rule{0.25em}{0ex}}\left\{0,1\right\},\phantom{\rule{0.25em}{0ex}}C{}_{2}\phantom{\rule{0.25em}{0ex}}\left(x\right),\varnothing \right),\\ \phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}R{}_{-}\phantom{\rule{0.25em}{0ex}}\left(X\right)\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\left(U,\phantom{\rule{0.25em}{0ex}}\left\{0,1\right\},\phantom{\rule{0.25em}{0ex}}C{}_{2}\phantom{\rule{0.25em}{0ex}}\left(x\right),\phantom{\rule{0.25em}{0ex}}\varnothing \right)。\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\left(15\right)\end{array}$

3) 模糊粗集[10,11]和广义粗集[12]。

$\begin{array}{l}X\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\left(U,\phantom{\rule{0.25em}{0ex}}\left\{0,1\right\},\phantom{\rule{0.25em}{0ex}}\mu {}_{1}\phantom{\rule{0.25em}{0ex}}\left(x\right),\phantom{\rule{0.25em}{0ex}}\varnothing \right),\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\\ R{}^{-}\phantom{\rule{0.25em}{0ex}}\left(X\phantom{\rule{0.25em}{0ex}}\right)\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\left(U,\phantom{\rule{0.25em}{0ex}}\left\{0,1\right\},\phantom{\rule{0.25em}{0ex}}\mu {}_{2}\phantom{\rule{0.25em}{0ex}}\left(x\right),\phantom{\rule{0.25em}{0ex}}\varnothing \right),\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\\ R{}_{-}\phantom{\rule{0.25em}{0ex}}\left(X\phantom{\rule{0.25em}{0ex}}\right)\phantom{\rule{0.25em}{0ex}}=\phantom{\rule{0.25em}{0ex}}\left(U,\phantom{\rule{0.25em}{0ex}}\left\{0,1\right\},\phantom{\rule{0.25em}{0ex}}\mu {}_{3}\phantom{\rule{0.25em}{0ex}}\left(x\right),\phantom{\rule{0.25em}{0ex}}\varnothing \right)。\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(16\right)\end{array}$

$S{}_{\text{r}\text{o}\text{u}\text{g}\text{h}}=\left(F,F×F,\left(F{}_{1}:F\to F,F{}_{2}:F\to F\right),J\right)。\text{ }\text{ }\text{ }\left(17\right)$

F为论域U上的所有的模糊集合, 映射F1F2也要变换成相应的模糊粗集定义的形式。并且约束中R也成为模糊的划分关系。

《2.4统一集上的运算》

2.4统一集上的运算

$S{}_{\text{Ρ}\left(\text{S}\right)}=\left(Ρ\left(S\right),\left\{0,1\right\},C\left(x\right)=1,\varnothing \right)。\text{ }\text{ }\text{ }\left(18\right)$

$\begin{array}{l}r{}_{1}=\left(x{}_{1},F{}_{1}\left(x{}_{1}\right)\right),r{}_{2}=\left(x{}_{2},F{}_{2}\left(x{}_{2}\right)\right),\\ x{}_{i}\in A,f{}_{i}\left(x{}_{i}\right)\in B,i=1,2。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(19\right)\end{array}$

r1S1中的元素, r2S2中的元素。如果要求x1=x2=x, 则σ作用在这2个元素上产生的像:

$r{}_{u}=\sigma \left(r{}_{1},r{}_{2}\right)=\left(x,F\left(x\right)\right),x\in A,F\left(x\right)\in B。\text{ }\text{ }\text{ }\left(20\right)$

$\sigma =\sigma {}_{\text{B}}‚\phantom{\rule{0.25em}{0ex}}\text{其}\text{中}\sigma {}_{\text{B}}:B×B\to B。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(21\right)$

$\begin{array}{l}\sigma \left(r{}_{1},r{}_{2}\right)=r{}_{\sigma }=\left(x{}_{\sigma },F\left(x{}_{\sigma }\right)\right)=\left(\sigma {}_{\text{A}}\left(x{}_{1},x{}_{2}\right),\\ \sigma {}_{\text{B}}\left(F\left(x{}_{1}\right),F\left(x{}_{2}\right)\right)\right),\text{ }\text{ }\text{ }\left(22\right)\end{array}$

$\sigma =\left(\sigma {}_{\text{A}},\sigma {}_{\text{B}}\right),\sigma {}_{\text{A}}:A×A\to A,\sigma {}_{\text{B}}:B×B\to B。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(23\right)$

《2.5关于界壳J的讨论》

2.5关于界壳J的讨论

S0n 维欧几里得空间Rn中的一个超球面, 其半径为1, 在S0上存在有限个界门Gm, m<∞, 则称S0构成了一个标准界壳

$J=\left\{S{}_{0}‚1‚G{}_{m}\right\}。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(24\right)$

$S=\left(U×U,U,\sigma ,J\right)\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(25\right)$

J1: ∅, 则S1为群胚。

J2: ∀a, b, cU, 有 (aσb) σc= (bσc) 即结合律, 则S2为半群。

J3: 满足J2的约束, 并且, ∃e, ∀aU, 有aσe=eσa, 那么S3就是幺半群。

J4: 满足J2, J3的约束, 且∀aU, ∃a-1U, 有aσa-1=a-1σa=e, 那么S4就是群。

J5: 满足J2, J3, J4中的约束, 且∀a, bU, 有aσb=bσa, 则S5为交换群。

《3 统一集在人工智能中的应用》

3 统一集在人工智能中的应用

《3.1统一集与模式识别》

3.1统一集与模式识别

$S{}_{i}=\left(A,B,F{}_{i}\left(x\right),J\right),i\in \left[1‚n\right]。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(26\right)$

《3.1.1 直接识别[15]》

3.1.1 直接识别[15]

《3.1.2 间接识别[15]》

3.1.2 间接识别[15]

《3.2统一集与聚类分析》

3.2统一集与聚类分析

$r=\left(\left(a{}_{1},\phantom{\rule{0.25em}{0ex}}a{}_{2}\right),\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}F\left(a{}_{1},\phantom{\rule{0.25em}{0ex}}a{}_{2}\right)\right)。\text{ }\text{ }\text{ }\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\phantom{\rule{0.25em}{0ex}}\left(27\right)$

《3.3统一集与逻辑推理》

3.3统一集与逻辑推理

《图7》

《图8》

$\begin{array}{l}S{}_{\text{r}\text{o}\text{u}\text{g}\text{h}}=\left(Ρ,Ρ×Ρ,\left(F{}_{1}:Ρ\to \\ Ρ,F{}_{2}:Ρ\to Ρ\right),R\right),\text{ }\text{ }\text{ }\left(30\right)\end{array}$

Table 1 Comparison of Γ and Ψ

《图9》

《3.4统一集与机器学习》

3.4统一集与机器学习

《3.5统一集与智能决策》

3.5统一集与智能决策

《图10》

《4 结语》

4 结语