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Abstract

From a cognitive perspective in the semantic space, we proposed the revelation of the semantics of the Four Color Theorem based on our proposed Existence Computation and Reasoning(EXCR) and Essence Computation and Reasoning(ESCR) mechanism following our previous revelation of the semantics of point, line and plane.
Existence Computation and Reasoning(EXCR) and Essence Computation and
Reasoning(ESCR) based Revelation of the Four Color Theorem
基于存在计算(EXCR)与语义计算(ESCR)的关于四色定理的语义空间(SCR)解释
By Yucong Duan,
DIKW research group, Hainan University
Email: duanyucong@hotmail.com
Abstract: From a cognitive perspective in the semantic space, we proposed the
revelation of the semantics of the Four Color Theorem based on our proposed
Existence Computation and Reasoning(EXCR) and Essence Computation and
Reasoning(ESCR) mechanism following our previous revelation of the semantics of
point, line and plane.
四色定理( Four Color Theorem)的语义解释:
四色定理语义等价于寻找一个平面 PL 上的所有区域 Z组合实例对应的本质色彩
区分数量 NUM(PL, Z, C)
为了解释平面 PL(X, Y)上的区域 Z的语义,在存在语义层面可以借助点、线、平
面的存在语义进行存在语义分析。
色彩 C在存在语义上就是一个区分语义 SM它区分了色彩实例 c填充的区域 CZ
和未填充的区域 NZ
SM(C):=(CZ,NZ)
:={(c,z)}
:={(cz,nz))}
:={(z(c),z(!c)))}
区域 Z的所有组合情形 Complex(Z)可以用构造的方式枚举解释:
0条本质存在的线定义的区域实例:
当欧式坐标空间平面 PL=COD(X, Y)中存在 0条线时,也即 L={}0个区域需要被
区分,也就需要 1个色彩 c1 来填充这个区域。
SMD(Z,L={})
=>NUM(Z)=0
=>NUM(SMD0 (C))=NUM(SMD0 (c1))=1
从存在语义的 EXCR 角度,NUM(SMD0 ({c1}))也等价于,在欧式坐标空间平面
PL=COD(X, Y)中在存在 0条线时最多对应 1个基本区域标记存在语义。
NUM(EXCR(COD(X, Y), {} )):=1
1条本质存在的线定义的区域实例:
当欧式坐标空间平面 PL=COD(X, Y)中存在 1条线,也即 L={l1}
经过坐标语义等价变换,线 l1=COD(X,R(ry))将平面 PL=COD(X, Y)区分为线两侧
Y=R(ry)其两侧的两个平面部分对应的区域。
Z(PL=COD(X, Y),l1=COD(X,R(ry)) )
:={Z1(PL=COD(X, Y<R(ry))), Z2(PL=COD(X, Y>R(ry)))}
:={Z1, Z2}
2个区域{Z1, Z2}需要用不同颜色标记,对应这个不同语义的 SMD1 的本质存在
就被确认了。SMD1 的基本标记需要 2个部分对应,我们用色彩{c1,c2}来标记并
对应这 2个区域。
SMD(Z, L={l1})
=>NUM({Z1, Z2})=2
=>NUM(SMD1 (C))=NUM(SMD1 ({c1,c2}))=2
从存在语义角度分析,2个区域 Z1(PL=COD(X, Y<R(ry)))Z2(PL=COD(X, Y>R(ry)))
的表达的区分完全是在 ASS(Y, R(ry))的语义范围上的 ASS(Y<R(ry), Y>R(ry))语义上。
从本质上 ASS(Y<R(ry), Y>R(ry))语义等价于 ASS(c1, c2)语义。
SMD(ASS(Y<R(ry), Y>R(ry))):=SMD(ASS(c1, c2))
从存在语义的 EXCR 角度,NUM(SMD3 ({c1,c2}))也等价于,在欧式坐标空间平面
PL=COD(X, Y)中在存在 1条线{l1}时最多对应 2个基本存在语义。
NUM(EXCR(COD(X, Y), {l1} )):=2
2条本质存在的线定义的区域实例:
当欧式坐标空间平面 PL=COD(X, Y)中在存在 1条线 l1 的基础上增加 1条线 l2
L={l1, l2}
经过坐标语义等价变换,线 l2=COD(X,R(rx))将平面对应的区域 Z{Z1, Z2}的中的每
个区域区分为 2个部分。
Z(PL={Z1, Z2},L{l1=COD(X,R(ry)),l2=COD(R(rx),Y)} )
:={Z11=COD(X<R(rx), Y<R(ry)),Z12=COD(X>R(rx), Y<R(ry)),Z21=COD(X<R(rx),
Y>R(ry)),Z22=COD(X>R(rx), Y>R(ry))}
:={Z11, Z12, Z21, Z22}
表面上这 4个区域{Z11, Z12, Z21, Z22}需要用不同颜色标记,也就需要 4个色彩
{c1,c2,c3,c4}来对应这 4个区域。
然而从存在语义角度分析,4个区域{Z11, Z12, Z21, Z22}的表达的区分完全是在
ASS(R(rx), R(ry))的语义范围上的具体 ASS(ASS(X<R(rx), Y<R(ry)),ASS(X>R(rx),
Y<R(ry)),ASS(X<R(rx), Y>R(ry)),ASS(X>R(rx), Y>R(ry)) )语义上。
ASS(ASS(X<R(rx), Y<R(ry)),ASS(X>R(rx), Y<R(ry)),ASS(X<R(rx), Y>R(ry)),ASS(X>R(rx),
Y>R(ry)) )
:=ASS(R(rx), R(ry))
:=ASS((X<R(rx), X>R(rx)), SMD1)
:=ASS(Z21(X<R(rx), SMD1) ,Z22(X>R(rx), SMD1) )
:=ASS(Z((X<R(rx), X>R(rx)), SMD1)) )
:=ASS(SMD2, SMD1)
本质上这 4个区域来源于第二条线 l2 的加入在原有存在语义空间 SMD1 的基础
上进行了进一步语义区分 SMD2。这个区分有多种存在解释,从存在的独立性角
度,第二条线 l2 的语义区分过程与第一条线 l1 的区分基础和区分过程没有任何
区别。因而和第一条线被引入时一样,确认了一个新的不同语义 SMD2 的本质存
在。SMD2 的基本标记需要 2个部分对应,但由于已有区域已经被原有色彩标记
过了,所以只需在原有色彩{c1,c2}的存在的基础上引入一个新色彩 c3 来标记并
对应这 4个区域。
SMD(Z, L={l1, l2})
=>NUM(SMD2, SMD1)=NUM(SMD2 ({c1,c2,c3}))=2+1=3
几何直观上,我们上面讨论的两条相交的线的情形下,被划分的每个区域都有且
仅有两条来源于承载了不同区分语义的线作为这个区域存在的本质依据。因而这
个区域也有且仅有对这两个边界的内外的的区分语义表达需求。对应这两个边界
的外部最多存在两个不同的外部区域。也就是两条交叉线的情形下确定的所有区
域最多需要标记其自身以及两个外部区域,也就是最多需要 3种独立存在的标
记,也即 3种色彩。
引入平行线的色彩标记情形:
上面讨论中未描述两条线平行的情形。当第二条线 l2 与第一条线 l1 平行时,l2
必存在于 l1 的一侧区域 zal1从而 l2 可以等价于 l1 的区分情形。由于 l2 的非介
l2 l1 之间的一侧区域 zbl2 l1 的另一侧区域 zbl1 不相邻,从而这两个区域
zbl2 zbl1 不存在区分需求。因而区域 zbl2 zbl1 可以使用与 zbl1 相同的颜色
进行填充。
定理(rZPL): 递归的,引入任何新的平行线的情形下不会引入对色彩标记的存在
意义上的增加需求,或需要的本质标记色数量不改变。
从存在语义的 EXCR 角度,NUM(SMD3 ({c1,c2,c3}))也等价于,在欧式坐标空间平
PL=COD(X, Y)中在存在 2条线{l1, l2}时最多对应 3个基本存在语义。
NUM(EXCR(COD(X, Y), {l1, l2} )):=3
3条本质存在的线定义的区域实例:
当欧式坐标空间平面 PL=COD(X, Y)中在存在 2条线{l1,l2}的基础上增加 1条线 l3
也即 L={l1, l2, l3}
经过坐标语义等价变换,可以将线{l1, l2}对应到线 l1=COD(X,R(ry))与线
l2=COD(X,R(rx))
线l3 对应于语义空间的 ASS(X, Y)的实例 ASS(R(rx), R(ry))ASS(R(rx), R(ry))将已有
区域{Z11, Z12, Z21, Z22}从平面几何角度表面上区分为 2个有本质区分语义的部
分。
Z(PL={Z11, Z12, Z21, Z22},L{l1, l2, l3} )
:=Z(PL, l3)
:={Z(PL,ASS(X, Y)< ASS(R(rx), R(ry))) , Z(PL,ASS(X, Y)> ASS(R(rx), R(ry))) }
:={
Z111=COD(X<R(rx), Y<R(ry), ASS(X,Y) < ASS(R(rx), R(ry))),
Z121=COD(X>R(rx), Y<R(ry)), ASS(X,Y) < ASS(R(rx), R(ry))),
Z211=COD(X<R(rx), Y>R(ry)), ASS(X,Y) < ASS(R(rx), R(ry))),
Z221=COD(X>R(rx), Y>R(ry)), ASS(X,Y) < ASS(R(rx), R(ry))),
Z112=COD(X<R(rx), Y<R(ry), ASS(X,Y) > ASS(R(rx), R(ry))),
Z122=COD(X>R(rx), Y<R(ry)), ASS(X,Y) > ASS(R(rx), R(ry))),
Z212=COD(X<R(rx), Y>R(ry)), ASS(X,Y) > ASS(R(rx), R(ry))),
Z222=COD(X>R(rx), Y>R(ry)), ASS(X,Y) > ASS(R(rx), R(ry))),
}
:={{Z111, Z121, Z211, Z221} ,{Z112, Z122, Z212, Z222}}
表面上这 8个区域{Z11, Z12, Z21, Z22}需要用不同颜色标记,也就需要 8个色彩来
对应这 8个区域。
然而从存在语义角度分析,8个区域{Z11, Z12, Z21, Z22}的表达的区分完全是在
ASS(R(rx), R(ry), ASS(R(rx), R(ry)))的语义范围上的具体语义上。
这里我们补充,ASS(R(rx), R(ry))ASS(X, Y)的实例 INS(ASS(X, Y), R)。这个实例的
获得是由 XY变量取值分别向实数域 R映射得到的。XY变量取值分别向实
数域 R映射也等价于 ASS(X, Y)向实数空间 R(X, Y)映射。ASS(X, Y)向实数空间 R(X, Y)
映射的结果也等价于,XY在实数操作加 R(+)与乘法 R(*)组合形成的所有 ASS(X,
Y)
ASS(R(rx), R(ry))
:=INS(ASS(X, Y), R)
:=R(X, Y)
:=ASS(ASS(X, Y), R(+),R(*))
理论上,只要 ASS(R(rx), R(ry))的存在语义不发生改变,ASS(ASS(X, Y), R(+),R(*))
以对应所有的平面连续线条
这里的连续的语义就是它们的存在一致性不被破坏。
面向本质分析,ASS(ASS(X, Y), R(+),R(*))的最基本形态包括 ASS(ASS(X, Y), R(+))
ASS(ASS(X, Y), R(*))。这些表达的存在意义上等价于 INS(X+Y=R(r))
ASS(R(rx), R(ry))
:=ASS(ASS(X, Y), R(+),R(*))
:=ASS(ESCR(ASS(X, Y), R(*)))
:=ASS(ESCR(ASS(X, Y), R(+)))
:=ASS((X, Y), R(+))
:=INS(X+Y=R(r))
ASS(R(rx), R(ry), ASS(R(rx), R(ry)))
:=ASS(ASS(X,Y) < ASS(R(rx), R(ry))), ASS(X,Y) > ASS(R(rx), R(ry))), ASS(SMD2, SMD1) )
:=ASS(SMD3, ASS(SMD2, SMD1))
本质上这 8个区域来源于第 3条线 l3 的加入在原有存在语义空间 SMD1 的基础
上进行了进一步语义区分 SMD3。这个区分有多种存在解释,从存在的独立性角
度,3条线 l3 的语义区分过程与第 1条线 l1 以及第 2条线 l2 的区分基础和区
分过程没有任何区别。
因而和{l1, l2}被引入时一样,确认了一个新的不同语义 SMD3 的本质存在。SMD3
的基本标记需要 2个部分对应,但由于已有区域已经被原有色彩标记过了,所以
只需在原有色彩{c1,c2,c3}的存在的基础上引入一个新色彩 c4 来标记并对应这 8
个区域。
SMD(Z, L={l1, l2, l3})
=>NUM(SMD3, SMD2, SMD1)=NUM(SMD3 ({c1,c2,c3,c4}))=3+1=4
从存在语义的 EXCR 角度,NUM(SMD3 ({c1,c2,c3,c4}))也等价于,在欧式坐标空间
平面 PL=COD(X, Y)中在存在 3条线{l1, l2, l3}时最多对应 4个基本存在语义。
NUM(EXCR(COD(X, Y), {l1, l2, l3} )):=4
3条本质存在的线定义的封闭区域 ZC
当欧式坐标空间平面 PL=COD(X, Y)中在存在 3条线,也即 L={l1, l2, l3}时,
经过坐标语义等价变换,可以将线{l1, l2}对应到线 l1=COD(X,R(ry))与线
l2=COD(X,R(rx))这样线{l1, l2}将可以将空间分别以其中一条线的一侧组合围出四
个存在语义上等价的区域。线 l3 对应到 l3=ASS( ASS((X, Y), R(+)), {l1, l2})线 l3
线{l1, l2}分别可以产生最多一个交点,分别是点 P1(ASS((X, Y), R(+)), {ASS(X,R(ry)))
和点 P2(ASS((X, Y), R(+)), {ASS(X,R(rx)))
l3
:=ASS( ASS((X, Y), R(+)), {l1, l2})
:=ASS( ASS((X, Y), R(+)), {COD(X,R(ry)), COD(X,R(rx))})
:=ASS( ASS((X, Y), R(+)), {ASS(X,R(ry)), ASS(X,R(rx))})
:=ASS( {(ASS((X, Y), R(+)), {ASS(X,R(ry))), (ASS((X, Y), R(+)), {ASS(X,R(rx)))})
=>INS(ASS(P1,P2))
如果 P1 P2 不存在的情形下,l3 的存在将等价于其不影响原有的色彩标记。
们考虑 P1 P2 都存在的情形下,线 l3=INS(ASS(P1,P2))将与线{l1, l2}形成一个封
闭区域 ZC
封闭区域在色彩标记的存在语义上有重要的意义:
封闭区域 ZC 的封闭也是存在语义层面的封闭,也就是 ZC 的任何实例将不会与非
形成封闭区域的不跨越封闭区域 ZC 的线 L的实例以外的已存线 L形成接触边界
LB
ASS(ZC, L, LB)=>LB={}
定理(rZCO): 用于标记 ZC(l1, l2, l3)的色彩 C(ZC)可以被用于标记在 ZC 区域以外(
跨越区域 ZC)引入的非平行任何已有线的线 L(x)所对应引入的标记需求。由于平
PL=COD(X, Y)中线总是被一条一条引入的,C(ZC)用于替代 C(L(x))是递归的。
C(ZC):=C(L(x))
定理(rZCI): ZC(l1, l2, l3)区域上(跨越区域 ZC)引入的非平行任何已有线的线 L(x)
可以与 ZC 的三条边界构成线中的任意两条,例如{l1, l2},构成一个包含在 ZC
的封闭区域 NZC=ZC(l1, l2, L(x))由于线 L(x) 与线 l3 的相对区分语义不存在,ZC(l1,
l2, l3) 可以被视作应用不影响存在语义的情形下做标记变换 MT(l3=L(x))
MT( L(x)=l3))之后等价于 NZC=ZC(l1, l2, L(x))这种变换后,ZC(l1, l2, l3)区域上(
越区域 ZC)引入的非平行任何已有线的线 L(x)被转换存在语义范畴等价于在 ZC
域以外(不跨越区域 ZC)引入的非平行任何已有线的线 L(x),进而可以应用定理
(rZCI)得到相同的处理。
4条及更多的线定义的区域实例:
当欧式坐标空间平面 PL=COD(X, Y)中在存在 3条线,也即 L={l1, l2, l3}时,引入任
何新的线 L(x)都将可以被纳入如下两种处理进行递归:
(1) 线L(x)平行于已存在线集合 L{l1, l2, l3}中任意一条的情形,可以对应应用定理
(rZPL)进行递归,从而否定增加存在意义上的标记色彩需求。
(2) 线L(x)不平行于已存在线集合 L{l1, l2, l3}中任意一条的情形,可以对应应用
理定理(rZCO)或定理(rZCI)进行递归,从而否定增加存在意义上的标记色彩需求。
综上,我们展示了基于存在计算(EXCR)与语义计算(ESCR)的关于四色定理的语义
空间(SCR)解释。
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Method
Full-text available
we believe that subjective semantics is objectively arranged. We'd like to explore a little bit on the usually viewed subjective math from a cognitive perspective in the semantic space. We reveal our intuition and lay the basis of our proposed definitions of: Semantic Computation and Reasoning Comprising Existence Computation and Reasoning, Essence Computation and Reasoning, Purpose Computation and Reasoning towards Resolution of Conceptual Computation and Reasoning.
Article
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We propose an outline of an approach to formalize semantic from conceptualization for both natural language (NL) and logic expression mechanisms. This goes beyond the level of discussions at conceptual level which has to either end in conscious/unconscious relativity of understanding or subjective enforcement in the form of definitions instead of expected objective semantic. This approach supports to view from a pure mathematical perspective, and explore and locate the fundamental problems. The semantic formalization mechanism realizes the integration of problem description and the solution expression at absolute semantic level. So a problem describing process is equivalent to the solution exploring process by integrating both in one. This essentially caters the ideology of proceeding with model refinement of model driven development. Other advantages include that it will reduce the need for validation for model migrations during a model driven development process, etc. Application is intended to cover specification refinement of both functional and quality requirement, and both static description and behavioral implementation, etc.
Conference Paper
after an investigation on existing ontology for three dimensional (3D) reconstructions of architectural objects from a cloud points achieved through 3D scanning, importance and imperativeness of completeness related issues is highlighted with the absence of logic connective “negation”. An EID-SCE based solution is proposed to fully model both humans side and machine side semantics to achieve completeness in the sense of integration and bridging of Closed world assumption (CWA) and Open world assumption (OWA). By consciously and explicitly modeling what is modeled previously implicitly and possibly unconsciously, decidability processes of object semantics are revealed with coherent traces of transformations from subjective Yes/No to objective True/False. Techniques of expressing behavioral/dynamic information and logic connective “and” without the need of extra constraints, with structural/static expressions, are also provided. Elementary implementation is shown with the creating an ontology for identification of 3D architectural objects.
Article
This work is motivated by the hypothesis that ideas of the Four Color Problem (FCP) could contain important implications on semantics identification in terms of the necessary and sufficient amounts of elements and contents inside a paradigm and among paradigms from ontology creation to model objects/relationships introduction/definition. Semantics ideas of EID-SCE are adopted for the semantics revelation on FCP. The key ideas include reduction based on the equal/“=” relationship for identification and existence. The revelation is extended towards covering all necessary situations at all possible stages in a constructive manner. The work aims to be part of introduction of mathematical achievement to software modeling practices. It is initially extended towards modeling applications. The result is expected to be beneficial to efforts of gaining improvement on reusability and efficiency.
Conference Paper
Archi3D is a successful practice well proved by engineering implementations. In this paper, both static class diagrams and dynamic constraint rules related to existing systems are investigated, optimization initiations are proposed with a semantics formalization approach called EID-SCE. Revelations and validations are extended at fundamental aspect of {OWA (open world assumption), CWA (closed world assumption)}, Yes/No vs. True/False, implicit vs. explicit, etc, to achieve formalization which could contribute to the semantics objects identification of D reconstruction processes. The expected result is gain efficiency of both description and implementation from the application of the proposed formalization.
Article
Firstly this article presents a thorough discussion of semantics formalization related issues in model driven engineering (MDE). Then motivated for the purpose of software implementation, and attempts to overcome the shortcomings of incompleteness and context-sensitivity in the existing models, we propose to study formalization of semantics from a cognitive background. Issues under study cover the broad scope of overlap vs. incomplete vs. complete, closed world assumption (CWA) vs. open world assumption (OWA), Y(Yes)/N(No) vs. T(True)/F(False), subjective (SUBJ) vs. objective (OBJ), static vs. dynamic, unconsciousness vs. conscious, human vs. machine aspects, etc. A semantics formalization approach called EID-SCE (Existence Identification Dualism-Semantics Cosmos Explosion) is designed to meet both the theoretical investigation and implementation of the proposed formalization goals. EID-SCE supports the measure/evaluation in a {complete, no overlap} manner whether a given concept or feature is an improvement. Some elementary cases are also shown to demonstrate the feasibility of EID-SCE.