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# Existence Computation and Reasoning(EXCR) and Essence Computation and Reasoning(ESCR) based Revelation of Collatz Conjecture

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## Abstract

From a cognitive perspective in the semantic space, we proposed the revelation of the semantics of Collatz Conjecture or the 3x+1 Problem based on our proposed Existence Computation and Reasoning(EXCR) and Essence Computation and Reasoning(ESCR) mechanism following our previous revelation of the semantics axioms of Conservation of Existence Set Axiom (CEX , Consistency of Compounded Essential Set Axiom (CES) and Inheritance of Existence Semantics Axiom (IHES ).
Existence Computation and Reasoning(EXCR) and Essence Computation and
Reasoning(ESCR) based Revelation of Collatz Conjecture

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 Collatz Conjecture or the 3x+1 Problem based on our
proposed Existence Computation and Reasoning(EXCR) and Essence Computation
and Reasoning(ESCR) mechanism following our previous revelation of the semantics
axioms of Conservation of Existence Set Axiom (CEX , Consistency of
Compounded Essential Set Axiom (CES) and Inheritance of Existence Semantics
Axiom (IHES ).
Collatz Conjecture 的语义解释：
(a)Collatz Conjecture 语义可以从类型的实例的语义角度等价于:

3再加 1获得 n:=3o+1n是偶数 e则对它除以 2获得 n:=e/2如此循环，

INS(N)
:=ASS({INS(O),INS(E)},{REL(+),REL(/)})
:=ASS({INS(O)*3+1, INS(E)/2})
:=ASS({INS(O)*3+1, INS(E)/2})
:=ASS({o*3+1, e/2}, 1)
:=ASS({n*3+1, n/2}, 1)
=>n->1
(b)Collatz Conjecture 语义可以从实例的整体类型的语义角度等价于:

INS(N):=ASS({INS(O),INS(E)},{REL(+),REL(/)})关联，根据存在计算与推理 EXCR 的基

INS(N):=ASS({INS(O),INS(E)},{REL(+),REL(/)})
=>TYPE(N):=ASS({TYPE(O),TYPE(E)},{REL(+),REL(/)})
TYPE(N):=ASS({TYPE(O),TYPE(E})蕴含类型层面自然数类型 N与奇数类型 O与偶数

TYPE(N):=ASS({TYPE(O),TYPE(E)},{REL(+),REL(/)})
=>EXCR(N):=EXCR(TYPE(O),TYPE(E))
=>EXCR(N):=EXCR(O,E)

ASS(TYPE(O),TYPE(E))
:=ASS((TYPE(O),TYPE(E)),{REL(+),REL(/)})
:=ASS(((TYPE(O),TYPE(E)),1),REL(+))
=>N(E):=N(O)+1

O与偶数类型 E之间的存在语义上的等价性。
N(E):=N(O)+1
=>EXCR(TYPE(O)):=EXCR(TYPE(E))
=>EXCR(O):=EXCR(E)
EXCR(N):=EXCR(O,E)以及 EXCR(O):=EXCR(E)我们可以可以依托存在计算与推理
EXCR 的基础假设公理存在的守恒公理 CEX 确定自然数类型 N与奇数类型 O与偶

ASS(EXCR(N):=EXCR(O,E), EXCR(O):=EXCR(E))
=>EXCR(N):=EXCR(O):=EXCR(E)

Conservation of Existence Set)与本质计算与推理 ESCR 的基础假设公理就是本质集

Semantics)：类型层面的存在语义在纯类型层面的语义处理过程中，对于具有存

B的所有存在语义 EX(A) {ex(a)}
IHES
ASS({EXCR(A):=EXCR(B), EXCR(A)=>EXCR(B)})
=>EX(B)=>EX(A)

Conservation of Existence Set)本质计算与推理 ESCR 的基础假设公理就是本质集

N的实例 INS(N)之间的自然数操作加 Z(+)、乘法 Z(*)、除法 Z(/)，也即
ASS(INS(N),{Z(+), Z(*), Z(/)} )，作用在自然数类型 N的连续实例 INS(N)的整体
{INS(N)}上不改变其中任何一个类型的本质语义关系 ASS(CES(EXCS(N)))的存在语

ASS(CES(EXCS(N)))
:=ASS(EXCR(CES(N)))
:=CES(ASS(EXCS(N)))
:=CES(EXCS(ASS(N)))
:=EXCS(CES(ASS(N)))

Z(*)、除法 Z(/)之后获得的边界 bound(ASS(n, n+1))会与 ASS(n-1, n)以及 ASS(n+1,
n+2)对应的边界 bound(ASS(n-1, n))以及 bound(ASS(n+1, n+2))相邻。这个相邻语义

1)=>bound(ASS({n*3+1, n/2}, 1))。由存在语义继承公理(IHES, Inheritance of
Existence Semantics)基于自然数类型 N与奇数类型 O与偶数类型 E整体之间的

EX(N)=>EX(E)代入 ASS({n*3+1, n/2}, 1)=>bound(ASS({n*3+1, n/2}, 1))
，即可得到 Collatz Conjecture 的有界语义 bound(ASS({O*3+1, E/2}, 1)):
ASS({n*3+1, n/2}, 1)=>bound(ASS({n*3+1, n/2}, 1))
=>
ASS({O*3+1, E/2}, 1)=>bound(ASS({O*3+1, E/2}, 1))

References:
(1) Yucong Duan: Towards a Periodic Table of conceptualization and formalization
on State, Style, Structure, Pattern, Framework, Architecture, Service and so on.
SNPD 2019: 133-138
(2) Yucong Duan: Existence Computation: Revelation on Entity vs. Relationship for
Relationship Defined Everything of Semantics. SNPD 2019: 139-144
(3) Yucong Duan: Applications of Relationship Defined Everything of Semantics on
Existence Computation. SNPD 2019: 184-189
(4) Yucong Duan, Xiaobing Sun, Haoyang Che, Chunjie Cao, Zhao Li, Xiaoxian Yang:
Modeling Data, Information and Knowledge for Security Protection of Hybrid
IoT and Edge Resources. IEEE Access 7: 99161-99176 (2019)
(5) 段玉聪等,跨界、DIKW 模态、介尺度内容主客观语义融合建模与处理研
.中国科技成果，2021 8498 期，45-48.
(6) Y. Duan, "Semantic Oriented Algorithm Design: A Case of Median Selection,"
2018 19th IEEE/ACIS International Conference on Software Engineering,
Artificial Intelligence, Networking and Parallel/Distributed Computing (SNPD),
2018, pp. 307-311, doi: 10.1109/SNPD.2018.8441053.
(7) Y. Duan, "A Constructive Semantics Revelation for Applying the Four Color
Problem on Modeling," 2010 Second International Conference on Computer
Modeling and Simulation, 2010, pp. 146-150, doi: 10.1109/ICCMS.2010.113.
(8) Yucong Duan, A dualism based semantics formalization mechanism for model
driven engineering, IJSSCI, vol. 1, no. 4, pp. 90-110, 2009.
(9) Yucong Duan, "Efficiency from Formalization: An Initial Case Study on Archi3D"
in Studies of Computing Intelligence, Springer, 2009.
(10) Yucong Duan, "Creation Ontology with Completeness for Identification of 3D
Architectural Objects" in ICCTD, IEEE CS press, pp. 447-455, 2009.
(11) Y. Huang and Y. Duan, "Towards Purpose Driven Content Interaction Modeling
and Processing based on DIKW," 2021 IEEE World Congress on Services
(SERVICES), 2021, pp. 27-32, doi: 10.1109/SERVICES51467.2021.00032.
(12) T. Hu and Y. Duan, "Modeling and Measuring for Emotion Communication
based on DIKW," 2021 IEEE World Congress on Services (SERVICES), 2021, pp.
21-26, doi: 10.1109/SERVICES51467.2021.00031.
(13) Duan Yucong, Christophe Cruz. Formalizing Semantic of Natural Language
through Conceptualization from Existence. International Journal of Innovation,
anagement and Technology, 2011, 2 (1), p. 37-42, ISSN: 2010-0248.
ffhal-00625002
(14) Y. Duan, "A stochastic revelation on the deterministic morphological change of
3x+1," 2017 IEEE 15th International Conference on Software Engineering
Research, Management and Applications (SERA), 2017, pp. 333-338, doi:
10.1109/SERA.2017.7965748.
(15) Yucong Duan, The end of "Objective" mathematics as a return to "Subjective".
February 2022.DOI:
10.13140/RG.2.2.36171.87841.https://www.researchgate.net/publication/
358 607773_The_end_of_Objective
_mathematics_as_a_return_to_Subjective/stats
(16) Yucong Duan, Existence Computation and Reasoning(EXCR) and Essence
Computation and Reasoning(ESCR) based revelation of the semantics of point,
line and plane. February 2022.
https://www.researchgate.net/publication/358608122_Existence_Computatio
n_and_ReasoningEXCR_and_Essence_Computation_and_ReasoningESCR_
base d_revelation_of_the_semantics_of_point_line_and_plane
(17) Yucong Duan, Existence Computation and Reasoning(EXCR) and Essence
Computation and Reasoning(ESCR) based Revelation of the Four Color
Theorem. February 2022.
https://www.researchgate.net/publication/358608147_Existence_Computatio
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base d_Revelation_of_the_Four_Color_Theorem
(18) Yucong Duan, Existence Computation and Reasoning(EXCR) and Essence
Computation and Reasoning(ESCR) based Revelation of the Goldbach's
conjecture. February 2022.
https://www.researchgate.net/publication/358637942_Existence_Computatio
n_and_ReasoningEXCR_and_Essence_Computation_and_ReasoningESCR_base
d_Revelation_of_the_Goldbach's_conjecture
(19) Yucong Duan, Identifying Objective True/False from Subjective Yes/No
Semantic based on OWA and CWA. July 2013. Journal of Computers 8(7)DOI:
10.4304/jcp.8.7.1847-1852.https://www.researchgate.net/publication/276240
420_Identifying_Objective_
TrueFalse_from_Subjective_YesNo_Semantic_based_on_OWA_and_CWA/c
itations
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Article
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