Existence Computation and Reasoning(EXCR)

Yucong Duan

Benefactor: Shiming Gong

AGI-AIGC-GPT Evaluation DIKWP (Global) Laboratory

DIKWP-AC Artificial Consciousness Standardization Committee

World Conference on Artificial Consciousness

World Artificial Consciousness Association

(Email：duanyucong@hotmail.com)

Catalog

1 Key concepts

2 The Implementation Process of EXCR

3 Specific case analysis

3.1 The EXCR Implementation of Collatz Conjecture

3.2 The EXCR Implementation of Goldbach Conjecture

4 Conclusion

References

Existence Computing and Reasoning (EXCR) is a method of revealing the essence and semantic relationships of complex mathematical and logical problems through semantic analysis and reasoning. EXCR is based on a series of semantic axioms and reasoning rules, aiming to provide new explanations for mathematical conjectures and problems from the perspectives of cognition and semantic space.

1 Key concepts

Conservation of Existence Set Axiom, CEX:

Any type of existence must remain conserved and will not disappear or arise out of thin air.

Mathematical expression:

EXCR(A):=EXCR(B)=>EX(A)=>EX(B)

Consistency of Compounded Essential Set Axiom, CES:

Any type of essential set must maintain consistency and integrity when combined.

Mathematical expression:

CES(ASS(A)):=ASS(CES(A))

Inheritance of Existence Semantics Axiom, IHES:

In the process of semantic processing at the pure type level, for target A and target B that have semantic dependency or equivalence relationships, target A inherits or retains all existing semantics of target B.

Mathematical expression:

ASS(EXCR(A):=EXCR(B),EXCR(A)=>EXCR(B))=>EX(B)=>EX(A)

2 The Implementation Process of EXCR

Semantic interpretation at the instance level:

Decompose complex problems into specific instances and infer through semantic associations between instances.

For example, in the Collatz conjecture, for any natural number N, the semantic explanation that N will eventually return to 1 is derived through examples of odd O and even E.

Semantic interpretation at the type level:

Analyze the problem from the overall semantic level of types and reveal the semantic equivalence relationship between different types.

For example, in the Collatz conjecture, the semantic equivalence between natural numbers N, odd numbers O, and even numbers E at the type level is derived.

Semantic reasoning and computation:

By applying the axioms of EXCR and IHES, the essence of the problem is revealed through semantic reasoning and computation.

For example, by using the Axiom of Existence Computing and Reasoning (EXCR) and the Axiom of Existence Semantic Inheritance (IHES), we derive the boundedness of the Collatz conjecture and prove that it will eventually return to 1.

3 Specific case analysis

3.1 The EXCR Implementation of Collatz Conjecture

The definition of the Collatz conjecture:

For any positive integer n, if n is odd, calculate 3n+1; If n is an even number, calculate n/2. This loop will eventually return to 1.

At the instance level:

For any instance of natural number N, INS (N)=n, which is either an odd number O instance INS (O)=o, or an even number E instance INS (E)=e.

When n is an odd number o, multiply it by 3 and then add 1 to obtain n:=3o+1; When n is an even number e, divide it by 2 to obtain n:=e/2. In this loop, we can ultimately obtain n=1.

Mathematical expression:

INS(N):=ASS(INS(O),INS(E),REL(+),REL(/))

:=ASS(INS(O)∗3+1,INS(E)/2):=ASS(o∗3+1,e/2,1):=ASS(n∗3+1,n/2,1)=>n→1

At the type level:

From the overall semantic level of a type, any instance of a natural number N, TYPE (INS (N))=n, can be associated by cross type odd O or even E instance level semantic INS (N):=ASS (INS (O), INS (E), REL (+), REL (/)) on the basis of confirming its own existence semantics.

Based on the basic assumption CEX of EXCR, the corresponding semantic association TYPE (N) at the type level is equivalently derived:=ASS (TYPE (O), TYPE (E)).

Mathematical expression:

INS(N):=ASS(INS(O),INS(E),REL(+),REL(/))

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

The type-level semantic association N(E):=N(O)+1 equivalently infers the existence of semantic equivalence between type-level odd types O and even types E, based on the assumption CEX underlying EXCR.

N(E):=N(O)+1=>EXCR(TYPE(O)):=EXCR(TYPE(E))=>EXCR(O):=EXCR(E)

Derivation results.

By EXCR and IHES, the boundedness of Collatz's conjecture is deduced, i.e., all operations eventually converge to 1.

Mathematical expression:

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))

3.2 The EXCR Implementation of Goldbach Conjecture

The definition of Goldbach's conjecture:

Any even number greater than 2 can be represented as the sum of two prime numbers.

At the instance level:

The even number E instance INS (E)=e can be represented by the sum of instances of two prime numbers P.

Mathematical expression:

INS(E):=ASS(INS(P),INS(P),REL(+)):=INS(P)+INS(P)=>e=p₁+p₂

At the type level:

From the overall semantic level of the type, the instance TYPE (INS (E))=E of even number E can infer the semantic relationship at the type level through its association with natural number type Z, based on the confirmation of its existence semantics.

Mathematical expression:

TYPE(E):=ASS(TYPE(Z),REL(+))=>E:=ASS(Z,REL(+))=>INS(E):=INS(Z)+INS(Z)

Derived results:

Derive the semantic interpretation of Goldbach conjecture through EXCR and ESCR.

Mathematical expression:

ASS(P,Z):=ASS(P,TYPE(Z)):=ASS(P,E)=>P+P=E

4 Conclusion

EXCR explains and derives complex mathematical and logical problems from the perspective of cognitive and semantic space through a series of semantic axioms and reasoning rules, providing a new perspective to understand the nature of these problems.EXCR is not only applicable to Collatz's conjecture, but can also be applied to other mathematical and logical problems, providing new methods and ideas for the study of these problems.

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、跨 DIKW 模态、介尺度内容主客观语义融合建模与处理研究." 中国科技成果，2021年8月498期，45-48.