Geodesic
The spatial logic of the conflict of convex sets
Nečetkie sistemy i mâgkie vyčisleniâ, Tome 18 (2023) no. 1, pp. 63-110

Voir la notice de l'article provenant de la source Math-Net.Ru

The article tests methods of logical branching of propositions of the theory of convex sets. The presented work is based on the concept of the Aristotelian Categorical Syllogism (ACS), which performs multiple entry into the result of logical inference using the conjunction -AND- . Formal inverting of the implications that make up ACS allows interpreting it to perform disjunctive branching of the predicate at the exit from it. As a result, we have a logical decomposition of the predicate into the following statements from it, grouped around the union -OR- . To complete the picture, we add two more types of branching: disjunctive entry and conjunctive exit, which are not difficult for understanding and applying. In mathematics, problems concentrate around ACS branches. Four types of logical branching in combination with Transfer of Control (ToC – same as GoTo) are demonstrated by solving the problem of splitting a vector space into two disjoint Convex Sets. It is shown that the intersection of their closures, hereinafter referred to as the «crack» of the vector space, is a hyperplane.
Mots-clés : syllogism, spatial algorithm, implication, quantifier, simplex, partition
Keywords: analysis, synthesis, logical scheme, block diagram, predicate, logical inference, algorithm, branching, hyperplane, semispace, linear manifold, convex set, local interior, projective independence, symmetry, Voronoi diagram. 
L. A. Pomortsev. The spatial logic of the conflict of convex sets. Nečetkie sistemy i mâgkie vyčisleniâ, Tome 18 (2023) no. 1, pp. 63-110. http://geodesic.mathdoc.fr/item/FSSC_2023_18_1_a3/
@article{FSSC_2023_18_1_a3,
     author = {L. A. Pomortsev},
     title = {The spatial logic of the conflict of convex sets},
     journal = {Ne\v{c}etkie sistemy i m\^agkie vy\v{c}isleni\^a},
     pages = {63--110},
     year = {2023},
     volume = {18},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FSSC_2023_18_1_a3/}
}
TY  - JOUR
AU  - L. A. Pomortsev
TI  - The spatial logic of the conflict of convex sets
JO  - Nečetkie sistemy i mâgkie vyčisleniâ
PY  - 2023
SP  - 63
EP  - 110
VL  - 18
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/FSSC_2023_18_1_a3/
LA  - ru
ID  - FSSC_2023_18_1_a3
ER  - 
%0 Journal Article
%A L. A. Pomortsev
%T The spatial logic of the conflict of convex sets
%J Nečetkie sistemy i mâgkie vyčisleniâ
%D 2023
%P 63-110
%V 18
%N 1
%U http://geodesic.mathdoc.fr/item/FSSC_2023_18_1_a3/
%G ru
%F FSSC_2023_18_1_a3

[1] Gilbert D., Bernajs P., Foundations of mathematics. Logical calculus and formalization of arithmetic, Nauka Publ., Moscow, 1979 (in Russian)

[2] Gilbert D., Bernajs P., Foundations of mathematics. Proof theory, Nauka Publ., Moscow, 1982 (in Russian)

[3] Pomortsev L. A., “Calculation of moments of ordinal statistics of consecutive sums of symmetrically dependent random variables”, Proceedings of the Moscow Mathematical Society, 46 (1983), 201–242 (in Russian) | MR | Zbl

[4] Pomortsev L. A., “Combinatorial-symmetric analysis of multidimensional random walks”, Discrete mathematics, 3:1 (1991), 21–41 (in Russian) | MR | Zbl

[5] Pomortsev L. A., “Algebraic interpretation of the completeness of the axioms of inference”, Fundamental and Applied Mathematics, 8:1 (2002), 195–219 (in Russian) | MR | Zbl

[6] Pomortsev L. A., “Functional dependency algebras in relational database theory”, Proceedings of the Institute of System Analysis of the Russian Academy of Sciences. Dynamics of inhomogeneous systems, 29:1 (2007), 169–183 (in Russian)

[7] Pomortsev L. A., “Systems Verbalization”, Global Journal of Computer Science and Technology (G), 19:4 (2019)

[8] Pomortsev L. A., Tsurkov V. I., “Algebraization of inferring functional dependences in relational databases”, Journal of Computer and Systems Sciences International, 58:2 (2019), 212–228 | DOI | DOI | MR | Zbl

[9] Pomortsev L. A., Tsurkov V. I., “SYNTHESIS AND ANALYSIS OF sequences of output of functional dependencies of tables”, Journal of Computer and Systems Sciences International, 6 (2020), 152–176 | DOI | MR | Zbl

[10] Pomortsev L. A., Tsurkov V. I., “Order alteration in a cascade-ordered set”, Fuzzy Systems and Soft Computing, 15:2 (2020), 96–115 (in Russian) | DOI

[11] Pomortsev L. A., Tsurkov V. I., “Hygiene of derivation sequences”, Fuzzy Systems and Soft Computing, 16:1 (2021), 34–57 (in Russian) | DOI

[12] Pomortsev L. A., “Algebraic apparatus of a Sequences of the Derivation”, Fuzzy Systems and Soft Computing, 16:2 (2021), 96–110 (in Russian) | DOI

[13] Pontryagin L. S., Fundamentals of combinatorial topology, Nauka Publ., Moscow (in Russian) | MR

[14] Kuratovskij K., Topology, v. 1, Mir Publ., Moscow, 1966 (in Russian)

[15] Elster K. Kh., Rejngardt R., Shojble M., Donat G., Introduction to Nonlinear Programming, Nauka Publ., Moscow, 1985 (in Russian)

[16] Leng S., Algebra, Mir Publ., Moscow, 1968 (in Russian)

[17] Mendelson E., Vvedenie v matematicheskuyu logiku, Nauka Publ., Moscow, 1971 (in Russian)

[18] Yablonskij S. V., Introduction to Discrete Mathematics, Nauka Publ., Moscow, 1986 (in Russian) | MR