Automation and Remote Control

, Volume 62, Issue 10, pp 1743–1755

Generalizations of the Continuous Logic

• V. I. Levin
Article

Abstract

An order logic is studied, i.e., a generalization of the continuous logic to the case in an arbitrary argument of order r is determined instead of the operations of determination of maximum (disjunction) and minimum (conjunction). The new operation is expressed as a superposition of disjunctions and conjunctions of the continuous logic. Different classes of logical determinants—numeric characteristics of matrices that can be expressed through the operations of the continuous logic—are studied. Order determinants that are the generalizations of order logic operations to arguments in matrix form are studied. Determinants with different types of constraints on matrix subsets defining the matrix characteristic are described. For logical determinants, the properties that partly resemble the properties of algebraic determinants and computation formulas based on the operations of continuous logic are described. A predicate decision algebra generalizing the continuous logic to modeling of discontinuous functions is elaborated. A hybrid logic algebra is generalized to hybrid (continuous and discrete) variables. A logical arithmetic algebra, which includes continuous logical operations along with the four arithmetical operations, is described. A complex logic algebra in which the carrier set C is a field of complex numbers is developed. For all these logical algebras, main laws are formulated and their similarity to and distinction from the laws of the continuous logic are described. Generalizations of continuous logic operations as operations over matrices and random and interval variables are investigated. Their fields of application are described.

Keywords

Arithmetical Operation Interval Variable Logic Operation Numeric Characteristic Order Logic
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

REFERENCES

1. 1.
Levin, V.I., Dinamika logicheskikh ustroistv i sistem (Dynamics of Logical Devices and Systems), Moscow: Energiya, 1980.Google Scholar
2. 2.
Levin, V.I., Beskonechnoznachnaya logika v zadachakh kibernetiki (Infinite-Valued Logic In Cybernetical Problems), Moscow: Radio i Svyaz', 1982.Google Scholar
3. 3.
Levin, V.I., Strukturno-logicheskie metody issledovaniya slozhnykh sistem s primeneniem EVM (Computerized Structural and Logical Research Methods for Complex Systems), Moscow: Nauka, 1987.Google Scholar
4. 4.
Volgin, L.I., Sintez ustroistv glya obrabotki i preobrozovaniya informatsii v elementnom bazise relyatorov (Synthesis of Devices for Information Processing and Transformation in Relator Elemental Base), Tallinn: Valgus, 1989.Google Scholar
5. 5.
Shimbirev, P.N., Gibridnye nepreryvno-logicheskie ustroistva (Hybrid Continuous Logic Devices), Moscow: Energoatomizdat, 1990.Google Scholar
6. 6.
Volgin, L.I. and Levin, V.I., Nepreryvnaya logika. Teoriya i primeneniya (Continuous Logic: Theory and Application), Tallinn: Akad. Nauk Estonii, 1989.Google Scholar
7. 7.
Levin, V.I., Continuous Logic: Its Generalizations and Applications. I, II, Avtom. Telemekh., 1990, no. 8.Google Scholar
8. 8.
Levin, V.I., Nondeterministic Infinite-Valued Logic, Kibern. Sist. Analiz, 1992, no. 3, pp. 51–63.Google Scholar
9. 9.
I Vseros. nauch. konf.Nepreryvnaya logika i ee primenenie v tekhnike, ekonomike i sotsiologii. Tez. dokl. (Proc. I Russian “Conf. Continuous Logic and Its Application in Engineering, Economics, and Sociology”), Levin, V.I., Ed., Penza: Penz. Dom Nauch.-Tekhn. Propagandy, 1994.Google Scholar
10. 10.
Nepreryvno-logicheskie metody i modeli v nauke, tekhnike i ekonomike.” Mater. mezhdynar. nauch.-tekhn. konf. (Proc. Int. Conf. “Continuous Logic Methods and Models in Science, Technology, and Economics”), Levin, V.I., Ed., Penza: Privolzh. Dom Znanii, 1995.Google Scholar
11. 11.
Nepreryvnaya i smezhnye logiki v tekhnike, ekonomike i sotsiologii.Mater. mezhdunar. konf. (Proc. Int. Conf. “Continuous and Mixed Logics in Technology, Economics, and Sociology”), Levin, V.I., Ed., Penza: Privolzh. Dom Znanii, 1996.Google Scholar
12. 12.
Neprebyvno-logicheskie sistemy, modeli i algoritmy.” Tr. mezhdunar. nauch.-tekhn. konf. (Proc. Int. Conf. “Continuous Logical Systems, Models, and Algorithms”), Levin, V.I., Ed., Ul'yanovsk: Ul'yan. Gos. Tekhn. Univ., 1995.Google Scholar
13. 13.
Vseros. nauch.-tekhn. konf. “Nepreryvnaya i smezhnye logiki v informatike, ekonomike i sotsiologii.” Mater. konf. (All Russian Conf. “Continuous and Mixed Logics in Informatics, Economics, and Sociology”), Levin, V.I., Ed., Penza: Privolzh. Dom Znanii, 1997.Google Scholar
14. 14.
Volgin, L.I., Relator and Continuous Logical Networks and Models, Tr. mezhdunar. konf. “Neironnye, relyatornye i nepreryvnologicheskie seti i modeli” (Proc. Int. Conf. “Neuron, Relator, and Continuous-Logic Networks and Models”), Ul'yanovsk: Ul'yan. Gos. Tekhn. Univ., 1998.Google Scholar
15. 15.
Logiko-matematicheskie metody v tekhnike, ekonomike i sotsiologii.” Mater. mezhdunar. nauch.-tekhn. konf. (Proc. Int. Conf. “Logical and Mathematical Methods in Technology, Economics, and Sociology”), Levin, V.I., Ed., Penza: Privolzh. Dom Znanii, 1998.Google Scholar