For citation:
Speranskiy D. V. Reversion of outputs of fuzzy discrete systems. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2022, vol. 22, iss. 1, pp. 112-122. DOI: 10.18500/1816-9791-2022-22-1-112-122, EDN: VMOPOL
Reversion of outputs of fuzzy discrete systems
The problem of reversion of unknown input sequences of fuzzy discrete systems by its observed outputs is investigated. As a mathematical model of fuzzy systems the fuzzy automata (FA) is used. This problem has been considered earlier for deterministic systems. Unambiguous solutions of the problem for such systems have been obtained using the model of finite automata, called the information lossless automata (IL-automata). In the article, for fuzzy discrete systems described by the FA model a similar problem is considered. Due to the specifics of functioning of such systems, unambiguous decoding of messages coming to their inputs is not always possible in principle. For this reason, there are problems of minimization of information lossless (according to various criteria) while solving the address problem. Automata are introduced, which allow solving such problems, called automata with minimized information lossless (FA MIL-automata). Solution of the problem of reversion for FA is a finite set of input words. Each such solution can be estimated according to various criteria — the cardinality of a set of words of the solution, the probability of appearance of these words on the system inputs, the complexity of obtaining different variants of the solutions. In order to minimize information lossless, the article formulates corresponding optimization tasks for FA and specifies possible ways of solving them. Different kinds of FA MIL-automata are considered. The obtained results show that the considered problems of reversion for fuzzy automata inputs are multi-criteria. It is known that solutions of such problems for discrete systems are traditionally evaluated by only one criterion.
- Kalman R., Falb P., Arbib M. Topics in Mathematical System Theory. New York, McGraw-Hill, 1969. 358 p. (Russ. ed.: Moscow, Editorial URSS, 2004. 400 p.).
- Ilyin A. V., Korovin S. K., Fomichev I. I. Metody robastnogo upravleniya dinamicheskikh sistem [Methods of Robust Control of Dynamical Systems]. Мoscow, Fizmatlit, 2009. 219 p. (in Russian).
- Eilenberg S. Automata, Languages and Machines. New York, London, Academy Press, 1974. 387 p.
- Коvalev F. М., Коzlovsky V. А., Scherbak V. F. Reversible dynamical systems of variable dimension in problems of cryptographic transformation of information. Applied Discrete Mathematic, 2008, no. 2, pp. 39–44 (in Russian).
- Pushkov S. G. Inversion of linear systems on the basis of state space realization. Journal of Computer and Systems Sciences International, 2018, vol. 57, pp. 7–17. https://doi.org/10.1134/S1064230717050094
- Gill А. Introduction to the Theory of Finite-State Machines. New York, McGrow-Hill, 1962. 218 p.
- Zadeh L. A. Fuzzy sets. Information and Control, 1965, vol. 8, iss. 3, pp. 338–353.
- Dubois D., Prade H. Fuzzy Sets and Systems: Theory and Applications. New York, Academy Press, 1980. 236 p.
- Speranskiy D. V. Experiments with fuzzy finite state machines. Automation and Remote Control, 2015, vol. 76, pp. 278–291. https://doi.org/10.1134/S0005117915020071
- Huffman D. A. Canonical forms for information-lossless finite-state logical machines. IRE Transactions on Circuit Theory, 1959, vol. 6, no. 5, pp. 41–59. https://doi.org/10.1109/TCT.1959.1086614
- Even S. On information lossless automata of finite order. IEEE Transactions on Electronic Computers, 1965, vol. EC-14, iss. 4, pp. 561–569. https://doi.org/10.1109/PGEC.1965.263996
- Speranskiy D. V. Lektsii po teorii experimentov s konechnymi avtomatami [Lectures on the Theory of Experiments with Finite Automata]. Мoscow, Internet-Universitet Informatsionnykh Tekhnologiy, 2010. 287 p. (in Russian).
- Bogomolov A. M., Grunsky I. S., Speransky D. V. Kontrol’ i preobrazovaniya diskretnykh avtomatov [Control and Transformation of Discrete Automata]. Kiev, Naukova dumka, 1975. 176 p. (in Russian).
- Kini R. L., Raifa H. Prinyatie resheniy pri mnogikh kriterijakh: predpochteniya i zameshcheniya [Decision-Making under Many Criteria: Preferences and Substitution]. Moscow, Radio i svyaz’, 1981. 347 p. (in Russian).
- Steuer R. E. Multiple Criteria Optimization: Theory, Computation and Application. New York, John Willey and Sons, Inc, 1986. 546 p.
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