Analysis of behaviour of automata

V. B. Kudryavtsev ; I. S. Grunskii ; V. A. Kozlovskii

Diskretnaya Matematika, Tome 21 (2009) no. 1, pp. 3-35.

Résumé

This survey contains results concerning behavioural and abstract theory of finite automata. Finite automata and related constructions are cornerstone concepts of discrete mathematics and mathematical cybernetics. The automata theory finds multiple applications in programming, technical cybernetics, theory of dynamic systems, etc. Classical problems of this theory are the direct problems: analysis of processes of transformation of information carried out by automata and properties of automata, and the inverse problems: synthesis of automata of prescribed properties and identification (restoring, recognition, deciphering, control and diagnosis) of an automaton by means of an experiment with it. The fundamental notion which underlies the study of the above problems is the notion of a fragment which is a natural extension of a great body of known fragments of particular forms possessing their characteristic properties. We introduce the fragment?automaton relation, consider properties of classes of fragments of a given automaton. We introduce the notion of a cofragment (forbidden fragment) of an automaton, consider properties of classes of automata which have a given fragment?cofragment pair. Another key notion is the notion of an identifier of unobservable components of functioning of an automaton, that is, of a fragment which allows for unambiguous determination of values of these components. It is demonstrated that identifiers provide us with a powerful tool for solving the problems we consider in this paper. Next, we introduce the general notion of representation of the reference automaton to within a given precision (similarity) relative to a priori class of automata as a pair (fragment, cofragment) of the reference automaton which can be a pair (fragment, cofragment) of an automaton in a priori class if and only if it is similar to the reference one. It is shown that this concept covers and generalises a series of particular notions (control, recognising experiments, questionnaire languages, $k$-sets, etc.) known in the automata theory. We suggest a natural classification of representations. We carry out a systematic study of conditions for existence and structure of representations. We obtain precise conditions for existence of representations of general form and their particular classes in terms of relations between properties of a priori class, class of automata similar to the reference one, and the class of automata indistinguishable from the reference automaton for the corresponding indistinguishability relation. We obtain conditions for existence of nontrivial representations for various classes, including questionnaire languages and control experiments. This survey contains a series of final results, but they can be treated as a regular step in studying problems of analysis and synthesis of automata by their behaviour, which are continuously filled with new content and require additional resources to solve them.

Bibliographie

[1] Kon P., Universalnaya algebra, Mir, Moskva, 1968 | MR

[2] Brauer R., Vvedenie v teoriyu konechnykh avtomatov, Radio i svyaz, Moskva, 1987 | MR

[3] Bogomolov A. M., Barashko A. S., Grunskii I. S., Eksperimenty s avtomatami, Naukova dumka, Kiev, 1973

[4] Bogomolov A. M., Grunskii I. S., Speranskii D. V., Kontrol i preobrazovaniya diskretnykh avtomatov, Naukova dumka, Kiev, 1975 | MR | Zbl

[5] Bogomolov A. M., Salii V. N., Algebraicheskie osnovy teorii diskretnykh sistem, Nauka, Moskva, 1997 | MR | Zbl

[6] Grunskii I. S., Kozlovskii V. A., Ponomarenko G. G., Predstavleniya konechnykh avtomatov fragmentami povedeniya, Naukova dumka, Kiev, 1990 | MR | Zbl

[7] Kudryavtsev V. B., Aleshin S. V., Podkolzin A. S., Vvedenie v teoriyu avtomatov, Nauka, Moskva, 1985 | MR | Zbl

[8] Trakhtenbrot B. A., Barzdin Ya. M., Konechnye avtomaty (povedenie i sintez), Nauka, Moskva, 1970 | MR | Zbl

[9] Kharari F., Teoriya grafov, Mir, Moskva, 1973 | MR

[10] Akho A., Khopkroft Dzh., Ulman Dzh., Postroenie i analiz vychislitelnykh algoritmov, Mir, Moskva, 1979 | MR | Zbl

[11] Grunskii I. S., Kozlovskii V. A., Sintez i identifikatsiya avtomatov, Naukova dumka, Kiev, 2004

[12] Gill A., Vvedenie v teoriyu konechnykh avtomatov, Nauka, Moskva, 1966 | MR | Zbl

[13] Glushkov V. M., “Abstraktnaya teoriya avtomatov”, Uspekhi matem. nauk, 16:5(101) (1961), 3–62 | MR | Zbl

[14] Mur E. F., “Umozritelnye eksperimenty s posledovatelnostnymi mashinami”, Avtomaty, IL, Moskva, 1956, 179–210

[15] Bhattacharyya A., Checking experiments on sequential machines, Wiley, New Delhi, 1989 | Zbl

[16] Kohavi Z., Switching and finite automata theory, McGraw–Hill, New York, 1970 | MR | Zbl

[17] Maksimenko I. K., Eksperimenty v finitno-opredelennykh metricheskikh prostranstvakh avtomatov, Avtoreferat dis. $\dots$ kand. fiz.-mat. nauk, Saratovskii gosuniversitet, Saratov, 2000

[18] Bogomolov S. A., O vosstanovlenii avtomatov po ikh sledam, Avtoreferat dis. $\dots$ kand. fiz.-mat. nauk, VTs AN SSSR, Moskva, 1986

[19] Spivak M. A., “K sintezu konechnogo avtomata po ego mnozhestvu eksperimentov”, Kibernetika, 1969, no. 5, 15–20 | MR | Zbl

[20] Buevich V. A., Kalandarishvili N. G., Tal A. A., “Ob opisanii konechnogo avtomata s pomoschyu konechnogo mnozhestva vkhod-vykhodnykh slov”, Avtomatika i telemekhanika, 1970, no. 1, 112–122 | MR | Zbl

[21] Ivanov N. N., Mikhailov G. I., Rudnev V. V., Tal A. A., Konechnye avtomaty – ekvivalentnost i povedenie, Nauka, Moskva, 1984 | MR

[22] Grunskii I. S., Analiz povedeniya konechnykh avtomatov, LGPU, Lugansk, 2003

[23] Evtushenko N. V., Lebedev A. V., Petrenko A. F., “O proveryayuschikh eksperimentakh s nedeterminirovannymi avtomatami”, Avtomatika i vychislitelnaya tekhnika, 1991, no. 6, 81–85

[24] Lukyanov B. D., “O razlichayuschikh i kontrolnykh eksperimentakh s nedeterminirovannymi avtomatami”, Kibernetika i sistemnyi analiz, 31:5 (1995), 69–76 | MR

[25] Lukyanov B. D., “Determinirovannye realizatsii nedeterminirovannykh avtomatov”, Kibernetika i sistemnyi analiz, 32:4 (1996), 34–50 | MR

[26] Skornyakov L. A., Obschaya algebra, T. 1, Nauka, Moskva, 1990 | Zbl

[27] Petrenko A. F., Avtomatnye metody analiza sovmestimosti sredstv vzaimodeistviya v otkrytykh setyakh, Avtoreferat dis. $\dots$ dokt. tekhn. nauk, IEP Latviiskoi SSR, Riga, 1988

[28] Petrenko A. F., “Eksperimenty nad protokolnymi ob'ektami”, Avtomatika i vychislitelnaya tekhnika, 1987, no. 1, 16–21

[29] Petrenko A. F., Yevtushenko N., Dssouli R., Grey-box FSM-based testing strategies, Dept. Publ. 991, Dept. IRO, Univ. de Montreal, 1994

[30] Zakharov V. N., Pospelov D. A., Khazatskii V. E., Sistemy upravleniya. Zadanie. Proektirovanie. Realizatsiya, Energiya, Moskva, 1972

[31] Kryvyi S. A., Matveeva L. E., “Formalnye metody analiza svoistv sistem”, Kibernetika i sistemnyi analiz, 39:2 (2003), 15–36 | MR | Zbl

[32] Parkhomenko P. P., Osnovy tekhnicheskoi diagnostiki, Energiya, Moskva, 1976

[33] Pradhan D. K. (ed.), Fault-tolerant computing: Theory and techniques, Vol. 1, Prentice Hall, Upper Saddle River, NJ, 1986

[34] Kornoushenko E. K., “Monitoring of logic-dynamic systems based on sampled performance data”, J. Comput. Syst. Sci. Int., 30:6 (1992), 107–117 | MR | Zbl

[35] Hennie F. C., “Fault detecting experiments for sequential circuits”, Proc. 5th Annual Symp. on Switching Circuit Theory and Logical Design, Princeton Univ., Princeton, NJ, 1964, 95–110

[36] Vasilevskii M. P., “O raspoznavanii neispravnostei avtomata”, Kibernetika, 1973, no. 4, 93–108 | MR

[37] Borodai S. Yu., Eksperimenty v effektivno zadannykh klassakh avtomatov, Avtoreferat dis. $\dots$ kand. fiz.-mat. nauk, SGU, Saratov, 1997

[38] Borodai S. Yu., “Uslovnye i bezuslovnye eksperimenty s klassami realizatsii ND-avtomatov”, Tezisy dokl. KhI Mezhd. konf. po problemam teor. kibernetiki, SVNTs, Ulyanovsk, 1996, 27–28

[39] Kuznetsov A. B., Trakhtenbrot B. A., “Issledovanie chastichno rekursivnykh operatorov sredstvami teorii berovskogo prostranstva”, Doklady AN SSSR, 105:5 (1955), 897–900 | Zbl

[40] Maksimenko I. I., “Raspoznavanie v effektivno-zadannykh klassakh avtomatov”, Trudy IPMM NAN Ukrainy, 2, 1998, 115–123 | MR

[41] Kornoushenko E. K., Diagnostirovanie diskretnykh dinamicheskikh sistem po nakoplennoi informatsii, Avtoreferat dis. ... dokt. tekhn. nauk, IPU, Moskva, 1992

[42] Kozlovskii V. A., “O raspoznavanii avtomata otnositelno lokalno porozhdennogo klassa”, Doklady AN SSSR, 258:5 (1981), 1047–1049 | MR

[43] Kozlovskii V. A., O raspoznavanii lokalnykh neispravnostei avtomata, Avtoreferat dis. $\dots$ kand. fiz.-mat. nauk, VTs AN SSSR, Moskva, 1981

[44] Kozlovskii V. A., “Lokalnye neispravnosti avtomata i ikh obnaruzhenie”, Matem. voprosy kibern., 3, 1991, 167–186 | MR | Zbl

[45] Kozlovskii V. A., Kopytova O. M., “Predstavleniya avtomatov otnositelno $m$-plotnykh klassov”, Materialy VIII Mezhd. Seminara “Diskretnaya matematika i ee prilozheniya”, MGU, Moskva, 2004, 277–280

[46] Kozlovskii V. A., “O predstavleniyakh gruppovykh avtomatov”, Kibernetika i sistemnyi analiz, 32:2 (1996), 21–28 | MR | Zbl

[47] Kozlovskii V. A., “O strukture kontrolnykh eksperimentov s avtomatom”, Kibernetika, 1978, no. 3, 19–33 | MR

[48] Kozlovskii V. A., “On the complexity of analyzing experiments for checking local faults of an automaton”, Lecture Notes Computer Sci., 278, 1987, 259–262

[49] Grunskii I. S., Kozlovskii V. A., Kopytova O. M., “Predstavleniya avtomatov i analiz atak na kriptosistemy”, Iskusstvennyi intellekt, 2004, no. 4, 764–775

[50] Evtushenko N. V., Matrosova A. Yu., “K sintezu kontroleprigodnykh avtomatnykh setei”, Tekhnicheskaya diagnostika, 1991, no. 3, 143–152 | Zbl

[51] Evtushenko N. V., Petrenko A. F., “Metod postroeniya proveryayuschikh eksperimentov dlya proizvolnogo nedeterminirovannogo avtomata”, Avtomatika i vychislitelnaya tekhnika, 1990, no. 5, 73–76

[52] Evtushenko N. V., Petrenko A. F., “O proveryayuschikh vozmozhnostyakh kratnykh eksperimentov”, Avtomatika i vychislitelnaya tekhnika, 1989, no. 3, 9–14

[53] Maksimenko I. I., “Eksperimenty v klasse realizatsii nedeterminirovannykh avtomatov”, Doklady NAN Ukrainy, 1999, no. 7, 95–99 | MR | Zbl

[54] Grunskii I. S., “Identifikatsiya upravlyayuschikh sistem avtomatnogo tipa”, Trudy ICIM' 98, TGPI, Taganrog, 1998, 107–109

[55] Grunskii I. S., Maksimenko I. I., “O raspoznavanii determinirovannykh avtomatov, zadavaemykh ND-avtomatami”, Trudy III Mezhd. Konf. “Diskretnye modeli v teorii upravlyayuschikh sistem” (Krasnovidovo), MGU, Moskva, 1998, 23–29

[56] Grunskii I. S., Maksimenko I. I., Eksperimenty s markirovannymi avtomatami, Preprint 96.02, IPMM NAN Ukrainy, Donetsk, 1998

[57] Grunskii I. S., Maksimenko I. I., “Ob eksperimentakh s avtomatami pri otsutstvii verkhnei otsenki chisla sostoyanii”, Kibernetika i sistemnyi analiz, 1999, no. 4, 59–71 | MR | Zbl