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New types of automata to solve fixed point problems. (English) Zbl 0973.68122

Summary: The aim of this article is to find the supremum of \(\{\text{X}\mid \text{AX}\subseteq \text{X}\cup \text{B}\) and \(\text{X}\subseteq \text{K}\}\), where A, B and K are three fixed, rational languages. This problem and its solution are formally inspired by retroaction in control theory. To get the solution, we introduce a new type of automata whose acceptance condition is a positive boolean formula. The time complexity of our algorithm depends linearly on the size of A, quadratically on the size of K and exponentially on the size of B, where the size is measured by the number of states of the minimal automaton.

MSC:

68Q45 Formal languages and automata
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[1] J. Berstel, C. Reutenauer, Les séries rationnelles et leurs langages, Masson, Paris, 1984 (English translation: Rational Series and Their Languages, Springer, Berlin, 1988).; J. Berstel, C. Reutenauer, Les séries rationnelles et leurs langages, Masson, Paris, 1984 (English translation: Rational Series and Their Languages, Springer, Berlin, 1988). · Zbl 0573.68037
[2] Blyth, T. S.; Janowitz, M. F., Residuation Theory (1972), Pergamon Press: Pergamon Press Oxford · Zbl 0301.06001
[3] Eilenberg, S., Automata, Languages and Machines, vol. A (1974), Academic Press: Academic Press New York · Zbl 0317.94045
[4] Engel, K., Sperner Theory (1997), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0868.05001
[5] Hopcroft, J.; Ullman, J., Introduction to Automata Theory, Languages, and Computation (1979), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0426.68001
[6] Kari, L., On language equations with invertible operations, Theoret. Comput. Sci., 132, 129-150 (1994) · Zbl 0821.68075
[7] Kari, L.; Thierrin, G., Maximal and minimal solutions to language equations, J. Comput. System Sci., 53, 487-496 (1996) · Zbl 0870.68094
[8] Leiss, E., On generalized language equations, Theoret. Comput. Sci., 14, 63-77 (1981) · Zbl 0454.68105
[9] Thomas, W., Languages, automata, and logic, (Rozenberg, G.; Salomaa, A., Handbook of Formal Language Theory, vol. III (1997), Springer: Springer New York), 389-455
[10] Wonham, W. M., Linear Multivariable Control - A Geometric Approach (1985), Springer: Springer Berlin · Zbl 0393.93024
[11] Wonham, W. M.; Ramadge, P. J., On the supremal controllable sublanguage of a given language, SIAM J. Control Optim., 25, 3, 637-659 (1987)
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