The state complexities of some basic operations on regular languages. (English) Zbl 0795.68112

Summary: We consider the state complexities of some basic operations on regular languages. We show that the number of states that is sufficiently and necessary in the worst case for a deterministic finite automaton (DFA) to accept the catenation of an \(m\)-state DFA language and an \(n\)-state DFA language is exactly \(m2^ n- 2^{n-1}\), for \(m,n\geq 1\). The result of \(2^{n-1}+ 2^{n-2}\) states is obtained for the star of an \(n\)-state DFA language, \(n>1\). State complexities for other basic operations and for regular languages over a one-letter alphabet are also studied.


68Q45 Formal languages and automata
68Q25 Analysis of algorithms and problem complexity
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[1] Chandra, A. K.; Kozen, D. C.; Stockmeyer, L. J., Alternation, J. ACM, 28, 114-133 (1981) · Zbl 0473.68043
[2] Fellah, A.; Jürgensen, H.; Yu, S., Constructions for alternating finite automata, Internat. J. Comput. Math., 35, 117-132 (1990) · Zbl 0699.68081
[3] Hopcroft, J. E.; Ullman, J. D., Introduction to Automata Theory, Languages, and Computation (1979), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0196.01701
[4] Jiang, T.; Ravikumar, B., Minimal NFA problems are hard, (Proc. 18th ICALP. Proc. 18th ICALP, Lecture Notes in Computer Science, Vol. 510 (1991), Springer: Springer Berlin), 629-640 · Zbl 0766.68063
[5] Leiss, E., Succinct representation of regular languages by boolean automata, Theoret. Comput., 13, 323-330 (1981) · Zbl 0458.68017
[6] Meyer, A. R.; Fischer, M. J., Economy of description by automata, grammars, and formal systems, FOCS, 12, 188-191 (1971)
[7] Ravikumar, B., Some applications of a technique of Sakoda and Sipser, SIGACT News, 21, 4, 73-77 (1990)
[8] Ravikumar, B.; Ibarra, O. H., Relating the type of ambiguity of finite automata to the succinctness of their representation, SIAM J. Comput., 18, 6, 1263-1282 (1989) · Zbl 0692.68049
[9] Salomaa, A., Theory of Automata (1969), Pergamon: Pergamon Oxford · Zbl 0193.32901
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