The complexity of concatenation on deterministic and alternating finite automata
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 52 (2018) no. 2-3-4, pp. 153-168.

We study the state complexity of the concatenation operation on regular languages represented by deterministic and alternating finite automata. For deterministic automata, we show that the upper bound m 2 n - k 2 n - 1 on the state complexity of concatenation can be met by ternary languages, the first of which is accepted by an m -state DFA with k final states, and the second one by an n -state DFA with final states for arbitrary integers m , n , k , with 1 k m 1 and 1 n 1 . In the case of k m 2 , we are able to provide appropriate binary witnesses. In the case of k = m 1 and 2 , we provide a lower bound which is smaller than the upper bound just by one. We use our binary witnesses for concatenation on deterministic automata to describe binary languages meeting the upper bound 2 m + n + 1 for the concatenation on alternating finite automata. This solves an open problem stated by Fellah 𝑒𝑡 𝑎𝑙 . [ 𝐼𝑛𝑡 . 𝐽 . 𝐶𝑜𝑚𝑝𝑢𝑡 . 𝑀𝑎𝑡ℎ . 35 (1990) 117–132].

DOI : 10.1051/ita/2018011
Classification : 68Q19, 68Q45
Mots clés : Regular languages, finite automata, concatenation, state complexity
Hospodár, Michal 1 ; Jirásková, Galina 1

1
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Hospodár, Michal; Jirásková, Galina. The complexity of concatenation on deterministic and alternating finite automata. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 52 (2018) no. 2-3-4, pp. 153-168. doi : 10.1051/ita/2018011. http://archive.numdam.org/articles/10.1051/ita/2018011/

[1] J. Brzozowski and E. Leiss, On equations for regular languages, finite automata, and sequential networks. Theoret. Comput. Sci. 10 (1980) 19–35 | DOI | MR | Zbl

[2] E. Dorčák, Concatenation of regular languages and state complexity, ŠVK thesis. P. J. Šafárik University, Košice (2015)

[3] A. Fellah, H. Jürgensen, and S. Yu, Constructions for alternating finite automata. Int. J. Comput. Math. 35 (1990) 117–132 | DOI | Zbl

[4] J.E. Hopcroft, and J.D. Ullman, Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979) | MR | Zbl

[5] J. Jirásek, G. Jirásková, and A. Szabari, State complexity of concatenation and complementation. Int. J. Found. Comput. Sci. 16 (2005) 511–529 | DOI | MR | Zbl

[6] G. Jirásková, State complexity of some operations on binary regular languages. Theoret. Comput. Sci. 330 (2005) 287–298 | DOI | MR | Zbl

[7] G. Jirásková, Descriptional complexity of operations on alternating and boolean automata. In CSR 2012, edited by E. Hirsch et al. Vol. 7353 of Lect. Notes Comput Sci. Springer (2012) 196–204 | MR | Zbl

[8] E. Leiss, Succinct representation of regular languages by boolean automata. Theoret. Comput. Sci. 13 (1981) 323–330 | DOI | MR | Zbl

[9] E. Leiss, On generalized language equations. Theoret. Comput. Sci. 14 (1981) 63–77 | DOI | MR | Zbl

[10] A.N. Maslov, Estimates of the number of states of finite automata. Soviet Math. Doklady 11 (1970) 1373–1375 | MR | Zbl

[11] M. Rabin and D. Scott, Finite automata and their decision problems. IBM Res. Develop. 3 (1959) 114–129 | DOI | MR | Zbl

[12] M. Sipser, Introduction to the theory of computation. Cengage Learning, Boston (2012)

[13] S. Yu, Regular languages, Chapter 2. In vol. I of Handbook of Formal Languages, edited by G. Rozenberg and A. Salomaa. Springer Heidelberg (1997) 41–110 | DOI | MR | Zbl

[14] S. Yu, Q. Zhuang, and K. Salomaa, The state complexity of some basic operations on regular languages. Theoret. Comput. Sci. 125 (1994) 315–328 | DOI | MR | Zbl

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