Deterministic blow-ups of minimal NFA's

Jirásková, Galina

RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 40 (2006) no. 3, pp. 485-499.

The paper treats the question whether there always exists a minimal nondeterministic finite automaton of $n$ states whose equivalent minimal deterministic finite automaton has $\alpha$ states for any integers $n$ and $\alpha$ with $n\le \alpha \le 2^n.$ Partial answers to this question were given by Iwama, Kambayashi, and Takaki (2000) and by Iwama, Matsuura, and Paterson (2003). In the present paper, the question is completely solved by presenting appropriate automata for all values of $n$ and $\alpha$. However, in order to give an explicit construction of the automata, we increase the input alphabet to exponential sizes. Then we prove that $2n$ letters would be sufficient but we describe the related automata only implicitly. In the last section, we investigate the above question for automata over binary and unary alphabets.

Classification : 68Q45, 68Q19
Mots clés : regular languages, deterministic finite automata, nondeterministic finite automata, state complexity

[1] J.C. Birget, Intersection and union of regular languages and state complexity. Inform. Process. Lett. 43 (1992) 185-190. | Zbl

[2] J.C. Birget, Partial orders on words, minimal elements of regular languages, and state complexity. Theoret. Comput. Sci. 119 (1993) 267-291. | Zbl

[3] P. Berman and A. Lingas, On the complexity of regular languages in terms of finite automata. Technical Report 304, Polish Academy of Sciences (1977). | Zbl

[4] M. Chrobak, Finite automata and unary languages. Theoret. Comput. Sci. 47 (1986) 149-158. | Zbl

[5] M. Chrobak, Errata to: “Finite automata and unary languages ” [Theoret. Comput. Sci. 47 (1986) 149-158]. Theoret. Comput. Sci. 302 (2003) 497-498. | Zbl

[6] I. Glaister and J. Shallit, A lower bound technique for the size of nondeterministic finite automata. Inform. Process. Lett. 59 (1996) 75-77. | Zbl

[7] M. Holzer and M. Kutrib, Nondeterministic descriptional complexity of regular languages. Internat. J. Found. Comput. Sci. 14 (2003) 1087-1102. | Zbl

[8] J. Hromkovič, Communication Complexity and Parallel Computing. Springer-Verlag, Berlin, Heidelberg (1997). | MR | Zbl

[9] J. Hromkovič, Descriptional complexity of finite automata: concepts and open problems. J. Autom. Lang. Comb. 7 (2002) 519-531. | Zbl

[10] J. Hromkovič, S. Seibert, J. Karhumäki, H. Klauck and G. Schnitger, Communication complexity method for measuring nondeterminism in finite automata. Inform. Comput. 172 (2002) 202-217. | Zbl

[11] K. Iwama, Y. Kambayashi and K. Takaki, Tight bounds on the number of states of DFAs that are equivalent to $n$-state NFAs. Theoret. Comput. Sci. 237 (2000) 485-494. | Zbl

[12] K. Iwama, A. Matsuura and M. Paterson, A family of NFAs which need $2^n-\alpha$ deterministic states. Theoret. Comput. Sci. 301 (2003) 451-462. | Zbl

[13] G. Jirásková, Note on minimal automata and uniform communication protocols, in Grammars and Automata for String Processing: From Mathematics and Computer Science to Biology, and Back, edited by C. Martin-Vide, V. Mitrana, Taylor and Francis, London (2003) 163-170. | Zbl

[14] G. Jirásková, State complexity of some operations on regular languages, in Proc. 5th Workshop Descriptional Complexity of Formal Systems, edited by E. Csuhaj-Varjú, C. Kintala, D. Wotschke, Gy. Vaszil, MTA SZTAKI, Budapest (2003) 114-125.

[15] O.B. Lupanov, A comparison of two types of finite automata. Problemy Kibernetiki 9 (1963) 321-326 (in Russian).

[16] A.R. Meyer and M.J. Fischer, Economy of description by automata, grammars and formal systems, in Proc. 12th Annual Symposium on Switching and Automata Theory (1971) 188-191.

[17] F.R. Moore, On the bounds for state-set size in the proofs of equivalence between deterministic, nondeterministic, and two-way finite automata. IEEE Trans. Comput. 20 (1971) 1211-1214. | Zbl

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

[19] M. Sipser, Lower bounds on the size of sweeping automata. J. Comput. System Sci. 21 (1980) 195-202. | Zbl

[20] M. Sipser, Introduction to the Theory of Computation. PWS Publishing Company, Boston (1997).

[21] S. Yu, Chapter 2: Regular languages, in Handbook of Formal Languages - Vol. I, edited by G. Rozenberg, A. Salomaa, Springer-Verlag, Berlin, New York (1997) 41-110.

[22] S. Yu, A renaissance of automata theory? Bull. Eur. Assoc. Theor. Comput. Sci. EATCS 72 (2000) 270-272.