Abstract
Finite Deterministic Cover Automata (DFCA) can be obtained from Deterministic Finite Automata (DFA) using the similarity relation. Since the similarity relation is not an equivalence relation, the minimal DFCA for a finite language is usually not unique. We count the number of minimal DFCA that can be obtained from a given minimal DFA with n states by merging the similar states in the given DFA. We compute an upper bound for this number and prove that in the worst case (for a non-unary alphabet) it is [4n-9+√8n+1!/8]/(2[4n-9+√8n+1/8] - n + 1)!
We prove that this upper bound is reached, i.e. for any given positive integer n we find a minimal DFA with n states, which has the number of minimal DFCA obtained by merging similar states equal to this maximum.
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References
C. Câmpeanu, N. Sântean and S. Yu, “Minimal Cover-Automata for Finite Languages”, Proceedings of the Third International Workshop on Implementing Automata WIA’98 (1998), 32–42 and TCS vol 267 (2001), 3-16.
C. Dwork and L. Stockmeyer, “A Time Complexity Gap for Two-Way Probabilistic Finite-State Automata”, SIAM Journal on Computing, vol.19 (1990), 1011–1023.
J. E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages and Computation Addison-Wesley, (1979).
J. Kaneps, R. Frievalds, “Running Time to Recognize Non-Regular Languages by 2-Way Probabilistic Automata”, in ICALP’91, LNCS, Springer-Verlag, New-York/Berlin (1991) vol 510, 174–185.
A. Păun, N. Sântean and Sheng Yu, “An O(n 2) algorithm for Minimal Cover-Automata for Finite Languages”, Proceedings of the 5th International Conference on Implementation and Application of Automata CIAA’00 (2000), 243–251.
N. Sântean, Towards a Minimal Representation for Finite Languages: Theory and Practice, MSc Thesis, Department of Computer Science, The University of Western Ontario, (2000).
J. M. Champarnaud and D. Maurel, Automata Implementation, Proceedings of Third International Workshop on Implementing Automata, LNCS 1660, Springer, (1999).
A. Salomaa, Formal Languages Academic Press, (1973).
S. Yu, “Regular languages”, in Handbook of Formal Languages, Vol I, eds. G. Rozenberg and A. Salomaa, Springer-Verlag, (1997), 41–110.
D. Wood and S. Yu, Automata Implementation, Proceedings of Second International Workshop on Implementing Automata, LNCS 1436, Springer, (1998).
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Cămpeanu, C., Păun, A. (2003). The Number of Similarity Relations and the Number of Minimal Deterministic Finite Cover Automata. In: Champarnaud, JM., Maurel, D. (eds) Implementation and Application of Automata. CIAA 2002. Lecture Notes in Computer Science, vol 2608. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44977-9_6
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DOI: https://doi.org/10.1007/3-540-44977-9_6
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