Hyper-minimizing minimized deterministic finite state automata
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 69-94.

We present the first (polynomial-time) algorithm for reducing a given deterministic finite state automaton (DFA) into a hyper-minimized DFA, which may have fewer states than the classically minimized DFA. The price we pay is that the language recognized by the new machine can differ from the original on a finite number of inputs. These hyper-minimized automata are optimal, in the sense that every DFA with fewer states must disagree on infinitely many inputs. With small modifications, the construction works also for finite state transducers producing outputs. Within a class of finitely differing languages, the hyper-minimized automaton is not necessarily unique. There may exist several non-isomorphic machines using the minimum number of states, each accepting a separate language finitely-different from the original one. We will show that there are large structural similarities among all these smallest automata.

DOI : https://doi.org/10.1051/ita:2007061
Classification : 68Q70
Mots clés : finite state automata, regular languages
@article{ITA_2009__43_1_69_0,
     author = {Badr, Andrew and Geffert, Viliam and Shipman, Ian},
     title = {Hyper-minimizing minimized deterministic finite state automata},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {69--94},
     publisher = {EDP-Sciences},
     volume = {43},
     number = {1},
     year = {2009},
     doi = {10.1051/ita:2007061},
     zbl = {1170.68023},
     mrnumber = {2483445},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ita:2007061/}
}
TY  - JOUR
AU  - Badr, Andrew
AU  - Geffert, Viliam
AU  - Shipman, Ian
TI  - Hyper-minimizing minimized deterministic finite state automata
JO  - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications
PY  - 2009
DA  - 2009///
SP  - 69
EP  - 94
VL  - 43
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/ita:2007061/
UR  - https://zbmath.org/?q=an%3A1170.68023
UR  - https://www.ams.org/mathscinet-getitem?mr=2483445
UR  - https://doi.org/10.1051/ita:2007061
DO  - 10.1051/ita:2007061
LA  - en
ID  - ITA_2009__43_1_69_0
ER  - 
Badr, Andrew; Geffert, Viliam; Shipman, Ian. Hyper-minimizing minimized deterministic finite state automata. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 43 (2009) no. 1, pp. 69-94. doi : 10.1051/ita:2007061. http://www.numdam.org/articles/10.1051/ita:2007061/

[1] A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms. Addison-Wesley (1976). | MR 413592 | Zbl 0326.68005

[2] A. Bertoni, C. Mereghetti and G. Pighizzini, An optimal lower bound for nonregular languages. Inform. Process. Lett. 50 (1994) 289-292. (Corrigendum: Inform. Process. Lett. 52 (1994) 339). | MR 1286605 | Zbl 0810.68089

[3] G. Brassard and P. Bratley, Fundamentals of Algorithmics. Prentice Hall (1996). | MR 1345611 | Zbl 0847.68046

[4] M. Chrobak, Finite automata and unary languages. Theoret. Comput. Sci. 47 (1986) 149-158. (Corrigendum: Theoret. Comput. Sci. 302 (2003) 497-498). | MR 881208 | Zbl 0638.68096

[5] V. Geffert, (Non)determinism and the size of one-way finite automata, in Proc. Descr. Compl. Formal Syst. IFIP & Univ. Milano (2005) 23-37.

[6] V. Geffert, Magic numbers in the state hierarchy of finite automata, in Proc. Math. Found. Comput. Sci., Springer-Verlag. Lect. Notes Comput. Sci. 4162 (2006) 412-423. | MR 2298196 | Zbl 1132.68444

[7] V. Geffert, C. Mereghetti and G. Pighizzini, Converting two-way nondeterministic unary automata into simpler automata. Theoret. Comput. Sci. 295 (2003) 189-203. | MR 1964665 | Zbl 1045.68080

[8] J. Hopcroft, R. Motwani and J. Ullman, Introduction to Automata Theory, Languages and Computation. Addison-Wesley (2001). | MR 645539 | Zbl 0980.68066

[9] J.E. Hopcroft, An nlogn algorithm for minimizing the states in a finite automaton, in The Theory of Machines and Computations, edited by Z. Kohave, pp. 189-196. Academic Press (1971). | MR 403320 | Zbl 0293.94022

[10] J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979). | MR 645539 | Zbl 0426.68001

[11] D.A. Huffman, The synthesis of sequential switching circuits. J. Franklin Inst. 257 (1954) 161-190 and 275-303. | MR 62648 | Zbl 0166.27201

[12] C. Mereghetti and G. Pighizzini, Optimal simulations between unary automata. SIAM J. Comput. 30 (2001) 1976-1992. | MR 1856565 | Zbl 0980.68048

[13] E.F. Moore, Gedanken experiments on sequential machines, in Automata Studies, edited by C.E. Shannon and J. McCarthy, pp. 129-153. Princeton University Press (1956). | MR 78059

Cité par Sources :