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.
@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},
volume = {43},
year = {2009},
pages = {69-94},
doi = {10.1051/ita:2007061},
mrnumber = {2483445},
zbl = {1170.68023},
language = {en},
url = {http://dml.mathdoc.fr/item/ITA_2009__43_1_69_0}
}
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) pp. 69-94. doi : 10.1051/ita:2007061. http://gdmltest.u-ga.fr/item/ITA_2009__43_1_69_0/
[1] , and , The Design and Analysis of Computer Algorithms. Addison-Wesley (1976). | MR 413592 | Zbl 0326.68005
[2] , and , An optimal lower bound for nonregular languages. Inform. Process. Lett. 50 (1994) 289-292. (Corrigendum: Inform. Process. Lett. 52 (1994) 339). | MR 1286605 | Zbl 0834.68062 | Zbl 0810.68089
[3] and , Fundamentals of Algorithmics. Prentice Hall (1996). | MR 1345611 | Zbl 0847.68046
[4] , 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] , 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] , and , Converting two-way nondeterministic unary automata into simpler automata. Theoret. Comput. Sci. 295 (2003) 189-203. | MR 1964665 | Zbl 1045.68080
[8] , and , Introduction to Automata Theory, Languages and Computation. Addison-Wesley (2001). | MR 645539 | Zbl 0980.68066
[9] , An 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] and , Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979). | MR 645539 | Zbl 0426.68001
[11] , The synthesis of sequential switching circuits. J. Franklin Inst. 257 (1954) 161-190 and 275-303. | MR 62648 | Zbl 0166.27201
[12] and , Optimal simulations between unary automata. SIAM J. Comput. 30 (2001) 1976-1992. | MR 1856565 | Zbl 0980.68048
[13] , 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