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Hyper-minimizing minimized deterministic finite state automata

Published online by Cambridge University Press:  20 December 2007

Andrew Badr,
Viliam Geffert and
Ian Shipman
Andrew Badr
Affiliation:
3210 Acklen Ave., Nashville, TN 37212, USA; badr@uiuc.edu
Viliam Geffert
Affiliation:
Department of Computer Science, P. J. Šafárik University, Jesenná 5, 04001 Košice, Slovakia; viliam.geffert@upjs.sk
Ian Shipman
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA; ics@math.uchicago.edu
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Abstract

We present the first (polynomial-time) algorithm for reducinga given deterministic finite state automaton (DFA) intoa hyper-minimized DFA, which may have fewer states thanthe classically minimized DFA. The price we pay is that thelanguage recognized by the new machine can differ from theoriginal on a finite number of inputs. These hyper-minimizedautomata are optimal, in the sense that every DFA with fewerstates must disagree on infinitely many inputs. With smallmodifications, the construction works also for finite statetransducers producing outputs. Within a class of finitely differing languages, thehyper-minimized automaton is not necessarily unique. There mayexist several non-isomorphic machines using the minimum number ofstates, each accepting a separate language finitely-differentfrom the original one. We will show that there are largestructural similarities among all these smallest automata.

Keywords

Finite state automataregular languages.

Information

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 43 , Issue 1 , January 2009, pp. 69 - 94
Copyright
© EDP Sciences, 2008

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References

A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms. Addison-Wesley (1976).
Bertoni, A., Mereghetti, C. and Pighizzini, G., An optimal lower bound for nonregular languages. Inform. Process. Lett. 50 (1994) 289–292. (Corrigendum: Inform. Process. Lett. 52 (1994) 339). CrossRef
G. Brassard and P. Bratley, Fundamentals of Algorithmics. Prentice Hall (1996).
Chrobak, M., Finite automata and unary languages. Theoret. Comput. Sci. 47 (1986) 149–158. (Corrigendum: Theoret. Comput. Sci. 302 (2003) 497–498). CrossRef
V. Geffert, (Non)determinism and the size of one-way finite automata, in Proc. Descr. Compl. Formal Syst. IFIP & Univ. Milano (2005) 23–37.
Geffert, V., 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. CrossRef
Geffert, V., Mereghetti, C. and Pighizzini, G., Converting two-way nondeterministic unary automata into simpler automata. Theoret. Comput. Sci. 295 (2003) 189–203. CrossRef
J. Hopcroft, R. Motwani and J. Ullman, Introduction to Automata Theory, Languages and Computation. Addison-Wesley (2001).
J.E. Hopcroft, An $n\log n$ 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).
J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979).
Huffman, D.A., The synthesis of sequential switching circuits. J. Franklin Inst. 257 (1954) 161–190 and 275–303. CrossRef
Mereghetti, C. and Pighizzini, G., Optimal simulations between unary automata. SIAM J. Comput. 30 (2001) 1976–1992. CrossRef
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).