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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; [email protected]
Viliam Geffert
Affiliation:
Department of Computer Science, P. J. Šafárik University, Jesenná 5, 04001 Košice, Slovakia; [email protected]
Ian Shipman
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Ave., Chicago, IL 60637, USA; [email protected]
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Abstract

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.

Keywords

Finite state automataregular languages.
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).