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Hyper-minimizing minimized deterministic finite state automata

  • Andrew Badr (a1), Viliam Geffert (a2) and Ian Shipman (a3)

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.

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[1] A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms. Addison-Wesley (1976).
[2] Bertoni A., Mereghetti C. and Pighizzini G., An optimal lower bound for nonregular languages. Inform. Process. Lett. 50 (1994) 289292. (Corrigendum: Inform. Process. Lett. 52 (1994) 339).
[3] G. Brassard and P. Bratley, Fundamentals of Algorithmics. Prentice Hall (1996).
[4] Chrobak M., Finite automata and unary languages. Theoret. Comput. Sci. 47 (1986) 149158. (Corrigendum: Theoret. Comput. Sci. 302 (2003) 497–498).
[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] 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) 412423.
[7] Geffert V., Mereghetti C. and Pighizzini G., Converting two-way nondeterministic unary automata into simpler automata. Theoret. Comput. Sci. 295 (2003) 189203.
[8] J. Hopcroft, R. Motwani and J. Ullman, Introduction to Automata Theory, Languages and Computation. Addison-Wesley (2001).
[9] 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).
[10] J.E. Hopcroft and J.D. Ullman, Introduction to Automata Theory, Languages and Computation. Addison-Wesley (1979).
[11] Huffman D.A., The synthesis of sequential switching circuits. J. Franklin Inst. 257 (1954) 161190 and 275–303.
[12] Mereghetti C. and Pighizzini G., Optimal simulations between unary automata. SIAM J. Comput. 30 (2001) 19761992.
[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).
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RAIRO - Theoretical Informatics and Applications
  • ISSN: 0988-3754
  • EISSN: 1290-385X
  • URL: /core/journals/rairo-theoretical-informatics-and-applications
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