Hyper-minimizing minimized deterministic finite state automata

Andrew Badr; Viliam Geffert; Ian Shipman

RAIRO - Theoretical Informatics and Applications (2007)

  • Volume: 43, Issue: 1, page 69-94
  • ISSN: 0988-3754

Abstract

top
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.

How to cite

top

Badr, Andrew, Geffert, Viliam, and Shipman, Ian. "Hyper-minimizing minimized deterministic finite state automata." RAIRO - Theoretical Informatics and Applications 43.1 (2007): 69-94. <http://eudml.org/doc/92908>.

@article{Badr2007,
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. },
author = {Badr, Andrew, Geffert, Viliam, Shipman, Ian},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Finite state automata; regular languages.; finite state automata; regular languages},
language = {eng},
month = {12},
number = {1},
pages = {69-94},
publisher = {EDP Sciences},
title = {Hyper-minimizing minimized deterministic finite state automata},
url = {http://eudml.org/doc/92908},
volume = {43},
year = {2007},
}

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
DA - 2007/12//
PB - EDP Sciences
VL - 43
IS - 1
SP - 69
EP - 94
AB - 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.
LA - eng
KW - Finite state automata; regular languages.; finite state automata; regular languages
UR - http://eudml.org/doc/92908
ER -

References

top
  1. A.V. Aho, J.E. Hopcroft and J.D. Ullman, The Design and Analysis of Computer Algorithms. Addison-Wesley (1976).  
  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).  
  3. G. Brassard and P. Bratley, Fundamentals of Algorithmics. Prentice Hall (1996).  
  4. M. Chrobak, Finite automata and unary languages. Theoret. Comput. Sci.47 (1986) 149–158. (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. 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.  
  7. V. Geffert, C. Mereghetti and G. Pighizzini, Converting two-way nondeterministic unary automata into simpler automata. Theoret. Comput. Sci.295 (2003) 189–203.  
  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. D.A. Huffman, The synthesis of sequential switching circuits. J. Franklin Inst.257 (1954) 161–190 and 275–303.  
  12. C. Mereghetti and G. Pighizzini, Optimal simulations between unary automata. SIAM J. Comput.30 (2001) 1976–1992.  
  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).  

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.