Binary operations on automatic functions

Juhani Karhumäki; Jarkko Kari; Joachim Kupke

RAIRO - Theoretical Informatics and Applications (2007)

  • Volume: 42, Issue: 2, page 217-236
  • ISSN: 0988-3754

Abstract

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Real functions on the domain [0,1)n – often used to describe digital images – allow for different well-known types of binary operations. In this note, we recapitulate how weighted finite automata can be used in order to represent those functions and how certain binary operations are reflected in the theory of these automata. Different types of products of automata are employed, including the seldomly-used full Cartesian product. We show, however, the infeasibility of functional composition; simple examples yield that the class of automatic functions (i.e., functions computable by automata) is not closed under this operation.

How to cite

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Karhumäki, Juhani, Kari, Jarkko, and Kupke, Joachim. "Binary operations on automatic functions." RAIRO - Theoretical Informatics and Applications 42.2 (2007): 217-236. <http://eudml.org/doc/92868>.

@article{Karhumäki2007,
abstract = { Real functions on the domain [0,1)n – often used to describe digital images – allow for different well-known types of binary operations. In this note, we recapitulate how weighted finite automata can be used in order to represent those functions and how certain binary operations are reflected in the theory of these automata. Different types of products of automata are employed, including the seldomly-used full Cartesian product. We show, however, the infeasibility of functional composition; simple examples yield that the class of automatic functions (i.e., functions computable by automata) is not closed under this operation. },
author = {Karhumäki, Juhani, Kari, Jarkko, Kupke, Joachim},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Automatic functions; weighted finite automata; full Cartesian product; digital images},
language = {eng},
month = {12},
number = {2},
pages = {217-236},
publisher = {EDP Sciences},
title = {Binary operations on automatic functions},
url = {http://eudml.org/doc/92868},
volume = {42},
year = {2007},
}

TY - JOUR
AU - Karhumäki, Juhani
AU - Kari, Jarkko
AU - Kupke, Joachim
TI - Binary operations on automatic functions
JO - RAIRO - Theoretical Informatics and Applications
DA - 2007/12//
PB - EDP Sciences
VL - 42
IS - 2
SP - 217
EP - 236
AB - Real functions on the domain [0,1)n – often used to describe digital images – allow for different well-known types of binary operations. In this note, we recapitulate how weighted finite automata can be used in order to represent those functions and how certain binary operations are reflected in the theory of these automata. Different types of products of automata are employed, including the seldomly-used full Cartesian product. We show, however, the infeasibility of functional composition; simple examples yield that the class of automatic functions (i.e., functions computable by automata) is not closed under this operation.
LA - eng
KW - Automatic functions; weighted finite automata; full Cartesian product; digital images
UR - http://eudml.org/doc/92868
ER -

References

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  1. J. Berstel and M. Morcrette, Compact representation of patterns by finite automata, in Proc. Pixim '89, Paris (1989) 387–402.  
  2. V. Blondel, J. Theys and J. Tsitsiklis, When is a pair of matrices stable? Problem 10.2 in Unsolved problems in Mathematical Systems and Control Theory. Princeton Univ. Press (2004) 304–308.  
  3. K. Culik II and S. Dube, Rational and affine expressions for image descriptions. Discrete Appl. Math.41 (1993) 85–120.  
  4. K. Culik II and I. Friš, Weighted finite transducers in image processing. Discrete Appl. Math.58 (1995) 223–237.  
  5. K. Culik II and J. Karhumäki, Finite automata computing real functions. SIAM J. Comput.23 (1994) 789–814.  
  6. K. Culik II and J. Kari, Image compression using weighted finite automata. Comput. Graph.17 (1993) 305–313.  
  7. K. Culik II and J. Kari, Efficient inference algorithms for weighted finite automata, in Fractal Image Compression, edited by Y. Fisher, Springer (1994).  
  8. K. Culik II and J. Kari, Digital Images and Formal Languages, in Handbook of Formal Languages, Vol. III, edited by G. Rozenberg and A. Salomaa, Springer (1997) 599–616.  
  9. D. Derencourt, J. Karhumäki, M. Latteux and A. Terlutte, On computational power of weighted finite automata, in Proc. 17th MFCS. Lect. Notes Comput. Sci.629 (1992) 236–245.  
  10. D. Derencourt, J. Karhumäki, M. Latteux and A. Terlutte, On continuous functions computed by finite automata. RAIRO-Theor. Inf. Appl.29 (1994) 387–403.  
  11. J. Karhumäki, W. Plandowski and W. Rytter, The complexity of compressing subsegments of images described by finite automata. Discrete Appl. Math.125 (2003) 235–254.  
  12. K. Knopp, Infinite Sequences and Series. Dover publications (1956).  
  13. J. Kupke, On Separating Constant from Polynomial Ambiguity of Finite Automata, in Proc. 32nd SOFSEM. Lect. Notes Comput. Sci.3831 (2006) 379–388.  
  14. J. Kupke, Limiting the Ambiguity of Non-Deterministic Finite Automata. PhD. Thesis. Aachen University, 2002. Available online at  URIhttp://www-i1.informatik.rwth-aachen.de/~joachimk/ltaondfa.ps

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