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Binary operations on automatic functions

Published online by Cambridge University Press:  13 December 2007

Juhani Karhumäki ,
Jarkko Kari  and
Joachim Kupke
Juhani Karhumäki
Affiliation:
University of Turku, Finland; [email protected], [email protected]
Jarkko Kari
Affiliation:
University of Turku, Finland; [email protected], [email protected]
Joachim Kupke
Affiliation:
ETH Zurich, Switzerland; [email protected]
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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.

Keywords

Automatic functionsweighted finite automatafull Cartesian product
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 42 , Issue 2 , April 2008 , pp. 217 - 236
Copyright
© EDP Sciences, 2007

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References

J. Berstel and M. Morcrette, Compact representation of patterns by finite automata, in Proc. Pixim '89, Paris (1989) 387–402.
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.
Culik II, K. and Dube, S., Rational and affine expressions for image descriptions. Discrete Appl. Math. 41 (1993) 85–120. CrossRef
Culik II, K. and Friš, I., Weighted finite transducers in image processing. Discrete Appl. Math. 58 (1995) 223–237. CrossRef
Culik II, K. and Karhumäki, J., Finite automata computing real functions. SIAM J. Comput. 23 (1994) 789–814. CrossRef
Culik II, K. and Kari, J., Image compression using weighted finite automata. Comput. Graph. 17 (1993) 305–313. CrossRef
K. Culik II and J. Kari, Efficient inference algorithms for weighted finite automata, in Fractal Image Compression, edited by Y. Fisher, Springer (1994).
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.
Derencourt, D., Karhumäki, J., Latteux, M. and Terlutte, A., On computational power of weighted finite automata, in Proc. 17th MFCS. Lect. Notes Comput. Sci. 629 (1992) 236–245. CrossRef
Derencourt, D., Karhumäki, J., Latteux, M. and Terlutte, A., On continuous functions computed by finite automata. RAIRO-Theor. Inf. Appl. 29 (1994) 387–403. CrossRef
Karhumäki, J., Plandowski, W. and Rytter, W., The complexity of compressing subsegments of images described by finite automata. Discrete Appl. Math. 125 (2003) 235–254. CrossRef
K. Knopp, Infinite Sequences and Series. Dover publications (1956).
Kupke, J., On Separating Constant from Polynomial Ambiguity of Finite Automata, in Proc. 32nd SOFSEM. Lect. Notes Comput. Sci. 3831 (2006) 379–388. CrossRef
J. Kupke, Limiting the Ambiguity of Non-Deterministic Finite Automata. PhD. Thesis. Aachen University, 2002. Available online at http://www-i1.informatik.rwth-aachen.de/~joachimk/ltaondfa.ps http://www-i1.informatik.rwth-aachen.de/~joachimk/ltaondfa.ps