Binary operations on automatic functions
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 42 (2008) no. 2, p. 217-236

Real functions on the domain ${\left[0,1\right)}^{n}$ - 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.

DOI : https://doi.org/10.1051/ita:2007036
Classification:  68Q45,  68Q10,  68U10
Keywords: automatic functions, weighted finite automata, full cartesian product
@article{ITA_2008__42_2_217_0,
author = {Karhum\"aki, Juhani and Kari, Jarkko and Kupke, Joachim},
title = {Binary operations on automatic functions},
journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
publisher = {EDP-Sciences},
volume = {42},
number = {2},
year = {2008},
pages = {217-236},
doi = {10.1051/ita:2007036},
zbl = {1144.68034},
mrnumber = {2401259},
language = {en},
url = {http://www.numdam.org/item/ITA_2008__42_2_217_0}
}

Karhumäki, Juhani; Kari, Jarkko; Kupke, Joachim. Binary operations on automatic functions. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 42 (2008) no. 2, pp. 217-236. doi : 10.1051/ita:2007036. http://www.numdam.org/item/ITA_2008__42_2_217_0/

[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. | MR 1198549 | Zbl 0784.68058

[4] K. Culik Ii and I. Friš, Weighted finite transducers in image processing. Discrete Appl. Math. 58 (1995) 223-237. | MR 1326048 | Zbl 0818.68145

[5] K. Culik Ii and J. Karhumäki, Finite automata computing real functions. SIAM J. Comput. 23 (1994) 789-814. | MR 1283575 | Zbl 0820.68061

[6] K. Culik Ii and J. Kari, Image compression using weighted finite automata. Comput. Graph. 17 (1993) 305-313. | MR 1265078

[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. | MR 1470027

[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. | MR 1255139

[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. | Numdam | MR 1282454 | Zbl 0883.68095

[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. | MR 1943115 | Zbl 1010.68075

[12] K. Knopp, Infinite Sequences and Series. Dover publications (1956). | MR 79110 | Zbl 0070.05807

[13] J. Kupke, On Separating Constant from Polynomial Ambiguity of Finite Automata, in Proc. 32nd SOFSEM. Lect. Notes Comput. Sci. 3831 (2006) 379-388. | Zbl pre05072088

[14] 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/$\sim$joachimk/ltaondfa.ps