Abstract
We represent a new fast algorithm for checking multiplicity equivalence for finite Z - Σ* - automata. The classical algorithm of Eilenberg [1]is exponential one. On the other hand, for the finite deterministic automata an analogous Aho - Hopcroft - Ullman [2] algorithm has “almost” linear time complexity. Such a big gap leads to the idea of the existence of an algorithm which is more faster than an exponential one. Hunt and Stearns [3] announced the existence of polynomial algorithm for the Z-Σ* - automata of finite degree of ambiguity only. Tzeng [4] created O(n 4) algorithm for the general case. Diekert [5] informed us Tzeng's algorithm can be implemented in O(n 3) using traingular matrices. Any way, we propose a new O(n 3) algorithm (for the general case). Our algorithm utilizes recent results of Siberian mathematical school (Gerasimov [6], Valitckas [7]) about the structure of rings and does not share common ideas with Eilenberg [1] and Tzeng [4].
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References
Eilenberg, S.: Automata, Languages, and Machines, Vol. A. Acad. Press (1974)
Aho, A., Hopcroft, J.E., Ullman, J.D.:The Design and Analysis of Computer Algorithms. Reading, Mass. (1976)
Hunt, H.B., Stearns, R.E.:On the Complexity of Equivalence, Nonlinear Algebra, and Optimization on Rings, Semirings, and Lattices. (Extended Abstract), Technical Report 86-23, Computer Science Dept, State University of NY at Albany (1986)
Tzeng, W.-G.:A Polynomial-time Algorithm for the Equivalence of Probabilistic Automata. FOCS (1989)
Diekert, V., Stuttgart University, private E-mail, December 12, (2001)
Gerasimov, V.N.:Localizations in the Associative Rings. Siberian Mathematical Journal, Vol.XXIII, N 6(1982) 36–54 (in Russian).
Valitckas, A.I., Institute of Mathematics, Siberian Division of Russian Academy, private communication (1992).
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Archangelsky, K. (2003). Efficient Algorithm for Checking Multiplicity Equivalence for the Finite Z - Σ*-Automata. In: Ito, M., Toyama, M. (eds) Developments in Language Theory. DLT 2002. Lecture Notes in Computer Science, vol 2450. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45005-X_24
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DOI: https://doi.org/10.1007/3-540-45005-X_24
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