Abstract
We propose a quasi-equational sound axiomatization of regular tree languages, called monodic tree Kleene algebra. The algebra is weaker than Kleene algebra introduced by Kozen. We find a subclass of regular tree languages, for which monodic tree Kleene algebra is complete. While regular tree expressions may have two or more kinds of place holders, the subclass can be equipped with only one kind of them. Along the lines of the original proof by Kozen, we prove the completeness theorem based on determinization and minimization of tree automata represented by matrices on monodic tree Kleene algebra.
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References
Comon, H., Dauchet, M., Gilleron, R., Jacquemard, F., Lugiez, D., Tison, S., Tommasi, M.: Tree automata techniques and applications (1997), http://www.grappa.univ-lille3.fr/tata/
Ésik, Z.: Axiomatizing the equational theory of regular tree languages (extended anstract). In: Morvan, M., Meinel, C., Krob, D. (eds.) STACS 1998. LNCS, vol. 1373, pp. 455–465. Springer, Heidelberg (1998)
Takai, T., Furusawa, H., Kahl, W.: Reasoning about term rewriting in Kleene categories with converse. In: Düntsch, I., Winter, M. (eds.) Proceedings of the 3rd Workshop on Applications of Kleene algebera, pp. 259–266 (2005)
Gécseg, F., Steinby, M.: Tree languages. In: Handbook of formal languages, 3: beyond words, pp. 1–68. Springer, Heidelberg (1997)
Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. In: Kahn, G. (ed.) Proceedings of the Sixth Annual IEEE Symp. on Logic in Computer Science, LICS 1991, pp. 214–225. IEEE Computer Society Press, Los Alamitos (1991)
Kozen, D.: A completeness theorem for Kleene algebras and the algebra of regular events. Information and Computation 110(2), 366–390 (1994)
Möller, B.: Lazy Kleene algebra. In: Kozen, D. (ed.) MPC 2004. LNCS, vol. 3125, pp. 252–273. Springer, Heidelberg (2004)
McIver, A., Weber, T.: Towards automated proof support for probabilistic distributed systems. In: Sutcliffe, G., Voronkov, A. (eds.) LPAR 2005. LNCS (LNAI), vol. 3835, pp. 534–548. Springer, Heidelberg (2005)
McIver, A.K., Cohen, E., Morgan, C.C.: Using probabilistic Kleene algebra for protocol verification. In: Schmidt, R.A. (ed.) RelMiCS/AKA 2006. LNCS, vol. 4136, pp. 296–310. Springer, Heidelberg (2006)
Möller, B., Struth, G.: Modal Kleene algebra and partial correctness. In: Rattray, C., Maharaj, S., Shankland, C. (eds.) AMAST 2004. LNCS, vol. 3116, pp. 379–393. Springer, Heidelberg (2004)
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Takai, T., Furusawa, H. (2006). Monodic Tree Kleene Algebra. In: Schmidt, R.A. (eds) Relations and Kleene Algebra in Computer Science. RelMiCS 2006. Lecture Notes in Computer Science, vol 4136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11828563_27
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DOI: https://doi.org/10.1007/11828563_27
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37873-0
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