Abstract
We consider systems of equations of polynomial weighted tree transformations over the max-plus (or: arctic) semiring ℝ max = ( ℝ + ∪ { − ∞ }, max , + , − ∞ ,0). We apply discounting with a parameter 0 ≤ d < 1 in order to guarantee the existence of the least solution, called least d-solution, of such systems. We compute least d-solutions under u-substitution mode, where u = [IO] or u = OI. We define a weighted relation over ℝ max to be u-d-equational, if it is a component of the least u-d-solution of such a system of equations in a pair of algebras. We mainly focus on u-d-equational weighted tree transformations which are equational relations obtained by considering the least u-d-solutions in pairs of term algebras. We also introduce u-d-equational weighted tree languages over ℝ max . We characterize u-d-equational weighted tree transformations in terms of weighted tree transformations defined by weighted d-bimorphisms, which are bimorphisms from d-recognizable weighted tree languages. Finally, we prove that a weighted relation is u-d-equational if and only if it is, roughly speaking, the morphic image of a weighted u-d-equational tree transformation.
This research was financially supported by the TÁMOP-4.2.1/B-09/1/KONV-2010-0005 program of the Hungarian National Development Agency.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Alexandrakis, A., Bozapalidis, S.: Weighted grammars and Kleene’s theorem. Inform. Process. Lett. 24, 1–4 (1987)
Arnold, A., Dauchet, M.: Bi-transductions de forets. In: Michaelson, S., Milner, R. (eds.) Proc. 3rd Int. Coll. Automata, Languages, and Programming, pp. 74–86. Edinburgh University Press, Edinburgh (1976)
Arnold, A., Dauchet, M.: Morphismes et bimorphismes d’arbes. Theoret. Comput. Sci. 20, 33–93 (1982)
Berstel, J., Reutenauer, C.: Recognizable formal power series on trees. Theoret. Comput. Sci. 18, 115–148 (1982)
Bozapalidis, S.: Equational elements in additive algebras. Theory of Comput. Syst. 32, 1–33 (1999)
Bozapalidis, S., Fülöp, Z., Rahonis, G.: Equational tree transformations. Theoret. Comput. Sci. 412, 3676–3692 (2011)
Bozapalidis, S., Fülöp, F., Rahonis, G.: Equational weighted tree transformations (submitted, 2011)
Bozapalidis, S., Rahonis, G.: On the closure of recognizable tree series under tree homomorphisms. J. Autom. Lang. Comb. 10, 185–202 (2005)
Chatterjee, K., Doyen, L., Henzinger, T.A.: Quantitative languages. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 385–400. Springer, Heidelberg (2008)
Chatterjee, K., Doyen, L., Henzinger, T.A.: Composition and alternation for weighted automata. EPLF Technical Report MTC-REPORT-2008-004
Comon, H., Dauchet, M., Gilleron, R., Jacquema, F., Lugiez, D., Tison, S., Tommasi, M.: Tree Automata Techniques and Applications, http://tata.gforge.inria.fr/
Courcelle, B.: Equivalences and transformations of regular systems – Applications to recursive program schemes and grammars. Theoret. Comput. Sci. 42, 1–122 (1986)
Courcelle, B.: Basic notions of universal algebra for language theory and graph grammars. Theoret. Comput. Sci. 163, 1–54 (1996)
de Alfaro, L., Henzinger, T.A., Majumda, R.: Discounting the future in systems theory. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 1022–1037. Springer, Heidelberg (2003)
Droste, M., Rahonis, G.: Weighted automata and weighted logics with discounting. Theoret. Comput. Sci. 410, 3481–3494 (2009)
Droste, M., Sakarovitch, J., Vogler, H.: Weighted automata with discounting. Inform. Process. Lett. 108, 23–28 (2008)
Droste, M., Kuich, W.: Semirings and formal power series. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata. EATCS Monographs in Theoretical Computer Science, pp. 3–28. Springer, Heidelberg (2009)
Droste, M., Kuske, D.: Skew and infinitary formal power series. Theoret. Comput. Sci. 366, 189–227 (2006)
Engelfriet, J.: Bottom-up and top-down tree transformations - a comparison. Math. Systems Theory 9, 198–231 (1975)
Engelfriet, J., Schmidt, E.M.: IO and OI. I.;II. J. Comput. System Sci. 15, 328–353 (1977); 16, 67–99 (1978)
Engelfriet, J., Fülöp, Z., Vogler, H.: Bottom-up and top-down tree series transformations. J. Autom. Lang. Comb. 7, 11–70 (2002)
Ésik, Z., Kuich, W.: Formal tree series. J. Autom. Lang. Comb. 8, 219–285 (2003)
Filar, J., Vrieze, K.: Competitive Marcov Decision Processes. Springer, Heidelberg (1997)
Fülöp, Z., Maletti, A., Vogler, H.: Backward and forward application of extended tree series transformations. Fund. Inform. 112, 1–39 (2011)
Fülöp, Z., Vogler, H.: Weighted tree automata and tree transducers. In: Droste, M., Kuich, W., Vogler, H. (eds.) Handbook of Weighted Automata. EATCS Monographs in Theoretical Computer Science, pp. 313–404. Springer, Heidelberg (2009)
Fülöp, Z., Vogler, H.: Weighted tree transducers. J. Autom. Lang. Comb. 9, 31–54 (2004)
Gécseg, F., Steinby, M.: Tree Automata. Akadémiai Kiadó, Budapest (1984)
Gécseg, F., Steinby, M.: Tree languages. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. III, pp. 1–68. Springer, Heidelberg (1997)
Golan, J.S.: Semirings and their Applications. Kluwer Academic Publishers, Dordrecht (1999)
Graehl, J., Knight, K., May, J.: Training tree transducers. Computational Linguistics 34, 391–427 (2008)
Knight, K., Graehl, J.: An overview of probabilistic tree transducers for natural language processing. In: Gelbukh, A. (ed.) CICLing 2005. LNCS, vol. 3406, pp. 1–24. Springer, Heidelberg (2005)
Kuich, W.: Formal power series over trees. In: Bozapalidis, S. (ed.) Proceedings of DLT 1997, pp. 61–101. Aristotle University of Thessaloniki, Thessaloniki (1998)
Kuich, W.: Tree transducers and formal tree series. Acta Cybernet. 14, 135–149 (1999)
Kuich, W.: On skew formal power series. In: Bozapalidis, S., Kalampakas, A., Rahonis, G. (eds.) Proceedings of CAI 2005, pp. 7–30 (2005)
Maletti, A.: Relating tree series transducers and weighted tree automata. Internat. J. Found. Comput. Sci. 16, 723–741 (2005)
Maletti, A.: Compositions of tree series transformations. Theoret. Comput. Sci. 366, 248–271 (2006)
Maletti, A.: Compositions of extended top-down tree transducers. Inform. and Comput. 206, 1187–1196 (2008)
Maletti, A., Graehl, J., Hopkins, M., Knight, K.: The power of extended top-down tree transducers. SIAM J. Comput. 39(2), 410–430 (2008)
Mandrali, E., Rahonis, G.: Recognizable tree series with discounting. Acta Cybernet. 19, 411–439 (2009)
Mezei, J., Wright, J.B.: Algebraic automata and context-free sets. Inform. Control 11, 3–29 (1967)
Shapley, L.S.: Stochastic games. Roc. National Acad. of Sciences 39, 1095–1100 (1953)
Wechler, W.: Universal Algebra for Computer Scientists. EATCS Monographs on Theoretical Computer Science, vol. 25. Springer, Heidelberg (1992)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Fülöp, Z., Rahonis, G. (2011). Equational Weighted Tree Transformations with Discounting. In: Kuich, W., Rahonis, G. (eds) Algebraic Foundations in Computer Science. Lecture Notes in Computer Science, vol 7020. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24897-9_6
Download citation
DOI: https://doi.org/10.1007/978-3-642-24897-9_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24896-2
Online ISBN: 978-3-642-24897-9
eBook Packages: Computer ScienceComputer Science (R0)