Abstract
We introduce a weighted logic with discounting and we establish Büchi’s and Elgot’s theorem for weighted automata over finite words and arbitrary commutative semirings. Then we investigate Büchi and Muller automata with discounting over the max-plus and the min-plus semiring. We show their expressive equivalence with weighted MSO-sentences with discounting. In this case our logic has a purely syntactic definition. For the finite case, we obtain a purely syntactically defined weighted logic if the underlying semiring is additively locally finite.
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
Arnold, A.: Finite Transition Systems. International Series in Computer Science. Prentice Hall, Englewood Cliffs (1994)
Berstel, J., Reutenauer, C.: Rational Series and Their Languages. In: EATCS Monographs in Theoretical Computer Science, vol. 12, Springer, Heidelberg (1988)
Büchi, J.R.: Weak second-order arithmetic and finite automata. Z. Math. Logik Grundlager Math. 6, 66–92 (1960)
Büchi, J.R.: On a decision method in restricted second order arithmetic. In: Proc. 1960 Int. Congr. for Logic, Methodology and Philosophy of Science, pp. 1–11 (1962)
Culik II, K., Kari, J.: Image compression using weighted finite automata. Computer and Graphics 17, 305–313 (1993)
Culik, K., Karhumäki, J.: Finite automata computing real functions. SIAM J. Comput. 23(4), 789–814 (1994)
Cuninghame-Green, R.A.: Minimax algebra and applications. Adv. in Imaging Electron Phy. 90, 1–121 (1995)
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., Gastin, P.: Weighted automata and weighted logics. Theoret. Comput. Sci. 380, 69–86 (2007) extended abstract In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 513–525. Springer, Heidelberg (2005)
Droste, M., Kuske, D.: Skew and infinitary formal power series. Theoret. Comput. Sci. 366, 189–227 (2006). extended abstract In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds.) ICALP 2003. LNCS, vol. 2719, pp. 426–438. Springer, Heidelberg (2003)
Droste, M., Rahonis, G.: Weighted automata and weighted logics on infinite words. In: Ibarra, O.H., Dang, Z. (eds.) DLT 2006. LNCS, vol. 4036, pp. 49–58. Springer, Heidelberg (2006)
Droste, M., Vogler, H.: Weighted tree automata and weighted logics. Theoret. Comput. Sci. 366, 228–247 (2006)
Eilenberg, S.: Automata, Languages and Machines, vol. A. Academic Press, London (1974)
Elgot, C.: Decision problems of finite automata design and related arithmetics. Trans. Amer. Math. Soc. 98, 21–52 (1961)
Ésik, Z., Kuich, W.: A semiring-semimodule generalization of ω -regular languages I and II. Special issue on “Weighted automata”. In: Droste, M., Vogler, H. (eds.) J. of Automata Languages and Combinatorics 10, 203–242, 243–264 (2005)
Filar, J., Vrieze, K.: Competitive Marcov Decision Processes. Springer, Heidelberg (1997)
Gaubert, S., Plus, M.: Methods and applications of (max, +) linear algebra, Techical Report 3088, INRIA, Rocquencourt (January 1997)
Jiang, Z., Litow, B., de Vel, O.: Similarity enrichment in image compression through weighted finite automata. In: Du, D.-Z., Eades, P., Sharma, A.K., Lin, X., Estivill-Castro, V. (eds.) COCOON 2000. LNCS, vol. 1858, pp. 447–456. Springer, Heidelberg (2000)
Katritzke, F.: Refinements of data compression using weighted finite automata, PhD thesis, Universität Siegen, Germany (2001)
Khoussainov, B., Nerode, A.: Automata Theory and its Applications, Birkhäuser Boston (2001)
Knight, K., Graehl, J.: Machine transliteration. Comput. Linguist. 24(4), 599–612 (1998)
Krob, D.: The equality problem for rational series with multiplicities in the tropical semiring is undecidable. Intern. J. of Information and Computation 4, 405–425 (1994)
Kuich, W.: Semirings and formal power series: Their relevance to formal languages and automata theory. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, pp. 609–677. Springer, Heidelberg (1997)
Kuich, W.: On skew formal power series. In: Bozapalidis, S., Kalampakas, A., Rahonis, G. (eds.) Proceedings of the Conference on Algebraic Informatics, Thessaloniki, pp. 7–30 (2005)
Kuich, W., Salomaa, A.: Semirings, Automata, Languages. EATCS Monographs in Theoretical Computer Science, vol. 5. Springer, Heidelberg (1986)
Kurshan, R.P.: Computer-Aided Verification of Coordinating Processes. Princeton Series in Computer Science. Princeton University Press, Princeton (1994)
Mathissen, C.: Definable transductions and weighted logics for texts. In: Proceedings of DLT 2007. 11th International Conference on Developments in Language Theory (DLT) 2007. LNCS, vol. 4588, pp. 324–336 (2007)
Mäurer, I.: Weighted picture automata and weighted logics. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 313–324. Springer, Heidelberg (2006)
McMillan, K.: Symbolic Model Checking. Kluwer Academic Publishers, Dordrecht (1993)
Meinecke, I.: Weighted logics for traces. In: Grigoriev, D., Harrison, J., Hirsch, E.A. (eds.) CSR 2006. LNCS, vol. 3967, pp. 235–246. Springer, Heidelberg (2006)
Mohri, M., Pereira, F., Riley, M.: The design principles of a weighted finite-state transducer library. Theoret. Comput. Sci. 231, 17–32 (2000)
Perrin, D., Pin, J.E.: Infinite Words. Elsevier, Amsterdam (2004)
Salomaa, A., Soittola, M.: Automata-Theoretic Aspects of Formal Power Series. Texts and Monographs in Computer Science. Springer, Heidelberg (1978)
Schützenberger, M.: On the definition of a family of automata. Inf. Control 4, 245–270 (1961)
Shapley, L.S.: Stochastic games. Roc. National Acad. of Sciences 39, 1095–1100 (1953)
Thomas, W.: Automata on infinite objects. In: Leeuwen, J.v. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 135–191. Elsevier Science Publishers, Amsterdam (1990)
Thomas, W.: Languages, automata and logic. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 3, pp. 389–485. Springer, Heidelberg (1997)
Zimmermann, U.: Combinatorial Optimization in Ordered Algebraic Structures, Annals of Discrete Mathematics, vol. 10. North-Holland, Amsterdam (1981)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Droste, M., Rahonis, G. (2007). Weighted Automata and Weighted Logics with Discounting. In: Holub, J., Žďárek, J. (eds) Implementation and Application of Automata. CIAA 2007. Lecture Notes in Computer Science, vol 4783. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-76336-9_9
Download citation
DOI: https://doi.org/10.1007/978-3-540-76336-9_9
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-76335-2
Online ISBN: 978-3-540-76336-9
eBook Packages: Computer ScienceComputer Science (R0)