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Convergence of Newton’s Method over Commutative Semirings

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Language and Automata Theory and Applications (LATA 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7810))

Abstract

We give a lower bound on the speed at which Newton’s method (as defined in [5,6]) converges over arbitrary ω-continuous commutative semirings. From this result, we deduce that Newton’s method converges within a finite number of iterations over any semiring which is “collapsed at some k ∈ ℕ” (i.e. k = k + 1 holds) in the sense of [1]. We apply these results to (1) obtain a generalization of Parikh’s theorem, (2) to compute the provenance of Datalog queries, and (3) to analyze weighted pushdown systems. We further show how to compute Newton’s method over any ω-continuous semiring.

This work was partially funded by theDFG project “Polynomial Systems on Semirings: Foundations, Algorithms, Applications”.

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References

  1. Bloom, S.L., Ésik, Z.: Axiomatizing rational power series over natural numbers. Inf. Comput. 207(7), 793–811 (2009)

    Article  MATH  Google Scholar 

  2. Bouajjani, A., Esparza, J., Touili, T.: A generic approach to the static analysis of concurrent programs with procedures. Int. J. Found. Comput. Sci. 14(4), 551 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bozapalidis, S.: Equational elements in additive algebras. Theory Comput. Syst. 32(1), 1–33 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  4. Esparza, J., Ganty, P., Kiefer, S., Luttenberger, M.: Parikh’s theorem: A simple and direct automaton construction. Inf. Process. Lett. 111(12), 614–619 (2011)

    Article  MathSciNet  Google Scholar 

  5. Esparza, J., Kiefer, S., Luttenberger, M.: An Extension of Newton’s Method to ω-Continuous Semirings. In: Harju, T., Karhumäki, J., Lepistö, A. (eds.) DLT 2007. LNCS, vol. 4588, pp. 157–168. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  6. Esparza, J., Kiefer, S., Luttenberger, M.: On Fixed Point Equations over Commutative Semirings. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 296–307. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  7. Esparza, J., Kiefer, S., Luttenberger, M.: Newtonian program analysis. J. ACM 57(6), 33 (2010)

    Article  MathSciNet  Google Scholar 

  8. Etessami, K., Yannakakis, M.: Recursive markov chains, stochastic grammars, and monotone systems of nonlinear equations. J. ACM 56(1) (2009)

    Google Scholar 

  9. Flajolet, P., Raoult, J.C., Vuillemin, J.: The number of registers required for evaluating arithmetic expressions. Theor. Comput. Sci. 9, 99–125 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  10. Foster, J.N., Karvounarakis, G.: Provenance and data synchronization. IEEE Data Eng. Bull. 30(4), 13–21 (2007)

    Google Scholar 

  11. Ganty, P., Majumdar, R., Monmege, B.: Bounded Underapproximations. In: Touili, T., Cook, B., Jackson, P. (eds.) CAV 2010. LNCS, vol. 6174, pp. 600–614. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  12. Green, T.J., Karvounarakis, G., Tannen, V.: Provenance semirings. In: PODS, pp. 31–40 (2007)

    Google Scholar 

  13. Luttenberger, M., Schlund, M.: An extension of Parikh’s theorem beyond idempotence. Tech. rep., TU München (2011), http://arxiv.org/abs/1112.2864

  14. Mohri, M.: Semiring frameworks and algorithms for shortest-distance problems. J. Autom. Lang. Comb. 7(3), 321–350 (2002)

    MathSciNet  MATH  Google Scholar 

  15. Petre, I.: Parikh’s theorem does not hold for multiplicities. J. Autom. Lang. Comb. 4(1), 17–30 (1999)

    MathSciNet  MATH  Google Scholar 

  16. Pilling, D.L.: Commutative regular equations and Parikh’s theorem. J. London Math. Soc., 663–666 (1973)

    Google Scholar 

  17. Pivoteau, C., Salvy, B., Soria, M.: Algorithms for combinatorial structures: Well-founded systems and newton iterations. J. Comb. Theory, Ser. A 119(8), 1711–1773 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Reps, T.W., Schwoon, S., Jha, S., Melski, D.: Weighted pushdown systems and their application to interprocedural dataflow analysis. Sci. Comput. Program. 58(1-2), 206–263 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  19. Rozenberg, G.: Handbook of formal languages: Word, language, grammar, vol. 1. Springer (1997)

    Google Scholar 

  20. Schwoon, S., Jha, S., Reps, T.W., Stubblebine, S.G.: On generalized authorization problems. In: CSFW, pp. 202–218 (2003)

    Google Scholar 

  21. Verma, K.N., Seidl, H., Schwentick, T.: On the Complexity of Equational Horn Clauses. In: Nieuwenhuis, R. (ed.) CADE 2005. LNCS (LNAI), vol. 3632, pp. 337–352. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

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Authors and Affiliations

  1. Institut für Informatik, Technische Universität München, Boltzmannstr. 3, 85748, Garching, Germany

    Michael Luttenberger & Maximilian Schlund

Authors
  1. Michael Luttenberger
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  2. Maximilian Schlund
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Editor information

Editors and Affiliations

  1. Research Group on Mathematical Linguistics, Universitat Rovira i Virgili, Avinguda Catalunya, 35, 43002, Tarragona, Spain

    Adrian-Horia Dediu  & Carlos Martín-Vide  & 

  2. Fakultät für Informatik, Institut für Wissens- und Sprachverarbeitung, Otto-von-Guericke-Universität Magdeburg, Universitätsplatz 2, 39106, Magdeburg, Germany

    Bianca Truthe

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Luttenberger, M., Schlund, M. (2013). Convergence of Newton’s Method over Commutative Semirings. In: Dediu, AH., Martín-Vide, C., Truthe, B. (eds) Language and Automata Theory and Applications. LATA 2013. Lecture Notes in Computer Science, vol 7810. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-37064-9_36

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  • DOI: https://doi.org/10.1007/978-3-642-37064-9_36

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-37063-2

  • Online ISBN: 978-3-642-37064-9

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