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”.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bloom, S.L., Ésik, Z.: Axiomatizing rational power series over natural numbers. Inf. Comput. 207(7), 793–811 (2009)
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)
Bozapalidis, S.: Equational elements in additive algebras. Theory Comput. Syst. 32(1), 1–33 (1999)
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)
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)
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)
Esparza, J., Kiefer, S., Luttenberger, M.: Newtonian program analysis. J. ACM 57(6), 33 (2010)
Etessami, K., Yannakakis, M.: Recursive markov chains, stochastic grammars, and monotone systems of nonlinear equations. J. ACM 56(1) (2009)
Flajolet, P., Raoult, J.C., Vuillemin, J.: The number of registers required for evaluating arithmetic expressions. Theor. Comput. Sci. 9, 99–125 (1979)
Foster, J.N., Karvounarakis, G.: Provenance and data synchronization. IEEE Data Eng. Bull. 30(4), 13–21 (2007)
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)
Green, T.J., Karvounarakis, G., Tannen, V.: Provenance semirings. In: PODS, pp. 31–40 (2007)
Luttenberger, M., Schlund, M.: An extension of Parikh’s theorem beyond idempotence. Tech. rep., TU München (2011), http://arxiv.org/abs/1112.2864
Mohri, M.: Semiring frameworks and algorithms for shortest-distance problems. J. Autom. Lang. Comb. 7(3), 321–350 (2002)
Petre, I.: Parikh’s theorem does not hold for multiplicities. J. Autom. Lang. Comb. 4(1), 17–30 (1999)
Pilling, D.L.: Commutative regular equations and Parikh’s theorem. J. London Math. Soc., 663–666 (1973)
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)
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)
Rozenberg, G.: Handbook of formal languages: Word, language, grammar, vol. 1. Springer (1997)
Schwoon, S., Jha, S., Reps, T.W., Stubblebine, S.G.: On generalized authorization problems. In: CSFW, pp. 202–218 (2003)
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
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
eBook Packages: Computer ScienceComputer Science (R0)