Abstract
Fixed point equations x = F(x) over ω-continuous semirings are a natural mathematical foundation of interprocedural program analysis. Equations over the semiring of the real numbers can be solved numerically using Newton’s method. We generalize the method to any ω-continuous semiring and show that it converges faster to the least fixed point than the Kleene sequence 0, F(0), F(F(0)),... We prove that the Newton approximants in the semiring of languages coincide with finite-index approximations studied by several authors in the 1960s. Finally, we apply our results to the analysis of stochastic context-free grammars.
This work was partially supported by the DFG project Algorithms for Software Model Checking.
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Esparza, J., Kiefer, S., Luttenberger, M. (2007). An Extension of Newton’s Method to ω-Continuous Semirings. In: Harju, T., Karhumäki, J., Lepistö, A. (eds) Developments in Language Theory. DLT 2007. Lecture Notes in Computer Science, vol 4588. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73208-2_17
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DOI: https://doi.org/10.1007/978-3-540-73208-2_17
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