Abstract
Fixed point equations x = f(x) over ω-continuous semirings can be seen as the mathematical foundation of interprocedural program analysis. The sequence 0, f(0),f 2(0),... converges to the least fixed point μ f. The convergence can be accelerated if the underlying semiring is commutative. We show that accelerations in the literature, namely Newton’s method for the arithmetic semiring [4] and an acceleration for commutative Kleene algebras due to Hopkins and Kozen [5], are instances of a general algorithm for arbitrary commutative ω-continuous semirings. In a second contribution, we improve the \(\mathcal{O}(3^n)\) bound of [5] and show that their acceleration reaches μ f after n iterations, where n is the number of equations. Finally, we apply the Hopkins-Kozen acceleration to itself and study the resulting hierarchy of increasingly fast accelerations.
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References
Conway, J.H.: Regular Algebra and Finite Machines. Chapman and Hall, Boca Raton (1971)
Esparza, J., Kiefer, S., Luttenberger, M.: On fixed point equations over commutative semirings. Technical report (2006)
Esparza, J., Kučera, A., Mayr, R.: Model checking probabilistic pushdown automata. In: LICS 2004, IEEE Computer Society Press, Los Alamitos (2004)
Etessami, K., Yannakakis, M.: Recursive Markov chains, stochastic grammars, and monotone systems of nonlinear equations. In: STACS, pp. 340–352 (2005)
Hopkins, M.W., Kozen, D.: Parikh’s theorem in commutative Kleene algebra. In: Logic in Computer Science, pp. 394–401 (1999)
Kozen, D.: On Kleene algebras and closed semirings. In: Rovan, B. (ed.) Mathematical Foundations of Computer Science 1990. LNCS, vol. 452, pp. 26–47. Springer, Heidelberg (1990)
Kuich, W.: Semirings and Formal Power Series: Their Relevance to Formal Languages and Automata. In: Rozenberg, G., Salomaa, A. (eds.) Handbook of Formal Languages, vol. 1, p. 609. Springer, Heidelberg (1997)
Ortega, J.M.: Numerical Analysis: A Second Course. Academic Press, London (1972)
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Esparza, J., Kiefer, S., Luttenberger, M. (2007). On Fixed Point Equations over Commutative Semirings. In: Thomas, W., Weil, P. (eds) STACS 2007. STACS 2007. Lecture Notes in Computer Science, vol 4393. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70918-3_26
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DOI: https://doi.org/10.1007/978-3-540-70918-3_26
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-70917-6
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