Reference:

Javier Esparza, Stefan Kiefer, and Michael Luttenberger. Derivation tree analysis for accelerated fixed-point computation. In Masami Ito and Masafumi Toyama, editors, Proceedings of the 12th International Conference on Developments in Language Theory (DLT), volume 5257 of Lecture Notes in Computer Science, pages 301–313, Kyoto, Japan, 2008. Springer.

Abstract:

We show that for several classes of idempotent semirings the least fixed-point of a polynomial system of equations X = f(X) is equal to the least fixed-point of a linear system obtained by linearizing the polynomials of f in a certain way. Our proofs rely on derivation tree analysis, a proof principle that combines methods from algebra, calculus, and formal language theory, and was used to show that Newton's method over commutative and idempotent semirings converges in a linear number of steps. Our results lead to efficient generic algorithms for computing the least fixed-point. We use these algorithms to derive several consequences, including an O(N^3) algorithm for computing the throughput of a context-free grammar (obtained by speeding up the O(N^4) algorithm of Caucal et al.), and a generalization of Courcelle's result stating that the downward-closed image of a context-free language is regular.

Suggested BibTeX entry:

@inproceedings{EKL08:dlt,
    address = {Kyoto, Japan},
    author = {Javier Esparza and Stefan Kiefer and Michael Luttenberger},
    booktitle = {Proceedings of the 12th International Conference on Developments in Language Theory (DLT)},
    editor = {Masami Ito and Masafumi Toyama},
    pages = {301--313},
    publisher = {Springer},
    series = {Lecture Notes in Computer Science},
    title = {Derivation Tree Analysis for Accelerated Fixed-Point Computation},
    volume = {5257},
    year = {2008}
}