Document

**Published in:** LIPIcs, Volume 96, 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)

We present two structural results concerning the longest common prefixes of non-empty languages.
First, we show that the longest common prefix of the language generated by a context-free grammar of size N
equals the longest common prefix of the same grammar where the heights of the derivation trees are bounded by
4N.
Second, we show that each non-empty language L has a representative subset of at most three elements which behaves
like L w.r.t. the longest common prefix as well as w.r.t. longest common prefixes of L after unions or
concatenations with arbitrary other languages.
From that, we conclude
that the longest common prefix, and thus the longest common suffix, of a context-free language can be computed in polynomial time.

Michael Luttenberger, Raphaela Palenta, and Helmut Seidl. Computing the Longest Common Prefix of a Context-free Language in Polynomial Time. In 35th Symposium on Theoretical Aspects of Computer Science (STACS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 96, pp. 48:1-48:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{luttenberger_et_al:LIPIcs.STACS.2018.48, author = {Luttenberger, Michael and Palenta, Raphaela and Seidl, Helmut}, title = {{Computing the Longest Common Prefix of a Context-free Language in Polynomial Time}}, booktitle = {35th Symposium on Theoretical Aspects of Computer Science (STACS 2018)}, pages = {48:1--48:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-062-0}, ISSN = {1868-8969}, year = {2018}, volume = {96}, editor = {Niedermeier, Rolf and Vall\'{e}e, Brigitte}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2018.48}, URN = {urn:nbn:de:0030-drops-84828}, doi = {10.4230/LIPIcs.STACS.2018.48}, annote = {Keywords: longest common prefix, context-free languages, combinatorics on words} }

Document

**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

Monotone systems of polynomial equations (MSPEs) are systems of
fixed-point equations $X_1 = f_1(X_1, ldots, X_n),$ $ldots, X_n =
f_n(X_1, ldots, X_n)$ where each $f_i$ is a polynomial with
positive real coefficients. The question of computing the least
non-negative solution of a given MSPE $vec X = vec f(vec X)$
arises naturally in the analysis of stochastic models such as
stochastic context-free grammars, probabilistic pushdown automata,
and back-button processes. Etessami and Yannakakis have recently
adapted Newton's iterative method to MSPEs. In a previous paper we
have proved the existence of a threshold $k_{vec f}$ for strongly
connected MSPEs, such that after $k_{vec f}$ iterations of
Newton's method each new iteration computes at least 1 new bit of
the solution. However, the proof was purely existential. In this
paper we give an upper bound for $k_{vec f}$ as a function of the
minimal component of the least fixed-point $muvec f$ of $vec
f(vec X)$. Using this result we show that $k_{vec f}$ is at most
single exponential resp. linear for strongly connected MSPEs
derived from probabilistic pushdown automata resp. from
back-button processes. Further, we prove the existence of a
threshold for arbitrary MSPEs after which each new iteration
computes at least $1/w2^h$ new bits of the solution, where $w$ and
$h$ are the width and height of the DAG of strongly connected
components.

Javier Esparza, Stefan Kiefer, and Michael Luttenberger. Convergence Thresholds of Newton's Method for Monotone Polynomial Equations. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 289-300, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{esparza_et_al:LIPIcs.STACS.2008.1351, author = {Esparza, Javier and Kiefer, Stefan and Luttenberger, Michael}, title = {{Convergence Thresholds of Newton's Method for Monotone Polynomial Equations}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {289--300}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1351}, URN = {urn:nbn:de:0030-drops-13516}, doi = {10.4230/LIPIcs.STACS.2008.1351}, annote = {Keywords: Newton's Method, Fixed-Point Equations, Formal Verification of Software, Probabilistic Pushdown Systems} }

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