Chistikov, D;
Haase, C;
(2016)
The taming of the semi-linear set.
In: Chatzigiannakis, I and Mitzenmacher, M and Rabani, Y and Sangiorgi, D, (eds.)
43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016).
(pp. 128:1-128:13).
Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik: Dagstuhl, Germany.
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Abstract
Semi-linear sets, which are rational subsets of the monoid (Z^d,+), have numerous applications in theoretical computer science. Although semi-linear sets are usually given implicitly, by formulas in Presburger arithmetic or by other means, the effect of Boolean operations on semi-linear sets in terms of the size of description has primarily been studied for explicit representations. In this paper, we develop a framework suitable for implicitly presented semi-linear sets, in which the size of a semi-linear set is characterized by its norm—the maximal magnitude of a generator. We put together a toolbox of operations and decompositions for semi-linear sets which gives bounds in terms of the norm (as opposed to just the bit-size of the description), a unified presentation, and simplified proofs. This toolbox, in particular, provides exponentially better bounds for the complement and set-theoretic difference. We also obtain bounds on unambiguous decompositions and, as an application of the toolbox, settle the complexity of the equivalence problem for exponent-sensitive commutative grammars.
Type: | Proceedings paper |
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Title: | The taming of the semi-linear set |
Event: | 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016) |
ISBN-13: | 978-3-95977-013-2 |
Open access status: | An open access version is available from UCL Discovery |
DOI: | 10.4230/LIPIcs.ICALP.2016.128 |
Publisher version: | https://doi.org/10.4230/LIPIcs.ICALP.2016.128 |
Language: | English |
Additional information: | Copyright © Dmitry Chistikov and Christoph Haase; licensed under Creative Commons License CC-BY (http://creativecommons.org/licenses/by/3.0/). |
Keywords: | semi-linear sets, convex polyhedra, triangulations, integer linear programming, commutative grammars |
UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Engineering Science > Dept of Computer Science |
URI: | https://discovery.ucl.ac.uk/id/eprint/10088924 |
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