Cylindrical Algebraic Sub-Decompositions

D. J. Wilson, R. J. Bradford, J. H. Davenport, M. England

Research output: Contribution to journalArticle

13 Citations (Scopus)
10 Downloads (Pure)

Abstract

Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semi-algebraic sets. In this paper we introduce cylindrical algebraic sub-decompositions (sub-CADs), which are subsets of CADs containing all the information needed to specify a solution for a given problem. We define two new types of sub-CAD: variety sub-CADs which are those cells in a CAD lying on a designated variety; and layered sub-CADs which have only those cells of dimension higher than a specified value. We present algorithms to produce these and describe how the two approaches may be combined with each other and the recent theory of truth-table invariant CAD. We give a complexity analysis showing that these techniques can offer substantial theoretical savings, which is supported by experimentation using an implementation in Maple.


Original languageEnglish
Pages (from-to)263-288
Number of pages26
JournalMathematics in Computer Science
Volume8
Issue number2
Early online date13 Jun 2014
DOIs
Publication statusPublished - 2014
Externally publishedYes

Fingerprint

Decomposition
Decompose
Real Algebraic Geometry
Truth table
Semi-algebraic Sets
Complexity Analysis
Maple
Cell
Quantifiers
Experimentation
Higher Dimensions
Subset
Invariant
Geometry

Bibliographical note

Publisher Statement: The final publication is available at Springer via http://dx.doi.org/10.1007/s11786-014-0191-z

Keywords

  • Computer algebra
  • Cylindrical algebraic decomposition
  • Equational constraints
  • Real algebraic geometry
  • Symbolic computation

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Computational Theory and Mathematics

Cite this

Cylindrical Algebraic Sub-Decompositions. / Wilson, D. J.; Bradford, R. J.; Davenport, J. H.; England, M.

In: Mathematics in Computer Science, Vol. 8, No. 2, 2014, p. 263-288.

Research output: Contribution to journalArticle

Wilson, D. J. ; Bradford, R. J. ; Davenport, J. H. ; England, M. / Cylindrical Algebraic Sub-Decompositions. In: Mathematics in Computer Science. 2014 ; Vol. 8, No. 2. pp. 263-288.
@article{6be1875d93154bf9846697fba5fcee63,
title = "Cylindrical Algebraic Sub-Decompositions",
abstract = "Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semi-algebraic sets. In this paper we introduce cylindrical algebraic sub-decompositions (sub-CADs), which are subsets of CADs containing all the information needed to specify a solution for a given problem. We define two new types of sub-CAD: variety sub-CADs which are those cells in a CAD lying on a designated variety; and layered sub-CADs which have only those cells of dimension higher than a specified value. We present algorithms to produce these and describe how the two approaches may be combined with each other and the recent theory of truth-table invariant CAD. We give a complexity analysis showing that these techniques can offer substantial theoretical savings, which is supported by experimentation using an implementation in Maple.",
keywords = "Computer algebra, Cylindrical algebraic decomposition, Equational constraints, Real algebraic geometry, Symbolic computation",
author = "Wilson, {D. J.} and Bradford, {R. J.} and Davenport, {J. H.} and M. England",
note = "Publisher Statement: The final publication is available at Springer via http://dx.doi.org/10.1007/s11786-014-0191-z",
year = "2014",
doi = "10.1007/s11786-014-0191-z",
language = "English",
volume = "8",
pages = "263--288",
journal = "Mathematics in Computer Science",
issn = "1661-8270",
publisher = "Springer Verlag",
number = "2",

}

TY - JOUR

T1 - Cylindrical Algebraic Sub-Decompositions

AU - Wilson, D. J.

AU - Bradford, R. J.

AU - Davenport, J. H.

AU - England, M.

N1 - Publisher Statement: The final publication is available at Springer via http://dx.doi.org/10.1007/s11786-014-0191-z

PY - 2014

Y1 - 2014

N2 - Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semi-algebraic sets. In this paper we introduce cylindrical algebraic sub-decompositions (sub-CADs), which are subsets of CADs containing all the information needed to specify a solution for a given problem. We define two new types of sub-CAD: variety sub-CADs which are those cells in a CAD lying on a designated variety; and layered sub-CADs which have only those cells of dimension higher than a specified value. We present algorithms to produce these and describe how the two approaches may be combined with each other and the recent theory of truth-table invariant CAD. We give a complexity analysis showing that these techniques can offer substantial theoretical savings, which is supported by experimentation using an implementation in Maple.

AB - Cylindrical algebraic decompositions (CADs) are a key tool in real algebraic geometry, used primarily for eliminating quantifiers over the reals and studying semi-algebraic sets. In this paper we introduce cylindrical algebraic sub-decompositions (sub-CADs), which are subsets of CADs containing all the information needed to specify a solution for a given problem. We define two new types of sub-CAD: variety sub-CADs which are those cells in a CAD lying on a designated variety; and layered sub-CADs which have only those cells of dimension higher than a specified value. We present algorithms to produce these and describe how the two approaches may be combined with each other and the recent theory of truth-table invariant CAD. We give a complexity analysis showing that these techniques can offer substantial theoretical savings, which is supported by experimentation using an implementation in Maple.

KW - Computer algebra

KW - Cylindrical algebraic decomposition

KW - Equational constraints

KW - Real algebraic geometry

KW - Symbolic computation

U2 - 10.1007/s11786-014-0191-z

DO - 10.1007/s11786-014-0191-z

M3 - Article

VL - 8

SP - 263

EP - 288

JO - Mathematics in Computer Science

JF - Mathematics in Computer Science

SN - 1661-8270

IS - 2

ER -