### Abstract

In ISSAC 2013 the current authors presented an algorithm that can efficiently and directly construct a TTICAD for a list of formulae in which each has an equational constraint. This was achieved by generalising McCallum's theory of reduced projection operators. In this paper we present an extended version of our theory which can be applied to an arbitrary list of formulae, achieving savings if at least one has an equational constraint. We then explain how the theory of reduced projection operators can allow for further improvements to the lifting phase of CAD algorithms, even in the context of a single equational constraint.

The algorithm is implemented fully in Maple and we present both promising results from experimentation and a complexity analysis showing the benefits of our new contributions.

Original language | English |
---|---|

Pages (from-to) | 1-35 |

Number of pages | 35 |

Journal | Journal of Symbolic Computation |

Volume | 76 |

Early online date | 4 Nov 2015 |

DOIs | |

Publication status | Published - 1 Sep 2016 |

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### Keywords

- cylindrical algebraic decomposition
- equational constraint

### Cite this

*Journal of Symbolic Computation*,

*76*, 1-35. https://doi.org/10.1016/j.jsc.2015.11.002

**Truth table invariant cylindrical algebraic decomposition.** / Bradford, Russell; Davenport, James H.; England, Matthew; McCallum, Scott; Wilson, David.

Research output: Contribution to journal › Article

*Journal of Symbolic Computation*, vol. 76, pp. 1-35. https://doi.org/10.1016/j.jsc.2015.11.002

}

TY - JOUR

T1 - Truth table invariant cylindrical algebraic decomposition

AU - Bradford, Russell

AU - Davenport, James H.

AU - England, Matthew

AU - McCallum, Scott

AU - Wilson, David

PY - 2016/9/1

Y1 - 2016/9/1

N2 - When using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is likely not the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This observation motivates our article and definition of a Truth Table Invariant CAD (TTICAD).In ISSAC 2013 the current authors presented an algorithm that can efficiently and directly construct a TTICAD for a list of formulae in which each has an equational constraint. This was achieved by generalising McCallum's theory of reduced projection operators. In this paper we present an extended version of our theory which can be applied to an arbitrary list of formulae, achieving savings if at least one has an equational constraint. We then explain how the theory of reduced projection operators can allow for further improvements to the lifting phase of CAD algorithms, even in the context of a single equational constraint. The algorithm is implemented fully in Maple and we present both promising results from experimentation and a complexity analysis showing the benefits of our new contributions.

AB - When using cylindrical algebraic decomposition (CAD) to solve a problem with respect to a set of polynomials, it is likely not the signs of those polynomials that are of paramount importance but rather the truth values of certain quantifier free formulae involving them. This observation motivates our article and definition of a Truth Table Invariant CAD (TTICAD).In ISSAC 2013 the current authors presented an algorithm that can efficiently and directly construct a TTICAD for a list of formulae in which each has an equational constraint. This was achieved by generalising McCallum's theory of reduced projection operators. In this paper we present an extended version of our theory which can be applied to an arbitrary list of formulae, achieving savings if at least one has an equational constraint. We then explain how the theory of reduced projection operators can allow for further improvements to the lifting phase of CAD algorithms, even in the context of a single equational constraint. The algorithm is implemented fully in Maple and we present both promising results from experimentation and a complexity analysis showing the benefits of our new contributions.

KW - cylindrical algebraic decomposition

KW - equational constraint

U2 - 10.1016/j.jsc.2015.11.002

DO - 10.1016/j.jsc.2015.11.002

M3 - Article

VL - 76

SP - 1

EP - 35

JO - Journal of Symbolic Computation

JF - Journal of Symbolic Computation

SN - 0747-7171

ER -