### Abstract

This paper studies the expressivity and computational complexity of networks of constraints of topological relations together with convexity. We consider constraint networks whose nodes are regular regions (a regular region is one equal to the closure of its interior) and whose constraints have the following forms: (i) the eight "base relations" of [12], which describe binary topological relations of containment and adjacency between regions; (ii) the predicate, "X is convex." We establish tight bounds on the computational complexity of this language: Determining whether such a constraint network is consistent is decidable, but essentially as hard as determining whether a set of comparable size of algebraic constraints over the real numbers is consistent. We also show an important expressivity result for this language: If r and s are bounded, regular regions that are not related by an affine transformation, then they can be distinguished by a constraint network. That is, there is a constraint network and a particular node in that network such that there is a solution where the node is equal to r, but no solution where the node is equal to s.

Original language | English (US) |
---|---|

Pages (from-to) | 241-280 |

Number of pages | 40 |

Journal | Constraints |

Volume | 4 |

Issue number | 3 |

State | Published - 1999 |

### Fingerprint

### Keywords

- Complexity
- Constraints
- Convexity
- Expressivity
- RCC8
- Topology

### ASJC Scopus subject areas

- Hardware and Architecture
- Applied Mathematics

### Cite this

*Constraints*,

*4*(3), 241-280.

**Constraint Networks of Topological Relations and Convexity.** / Davis, Ernest; Gotts, Nicholas Mark; Cohn, Anthony G.

Research output: Contribution to journal › Article

*Constraints*, vol. 4, no. 3, pp. 241-280.

}

TY - JOUR

T1 - Constraint Networks of Topological Relations and Convexity

AU - Davis, Ernest

AU - Gotts, Nicholas Mark

AU - Cohn, Anthony G.

PY - 1999

Y1 - 1999

N2 - This paper studies the expressivity and computational complexity of networks of constraints of topological relations together with convexity. We consider constraint networks whose nodes are regular regions (a regular region is one equal to the closure of its interior) and whose constraints have the following forms: (i) the eight "base relations" of [12], which describe binary topological relations of containment and adjacency between regions; (ii) the predicate, "X is convex." We establish tight bounds on the computational complexity of this language: Determining whether such a constraint network is consistent is decidable, but essentially as hard as determining whether a set of comparable size of algebraic constraints over the real numbers is consistent. We also show an important expressivity result for this language: If r and s are bounded, regular regions that are not related by an affine transformation, then they can be distinguished by a constraint network. That is, there is a constraint network and a particular node in that network such that there is a solution where the node is equal to r, but no solution where the node is equal to s.

AB - This paper studies the expressivity and computational complexity of networks of constraints of topological relations together with convexity. We consider constraint networks whose nodes are regular regions (a regular region is one equal to the closure of its interior) and whose constraints have the following forms: (i) the eight "base relations" of [12], which describe binary topological relations of containment and adjacency between regions; (ii) the predicate, "X is convex." We establish tight bounds on the computational complexity of this language: Determining whether such a constraint network is consistent is decidable, but essentially as hard as determining whether a set of comparable size of algebraic constraints over the real numbers is consistent. We also show an important expressivity result for this language: If r and s are bounded, regular regions that are not related by an affine transformation, then they can be distinguished by a constraint network. That is, there is a constraint network and a particular node in that network such that there is a solution where the node is equal to r, but no solution where the node is equal to s.

KW - Complexity

KW - Constraints

KW - Convexity

KW - Expressivity

KW - RCC8

KW - Topology

UR - http://www.scopus.com/inward/record.url?scp=0000933941&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0000933941&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0000933941

VL - 4

SP - 241

EP - 280

JO - Constraints

JF - Constraints

SN - 1383-7133

IS - 3

ER -