## Undecidability of Plane Polygonal Mereotopology (1998)

Venue: | Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference (KR-98 |

Citations: | 20 - 0 self |

### BibTeX

@INPROCEEDINGS{Dornheim98undecidabilityof,

author = {Christoph Dornheim and Fachbereich Informatik},

title = {Undecidability of Plane Polygonal Mereotopology},

booktitle = {Principles of Knowledge Representation and Reasoning: Proceedings of the 6th International Conference (KR-98},

year = {1998},

pages = {342--353},

publisher = {Morgan Kaufman}

}

### Years of Citing Articles

### OpenURL

### Abstract

This paper presents a mereotopological model of polygonal regions of the Euclidean plane and an undecidability proof of its firstorder theory. Restricted to the primitive operations the model is a Boolean algebra. Its single primitive predicate defines simple polygons as the topologically simplest polygonal regions. It turns out that both the relations usually provided by mereotopologies and more subtle topological relations are elementarily definable in the model. Using these relations, Post's correspondence problem, known as undecidable, can be reduced to the decision problem of the model. 1 Introduction Formalizing commonsense concepts of space has received much attention both in the philosophical literature and in recent AI research. Mereotopological theories as well as most calculi for spatial reasoning deal with spatial regions, i.e. the parts of space occupied by physical bodies, and their topological relations as intuitive concepts of our commonsense space. Whereas mereotopolo...

### Citations

3938 |
Introduction to automata theory, languages, and computation
- Hopcroft, Ullman
- 1980
(Show Context)
Citation Context ...\Delta \Delta ff i k = fi i 1 \Delta \Delta \Delta fi i k . Post's correspondence problem is then to determine if a given PCP-instance has a solution. As it is well-known, this problem is undecidable =-=[11]-=-. The key in proving any T ae L SM 0 with TM ` T ` Th(M) to be undecidable is to reduce Post's correspondence problem to the decision problem of T , that is, to associate with any PCP-instance C a sen... |

582 | A spatial logic based on regions and connection
- Randell, Cui, et al.
- 1992
(Show Context)
Citation Context ...gical and topological primitives, we refer to [19] and [9]. The first approach in defining a mereotopology is due to Whitehead [20]; some other approaches are presented by Tarski [18], Randell et al. =-=[16]-=- and Borgo et al. [4]. Most mereotopologies are defined axiomatically by taking as axioms propositions that are considered intuitive for regions and their relations. The overall structure of these reg... |

109 | On the complexity of qualitative spatial reasoning: A maximal tractable fragment of the region connection calculus
- Renz, Nebel
- 1999
(Show Context)
Citation Context ... spatial reasoning. Since the underlying semantics of many calculi (e.g. [16], [2], [3]) are too weak to represent the intended regions, these calculi are of limited use, though they can be tractable =-=[17]-=-. Therefore, an appropriate mereotopological model M can serve equally as a formal semantics for calculi for spatial reasoning, because the statements on regions and relations, which are computed, can... |

105 | Toward a Geometry of Common Sense: A Semantics and a Complete Axiomatization of Mereotopology
- Asher, Vieu
- 1995
(Show Context)
Citation Context ...ng whether Th(M) can be axiomatized elementarily. So far, such questions on semantics have gained only little attention in the literature; some models for mereotopology are proposed by Asher and Vieu =-=[1]-=- and by Pratt and Lemon [14]. These considerations are also important for spatial reasoning. Since the underlying semantics of many calculi (e.g. [16], [2], [3]) are too weak to represent the intended... |

100 | Spatial reasoning with propositional logics
- Bennett
- 1994
(Show Context)
Citation Context ...mereotopology are proposed by Asher and Vieu [1] and by Pratt and Lemon [14]. These considerations are also important for spatial reasoning. Since the underlying semantics of many calculi (e.g. [16], =-=[2]-=-, [3]) are too weak to represent the intended regions, these calculi are of limited use, though they can be tractable [17]. Therefore, an appropriate mereotopological model M can serve equally as a fo... |

98 | Qualitative spatial representation and reasoning techniques
- Cohn
- 1997
(Show Context)
Citation Context ...relations, the research aim of qualitative spatial reasoningsis the development of calculi to reason about regions and their relations as efficiently as possible (a recent survey is presented by Cohn =-=[6]-=-). Mereotopologies are formal theories, mostly specified in predicate logic, that describe general properties The author's new address is: Institut fur Informatik, Albert-Ludwigs-Universitat Freiburg,... |

83 | A Pointless Theory of Space Based On Strong Connection and Congruence
- Borgo, Guarino, et al.
(Show Context)
Citation Context ...primitives, we refer to [19] and [9]. The first approach in defining a mereotopology is due to Whitehead [20]; some other approaches are presented by Tarski [18], Randell et al. [16] and Borgo et al. =-=[4]-=-. Most mereotopologies are defined axiomatically by taking as axioms propositions that are considered intuitive for regions and their relations. The overall structure of these regions, together with t... |

82 | Modal logics for qualitative spatial reasoning
- Bennett
- 1996
(Show Context)
Citation Context ...topology are proposed by Asher and Vieu [1] and by Pratt and Lemon [14]. These considerations are also important for spatial reasoning. Since the underlying semantics of many calculi (e.g. [16], [2], =-=[3]-=-) are too weak to represent the intended regions, these calculi are of limited use, though they can be tractable [17]. Therefore, an appropriate mereotopological model M can serve equally as a formal ... |

62 |
Process and Reality. An Essay on Cosmology
- Whitehead
- 1929
(Show Context)
Citation Context ...ereotopology, especially the ways of defining mereotopology using mereological and topological primitives, we refer to [19] and [9]. The first approach in defining a mereotopology is due to Whitehead =-=[20]-=-; some other approaches are presented by Tarski [18], Randell et al. [16] and Borgo et al. [4]. Most mereotopologies are defined axiomatically by taking as axioms propositions that are considered intu... |

43 |
Geometric topology
- Moise
- 1977
(Show Context)
Citation Context ... a i+1 ] and any point of fa 1 ; : : : ; an g belongs to at most two line segments, then we call the polygonal line simple. For simple closed polygonal lines the well-known Jordan curve theorem holds =-=[13]-=-. Theorem 1 Let p be a simple closed polygonal line. Then R 2 n p is a disjoint union of two open domains, a bounded inner domain D I p and an unbounded outer domain D O p , each of which has the boun... |

42 | A complete axiom system for polygonal mereotopology of the real plane
- Pratt, Schoop
- 1998
(Show Context)
Citation Context ...r regions of M: replace the squares shown in figure 4 with three-dimensional cubes arranged on the same plane. Probably, the closest mereotopological model to M is proposed by Pratt, Lemon and Schoop =-=[14, 15]-=-. Its complete axiomatization is grounded on an infinitary rule of inference allowing proofs of infinite length and is therefore without any consequence for its decision problem (in contrast to the el... |

31 |
W.: Classical mereology and restricted domains
- Eschenbach, Heydrich
- 1995
(Show Context)
Citation Context ...tirely by the part-whole relation. For an overview about mereology and mereotopology, especially the ways of defining mereotopology using mereological and topological primitives, we refer to [19] and =-=[9]-=-. The first approach in defining a mereotopology is due to Whitehead [20]; some other approaches are presented by Tarski [18], Randell et al. [16] and Borgo et al. [4]. Most mereotopologies are define... |

31 | Ontologies for plane, polygonal mereotopology
- Pratt, Lemon
- 1998
(Show Context)
Citation Context ...omatized elementarily. So far, such questions on semantics have gained only little attention in the literature; some models for mereotopology are proposed by Asher and Vieu [1] and by Pratt and Lemon =-=[14]-=-. These considerations are also important for spatial reasoning. Since the underlying semantics of many calculi (e.g. [16], [2], [3]) are too weak to represent the intended regions, these calculi are ... |

27 |
Undecidability of some topological theories
- Grzegorczyk
- 1951
(Show Context)
Citation Context ... boundedness. A method of defining finite sums can be obtained from Grzegorczyk's undecidability proof of closure algebra, i.e. Boolean algebra extended by some axioms for the predicate "x is clo=-=sed" [10]-=-. The main idea is to interpret the arithmetic of natural numbers, well-known as undecidable, by elementarily definable sets of regions and relations between regions. In particular, the natural number... |

25 |
General theory of boolean algebras
- Koppelberg
- 1989
(Show Context)
Citation Context ...t and complement, respectively: x + y := x [ y; x \Delta y := (x " y) ffi ; \Gammax := R 2 n x: As is well known, the structure RC(R 2 ) := hRC(R 2 ); +; \Delta; \Gamma; ;; R 2 i is a Boolean alg=-=ebra [12]-=-. With regard to condition 1, we have to ensure that M, restricted to these operations, turns out as a subalgebra of RC(R 2 ). The elements of RC(R 2 ) themselves appear to be too general to represent... |

20 |
part-whole relations: the prospects of mereotopology
- Parts
- 1996
(Show Context)
Citation Context ...efined entirely by the part-whole relation. For an overview about mereology and mereotopology, especially the ways of defining mereotopology using mereological and topological primitives, we refer to =-=[19]-=- and [9]. The first approach in defining a mereotopology is due to Whitehead [20]; some other approaches are presented by Tarski [18], Randell et al. [16] and Borgo et al. [4]. Most mereotopologies ar... |

1 |
Unvollstandigkeit topologischer Kalkule zum qualitativen raumlichen Schlieen
- Dornheim
- 1998
(Show Context)
Citation Context ... encode Post's correspondence problem. In the remainder of this paper we shall abbreviate terms like x \Delta y by xy. 1 The detailed proofs of the propositions given in this section are contained in =-=[7]-=-. Clearly, ` can be defined in M by the partial ordersusually associated with a Boolean algebra by xsy j def x + y = y. Also trivially, the set of polygons POL is definable in M, since polygons are ex... |

1 |
Foundations of the geometry of solids. Logic
- Tarski
- 1956
(Show Context)
Citation Context ...opology using mereological and topological primitives, we refer to [19] and [9]. The first approach in defining a mereotopology is due to Whitehead [20]; some other approaches are presented by Tarski =-=[18]-=-, Randell et al. [16] and Borgo et al. [4]. Most mereotopologies are defined axiomatically by taking as axioms propositions that are considered intuitive for regions and their relations. The overall s... |