## Tolerance spaces and approximative representational structures (2003)

Venue: | Proceedings 26th German Conference on Artificial Intelligence, volume 2821 of LNAI |

Citations: | 14 - 11 self |

### BibTeX

@INPROCEEDINGS{Doherty03tolerancespaces,

author = {Patrick Doherty},

title = {Tolerance spaces and approximative representational structures},

booktitle = {Proceedings 26th German Conference on Artificial Intelligence, volume 2821 of LNAI},

year = {2003},

pages = {475--489},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Abstract. In traditional approaches to knowledge representation, notions such as tolerance measures on data, distance between objects or individuals, and similarity measures between primitive and complex data structures are rarely considered. There is often a need to use tolerance and similarity measures in processes of data and knowledge abstraction because many complex systems which have knowledge representation components such as robots or software agents receive and process data which is incomplete, noisy, approximative and uncertain. This paper presents a framework for recursively constructing arbitrarily complex knowledge structures which may be compared for similarity, distance and approximativeness. It integrates nicely with more traditional knowledge representation techniques and attempts to bridge a gap between approximate and crisp knowledge representation. It can be viewed in part as a generalization of approximate reasoning techniques used in rough set theory. The strategy that will be used is to define tolerance and distance measures on the value sets associated with attributes or primitive data domains associated with particular applications. These tolerance and distance measures will be induced through the different levels of data and knowledge abstraction in complex representational structures. Once the tolerance and similarity measures are in place, an important structuring generalization can be made where the idea of a tolerance space is introduced. Use of these ideas is exemplified using two application domains related to sensor modeling and communication between agents. 1

### Citations

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(Show Context)
Citation Context ...ques and attempts to bridge a gap between approximate and crisp knowledge representation [2]. It can be viewed in part as a generalization of approximate reasoning techniques used in rough set theory =-=[5]-=- where an approximate relation is represented as having both an upper and lower approximation represented as classical sets and an individual in a domain of discourse has additional structure in terms... |

296 |
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(Show Context)
Citation Context ...ed as classical sets and an individual in a domain of discourse has additional structure in terms of attribute/value pairs. It also has connections to recent work by Gärdenfors with conceptual spaces =-=[4]-=-. Ontologically, the world is viewed as consisting of individual elements with associated sets of attribute/value pairs. Each attribute has a value set and tolerance relations will be associated with ... |

54 |
Tolerance approximation spaces
- Skowron, Stepaniuk
- 1996
(Show Context)
Citation Context ...ation. 3 Below, for any set X, by |X| we mean the cardinality of X. 3 A different approach, based on a notion of approximation spaces, object neighborhoods and rough inclusion, has been introduced in =-=[6]-=-.sTolerance Spaces and Approximative Representational Structures 7 Definition 4.2. Let U1, U2 ⊆ U. By the standard inclusion function we mean the function given by µ(U1, U2) def ⎧ ⎨ |U1 ∩ U2| if U1 �=... |

24 |
Multi-Sensor Fusion
- Brooks, Iyengar
- 1998
(Show Context)
Citation Context ... is that of a tolerance function. Let’s begin with a value set V and two elements x, y ∈ V . A tolerance function τ provides us with a distance measure between x and y normalized to the real interval =-=[0, 1]-=- where the higher the value, the closer in tolerance the two elements are. Given a parameter p ∈ [0, 1], a tolerance relation τ p is then introduced among individuals with a threshold p which tunes th... |

5 |
Combining rough and crisp knowledge in deductive databases
- Doherty, ̷Lukaszewicz, et al.
- 2003
(Show Context)
Citation Context ...sequence, we also obtain a characterization of objects that surely do not satisfy the property – finally one can apply various deduction mechanisms to reason about the considered concepts (see, e.g., =-=[2]-=-). 6 We often drop the superscripts and subscripts when the tolerance spaces and relations are known from context. 7 The equality between ν τ p o (U1, U2) and ν τp o (U2, U1) follows from the symmetry... |

4 | On mutual understanding among communicating agents
- Doherty, Lukaszewicz, et al.
- 2003
(Show Context)
Citation Context ...he following approximations: 〈〈Q TA + 2 , Q TA ⊕ 2 〉 TA + 1 , 〈Q TA + 2 , Q TA ⊕ 2 〉 ⊕ TA 〉. (3) 1 The notion of mutual understanding used by communicating agents of this type is developed in full in =-=[3]-=-. 8 Summary This paper presents a framework for recursively constructing arbitrarily complex knowledge structures which may be compared for similarity, distance and approximativeness. The techniques u... |