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Reasoning about Categories in Conceptual Spaces
 In Proceedings of the Fourteenth International Joint Conference of Artificial Intelligence
, 2001
"... Understanding the process of categorization is a primary research goal in artificial intelligence. The conceptual space framework provides a flexible approach to modeling contextsensitive categorization via a geometrical representation designed for modeling and managing concepts. In this paper we s ..."
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Cited by 16 (1 self)
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Understanding the process of categorization is a primary research goal in artificial intelligence. The conceptual space framework provides a flexible approach to modeling contextsensitive categorization via a geometrical representation designed for modeling and managing concepts. In this paper we show how algorithms developed in computational geometry, and the Region Connection Calculus can be used to model important aspects of categorization in conceptual spaces. In particular, we demonstrate the feasibility of using existing geometric algorithms to build and manage categories in conceptual spaces, and we show how the Region Connection Calculus can be used to reason about categories and other conceptual regions. 1
Spatial Agents Implemented in a Logical Expressible Language
 RoboCup99: Robot Soccer WorldCup III, LNAI 1856
, 1999
"... In this paper, we present a multilayered architecture for spatial and temporal agents. The focus is laid on the declarativity of the approach, which makes agent scripts expressive and well understandable. They can be realized as (constraint) logic programs. The logical description language is ab ..."
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Cited by 15 (9 self)
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In this paper, we present a multilayered architecture for spatial and temporal agents. The focus is laid on the declarativity of the approach, which makes agent scripts expressive and well understandable. They can be realized as (constraint) logic programs. The logical description language is able to express actions or plans for one and more autonomous and cooperating agents for the RoboCup (Simulator League). The system architecture hosts constraint technology for qualitative spatial reasoning, but quantitative data is taken into account, too. The basic (hardware) layer processes the agent's sensor information. An interface transfers this lowlevel data into a logical representation. It provides facilities to access the preprocessed data and supplies several basic skills. The second layer performs (qualitative) spatial reasoning. On top of this, the third layer enables more complex skills such as passing, offsidedetection etc. At last, the fourth layer establishes acting as a team both by emergent and explicit cooperation. Logic and deduction provide a clean means to specify and also to implement teamwork behavior.
The Ignorance of Bourbaki
, 1990
"... this article writes: "Which half of his brains did Bourbaki use ? My impression is, the left half. Perhaps I am projecting. The Bourbachistes were uncomfortable with the rightbrain mathematics of the Italian geometers, and for good reason: significant portions were suspect and might, if one ta ..."
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Cited by 4 (1 self)
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this article writes: "Which half of his brains did Bourbaki use ? My impression is, the left half. Perhaps I am projecting. The Bourbachistes were uncomfortable with the rightbrain mathematics of the Italian geometers, and for good reason: significant portions were suspect and might, if one takes `true' and `false' to be leftbrain notions and `right' and `wrong' to be rightbrain ones, be justifiably described as right, but false.
Constructive Geometry
, 2009
"... Euclidean geometry, as presented by Euclid, consists of straightedgeandcompass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions ” to “constructive mathematics ” leads to the development of a firstorder t ..."
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Cited by 2 (1 self)
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Euclidean geometry, as presented by Euclid, consists of straightedgeandcompass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions ” to “constructive mathematics ” leads to the development of a firstorder theory ECG of the “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid rather close in spirit to the Elements of Euclid. ECG is axiomatized in a quantifierfree, disjunctionfree way. Unlike previous intuitionistic geometries, it does not have apartness. Unlike previous algebraic theories of geometric constructions, it does not have a testforequality construction. We show that ECG is a good geometric theory, in the sense that with classical logic it is equivalent to textbook theories, and its models are (intuitionistically) planes over Euclidean fields. We then apply the methods of modern metamathematics to this theory, showing that if ECG proves an existential theorem, then the object proved to exist can be constructed from parameters, using the basic constructions of ECG (which correspond to the Euclidean straightedgeandcompass constructions). In particular, objects proved to exist in ECG depend continuously on parameters. We also study the formal relationships between several versions of Euclid’s parallel postulate, and show that each corresponds to a natural axiom system for Euclidean fields. 1 1
The Parallel Postulate in Constructive Geometry
, 2009
"... Euclidean geometry, as presented by Euclid, consists of straightedgeandcompass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions ” to “constructive mathematics ” leads to the development of a firstorder t ..."
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Euclidean geometry, as presented by Euclid, consists of straightedgeandcompass constructions and rigorous reasoning about the results of those constructions. A consideration of the relation of the Euclidean “constructions ” to “constructive mathematics ” leads to the development of a firstorder theory ECG of the “Euclidean Constructive Geometry”, which can serve as an axiomatization of Euclid rather close in spirit to the Elements of Euclid. ECG is axiomatized in a quantifierfree, disjunctionfree way. Unlike previous intuitionistic geometries, it does not have apartness. Unlike previous algebraic theories of geometric constructions, it does not have a testforequality construction. In previous work [3], we have shown that ECG corresponds well to Euclid’s reasoning, and that when it proves an existential theorem, then the things proved to exist can be constructed by Euclidean rulerandcompass constructions. In this paper we take up the the formal relationships between three versions of Euclid’s parallel postulate: Euclid’s own formulation in his Postulate 5, Playfair’s 1795 version, which is the one usually used in modern axiomatizations, and the version used in ECG. We completely settle the questions about which versions imply which others using only constructive logic: ECG’s version implies Euclid 5, which implies Playfair, and none of the reverse implications are provable. 1 1