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Interpolative and extrapolative reasoning in propositional theories using qualitative knowledge about conceptual spaces

Schockaert, Steven and Prade, Henri Interpolative and extrapolative reasoning in propositional theories using qualitative knowledge about conceptual spaces. (2013) Artificial Intelligence, 202. 86-131. ISSN 0004-3702

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Official URL: http://dx.doi.org/10.1016/j.artint.2013.07.001

Abstract

Many logical theories are incomplete, in the sense that non-trivial conclusions about particular situations cannot be derived from them using classical deduction. In this paper, we show how the ideas of interpolation and extrapolation, which are of crucial importance in many numerical domains, can be applied in symbolic settings to alleviate this issue in the case of propositional categorization rules. Our method is based on (mainly) qualitative descriptions of how different properties are conceptually related, where we identify conceptual relations between properties with spatial relations between regions in Gärdenfors conceptual spaces. The approach is centred around the view that categorization rules can often be seen as approximations of linear (or at least monotonic) mappings between conceptual spaces. We use this assumption to justify that whenever the antecedents of a number of rules stand in a relationship that is invariant under linear (or monotonic) transformations, their consequents should also stand in that relationship. A form of interpolative and extrapolative reasoning can then be obtained by applying this idea to the relations of betweenness and parallelism respectively. After discussing these ideas at the semantic level, we introduce a number of inference rules to characterize interpolative and extrapolative reasoning at the syntactic level, and show their soundness and completeness w.r.t. the proposed semantics. Finally, we show that the considered inference problems are PSPACE-hard in general, while implementations in polynomial time are possible under some relatively mild assumptions.

Item Type:Article
Additional Information:Thanks to Elsevier editor. The definitive version is available at http://www.sciencedirect.com The original PDF of the article can be found at Artificial Intelligence website : http://www.sciencedirect.com/science/article/pii/S0004370213000672
HAL Id:hal-01153936
Audience (journal):International peer-reviewed journal
Uncontrolled Keywords:
Institution:French research institutions > Centre National de la Recherche Scientifique - CNRS (FRANCE)
Université de Toulouse > Institut National Polytechnique de Toulouse - INPT (FRANCE)
Université de Toulouse > Université Toulouse III - Paul Sabatier - UPS (FRANCE)
Université de Toulouse > Université Toulouse - Jean Jaurès - UT2J (FRANCE)
Université de Toulouse > Université Toulouse 1 Capitole - UT1 (FRANCE)
Other partners > Cardiff Metropolitan University (UNITED KINGDOM)
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Deposited By: IRIT IRIT
Deposited On:23 Feb 2015 10:21

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