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

Schockaert, S; Prade, H

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

Schockaert, S Prade, H

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Publication Type:: Journal Article
Citation:: Artificial Intelligence, 2013, 202 pp. 86 - 131
Issue Date:: 2013-08-23

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Field	Value	Language
dc.contributor.author	Schockaert, S	en_US
dc.contributor.author	Prade, H https://orcid.org/0000-0003-4586-8527	en_US
dc.date.issued	2013-08-23	en_US
dc.identifier.citation	Artificial Intelligence, 2013, 202 pp. 86 - 131	en_US
dc.identifier.issn	0004-3702	en_US
dc.identifier.uri	http://hdl.handle.net/10453/32272
dc.description.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. © 2013 Elsevier B.V. All rights reserved.	en_US
dc.relation.ispartof	Artificial Intelligence	en_US
dc.relation.isbasedon	10.1016/j.artint.2013.07.001	en_US
dc.subject.classification	Artificial Intelligence & Image Processing	en_US
dc.title	Interpolative and extrapolative reasoning in propositional theories using qualitative knowledge about conceptual spaces	en_US
dc.type	Journal Article
utslib.citation.volume	202	en_US
utslib.for	0801 Artificial Intelligence and Image Processing	en_US
utslib.for	1702 Cognitive Sciences	en_US
utslib.for	0802 Computation Theory and Mathematics	en_US
pubs.embargo.period	Not known	en_US
pubs.organisational-group	/University of Technology Sydney
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology
pubs.organisational-group	/University of Technology Sydney/Faculty of Engineering and Information Technology/School of Software
pubs.organisational-group	/University of Technology Sydney/Strength - CAI - Centre for Artificial Intelligence
utslib.copyright.status	closed_access
pubs.publication-status	Published	en_US
pubs.volume	202	en_US

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. © 2013 Elsevier B.V. All rights reserved.

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http://hdl.handle.net/10453/32272