Homogeneous logical proportions: Their uniqueness and their role in similarity-based prediction

Prade, H; Richard, G

Homogeneous logical proportions: Their uniqueness and their role in similarity-based prediction

Prade, H

Richard, G

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Publication Type:: Conference Proceeding
Citation:: Proceedings of the International Workshop on Temporal Representation and Reasoning, 2012, pp. 402 - 412
Issue Date:: 2012-01-01

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Field	Value	Language
dc.contributor.author	Prade, H https://orcid.org/0000-0003-4586-8527	en_US
dc.contributor.author	Richard, G	en_US
dc.date.issued	2012-01-01	en_US
dc.identifier.citation	Proceedings of the International Workshop on Temporal Representation and Reasoning, 2012, pp. 402 - 412	en_US
dc.identifier.isbn	9781577355601	en_US
dc.identifier.uri	http://hdl.handle.net/10453/32682
dc.description.abstract	Given a 4-tuple of Boolean variables (a, b, c, d), logical proportions are modeled by a pair of equivalences relating similarity indicators (a∧b̄ and ā∧b̄), or dissimilarity indicators (a ∧ b̄ and ā ∧ b) pertaining to the pair (a, b), to the ones associated with the pair (c, d). Logical proportions are homogeneous when they are based on equivalences between indicators of the same kind. There are only 4 such homogeneous proportions, which respectively express that i) "a differs from b as c differs from d" (and "b differs from a as d differs from c"), ii) "a differs from b as d differs from c" (and "b differs from a as c differs from d"), iii) "what a and b have in common c and d have it also", iv) "what a and b have in common neither c nor d have it". We prove that each of these proportions is the unique Boolean formula (up to equivalence) that satisfies groups of remarkable properties including a stability property w.r.t. a specific permutation of the terms of the proportion. The first one (i) is shown to be the only one to satisfy the standard postulates of an analogical proportion. The paper also studies how two analogical proportions can be combined into a new one. We then examine how homogeneous proportions can be used for diverse prediction tasks. We particularly focus on the completion of analogical-like series, and on missing value abduction problems. Finally, the paper compares our approach with other existing works on qualitative prediction based on ideas of betweenness, or of matrix abduction. Copyright © 2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.	en_US
dc.relation.ispartof	Proceedings of the International Workshop on Temporal Representation and Reasoning	en_US
dc.title	Homogeneous logical proportions: Their uniqueness and their role in similarity-based prediction	en_US
dc.type	Conference Proceeding
utslib.for	0801 Artificial Intelligence and Image Processing	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

Abstract:

Given a 4-tuple of Boolean variables (a, b, c, d), logical proportions are modeled by a pair of equivalences relating similarity indicators (a∧b̄ and ā∧b̄), or dissimilarity indicators (a ∧ b̄ and ā ∧ b) pertaining to the pair (a, b), to the ones associated with the pair (c, d). Logical proportions are homogeneous when they are based on equivalences between indicators of the same kind. There are only 4 such homogeneous proportions, which respectively express that i) "a differs from b as c differs from d" (and "b differs from a as d differs from c"), ii) "a differs from b as d differs from c" (and "b differs from a as c differs from d"), iii) "what a and b have in common c and d have it also", iv) "what a and b have in common neither c nor d have it". We prove that each of these proportions is the unique Boolean formula (up to equivalence) that satisfies groups of remarkable properties including a stability property w.r.t. a specific permutation of the terms of the proportion. The first one (i) is shown to be the only one to satisfy the standard postulates of an analogical proportion. The paper also studies how two analogical proportions can be combined into a new one. We then examine how homogeneous proportions can be used for diverse prediction tasks. We particularly focus on the completion of analogical-like series, and on missing value abduction problems. Finally, the paper compares our approach with other existing works on qualitative prediction based on ideas of betweenness, or of matrix abduction. Copyright © 2012, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved.

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http://hdl.handle.net/10453/32682