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European Conference on Symbolic and Quantitative Approaches to Reasoning and Uncertainty

ECSQARU 2017: Symbolic and Quantitative Approaches to Reasoning with Uncertainty pp 10–21Cite as

Boolean Analogical Proportions - Axiomatics and Algorithmic Complexity Issues

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10369))

Abstract

Analogical proportions, i.e., statements of the form a is to b as c is to d, are supposed to obey 3 axioms expressing reflexivity, symmetry, and stability under central permutation. These axioms are not enough to determine a single Boolean model, if a minimality condition is not added. After an algebraic discussion of this minimal model and of related expressions, another justification of this model is given in terms of Kolmogorov complexity. It is shown that the 6 Boolean patterns that make an analogical proportion true have a minimal complexity with respect to an expression reflecting the intended meaning of the proportion.

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Notes

  1. 1.

    There are 3 companion approximations that involve the two additional patterns of \(A_K\):

    \( (a \equiv d) \wedge (b \equiv c) \quad \begin{array}{|cccc|} \hline 1 &{} 1 &{} 1&{} 1 \\ \hline 0 &{} 0 &{} 0 &{} 0 \\ \hline 1 &{} 0 &{} 0 &{} 1 \\ \hline 0 &{} 1 &{} 1 &{} 0 \\ \hline \end{array} \); \( (a \not \equiv b) \wedge (c \not \equiv d) \quad \begin{array}{|cccc|} \hline 1 &{} 0 &{} 0 &{} 1 \\ \hline 0 &{} 1 &{} 1 &{} 0 \\ \hline 1 &{} 0 &{} 1 &{} 0 \\ \hline 0 &{} 1 &{} 0 &{} 1 \\ \hline \end{array} \); \( (a \not \equiv c) \wedge (b \not \equiv d) \quad \begin{array}{|cccc|} \hline 1 &{} 1 &{} 0 &{} 0 \\ \hline 0 &{} 0 &{} 1 &{} 1 \\ \hline 1 &{} 0 &{} 0 &{} 1 \\ \hline 0 &{} 1 &{} 1 &{} 0 \\ \hline \end{array} \).

  2. 2.

    Note that lower approximations of analogical proportions miss at least one axiom.

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Authors and Affiliations

  1. IRIT, Toulouse University, Toulouse, France

    Henri Prade & Gilles Richard

  2. QCIS, University of Technology, Sydney, Australia

    Henri Prade

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  1. Henri Prade
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  2. Gilles Richard
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Correspondence to Henri Prade .

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Editors and Affiliations

  1. IDSIA, Lugano, Switzerland

    Alessandro Antonucci

  2. ONERA, Toulouse, France

    Laurence Cholvy

  3. Aix-Marseille University, Marseille, France

    Odile Papini

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Prade, H., Richard, G. (2017). Boolean Analogical Proportions - Axiomatics and Algorithmic Complexity Issues. In: Antonucci, A., Cholvy, L., Papini, O. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2017. Lecture Notes in Computer Science(), vol 10369. Springer, Cham. https://doi.org/10.1007/978-3-319-61581-3_2

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  • DOI: https://doi.org/10.1007/978-3-319-61581-3_2

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