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Bi-oriented graphs and four valued logic for preference modelling

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Abstract

In this paper we show the relations between 4-valued logics (and more precisely the DDT logic) and the use of bi-oriented graphs. Further on we focus on the use of bi-oriented graphs for non conventional preference modelling. More specifically, we show how bi-oriented graphs can be used in order to represent extended preference structures of the type definable using the DDT logic which has been created with the purpose of modelling hesitation in preference statements. We then study how transitive closure can be extended within such extended preference structures.

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Notes

  1. Including some cases of non conventional preference structures, using valued graphs (see Kacprzyk & Roubens 1988; for a general survey see Öztürk et al., 2005)

  2. glb: greatest lower bound, lub: least upper bound.

  3. Note that the remaining 8 cases do not provide a b-path, hence they are not concerned by transitive closure.

  4. Meaning that \({\textbf{T}}S(x,y)\) and \({\textbf{T}}S(y,z)\) implies \({\textbf{T}}S(x,z)\),...

  5. If we interpret this by classical logic, saying that the existence of an arc S(x, y) means that the affirmation S(x, y) is true and the absence S(x, y) is false, then we can say that the transitive closure replace the value false of S(x, z) by the value true if S(x, y) and S(y, z) are true.

  6. \(\forall x,y,z\) \(\lnot S(x,y) \wedge \lnot S(y,z) \implies \lnot S(x,z)\).

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Authors and Affiliations

  1. USTHB, Algers, Algerie

    Ouahiba Bessouf & Abdelkader Khelladi

  2. CNRS, PSL Research University, Université Paris Dauphine, LAMSADE, Paris, France

    Meltem Öztürk & Alexis Tsoukiàs

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  1. Ouahiba Bessouf
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  2. Abdelkader Khelladi
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  3. Meltem Öztürk
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  4. Alexis Tsoukiàs
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Correspondence to Meltem Öztürk.

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Deceased–Abdelkader Khelladi contributed to this work but passed away before approving the final version of this manuscript.

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Bessouf, O., Khelladi, A., Öztürk, M. et al. Bi-oriented graphs and four valued logic for preference modelling. Ann Oper Res 328, 1239–1262 (2023). https://doi.org/10.1007/s10479-023-05371-w

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