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Extended symbolic approximate reasoning based on linguistic modifiers

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Abstract

Approximate reasoning allows inferring with imperfect knowledge. It is based on a generalization of modus ponens (MP) known as generalized modus ponens (GMP). We are interested in approximate reasoning within symbolic multi-valued logic framework. In a previous work, we have proposed a new GMP based on linguistic modifiers in the multi-valued logic framework. The use of linguistic modifiers allows having a gradual reasoning; moreover, it allows checking axiomatics of approximate reasoning. In this paper, we extend our approximate reasoning to hold with complex rules, i.e., rules whose premises are conjunction or disjunction of propositions. For this purpose, we introduce a new operator that aggregates linguistic modifiers and verifies the required properties of logical connectives within the multi-valued logic framework.

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Notes

  1. Denoted mathematically by “\(X \in _{\alpha } A\)”: the object \(X\) belongs with a degree \(\alpha \) to the multi-set \(A\).

  2. With \(M\) a positive integer not null, which represents the number of truth-degrees in the scale \(\mathcal {L}_{M}\).

  3. Also denoted: If “\(X\) is \(A\)” is \(\tau _{\alpha }\)-true, then “\(Y\) is \(B\)” is \(\tau _{\beta }\)-true.

  4. The set \(\mathbb {N}^*\) refers to the set of all natural numbers excluding zero.

  5. Such a modifier \(m\) can be a modification operator or the composition of several operators.

  6. LCM is the least common multiple.

  7. Knowing that an elementary operator acts either on the degrees or on the bases, in the last case it can be seen as a vector in a one-dimensional space that transforms a base \(M\) to another \({M^{\prime }}\).

  8. Definition of M-norm is similar to that of T-norm [7]. We prefer the name M-norm because it is a T-norm defined on the set of modifiers and not on [0,1]. The new definition results in properties specific to our context.

  9. The stability of aggregators is shown farther.

  10. same mode and same nature.

  11. The notations \(A_{\wedge }\) and \(A_{\vee }\) are more general, as they can be associated with other degrees aggregators than T-norm and T-conorm and can belong to other set of modifiers than \(\mathcal {M}^{AR}\).

  12. We recall that \(\lnot \) is negation operator \(\lnot : \mathcal {L}_M \longrightarrow \mathcal {L}_M\) with \(\lnot \tau _\alpha = \tau _{M-1-\alpha }\) that T-norm is \(T: \mathcal {L}_M \times \mathcal {L}_M \longrightarrow \mathcal {L}_M\) (cf. definition 1). Our notation \(\lnot T\) is the composition \(\lnot \circ T\).

  13. \(T(\tau _{\gamma },\tau _{0}) = \tau _{0}\) for any T-norm \(T\).

  14. Similarly \(S(\tau _{\alpha },\tau _{\beta }) = \lnot T(\lnot \tau _{\alpha },\lnot \tau _{\beta })\).

  15. For any T-norm T, \(T(\tau _{\alpha },\tau _{\beta }) \le \min (\tau _{\alpha },\tau _{\beta })\).

  16. For any T-conorm S, \(S(\tau _{\alpha },\tau _{\beta }) \ge \max (\tau _{\alpha },\tau _{\beta })\).

  17. We remind that \(A_{T}\) is an other notation of \(A_{\wedge }\) which is associated with a T-norm and aggregate modifiers from \(\mathcal {M}^{AR}\) (idem for \(A_{S}\) and \(A_{\vee }\)).

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Correspondence to Saoussen Bel Hadj Kacem.

Appendix: Proofs of properties

Appendix: Proofs of properties

1.1 Commutativity of \(A_{T}\) and \(A_{S}\)

The commutativity of \(A_{T}\) and \(A_{S}\) can be easily remarked from the definitions. They are based on T-norm and T-conorm, and these latter are commutative themselves.

1.2 Monotonicity of \(A_{T}\) and \(A_{S}\)

Tables 4 and 5 show aggregation with \(A_{T}\) and \(A_{S}\) of all possible combinations within the set \(\mathcal {M}^{AR}\). Modifiers are ordered in an increasing order from left to right and top to bottom, according to the lattice of the set \(\mathcal {M}^{AR}\) given in Fig. . We remark that if we take four modifiers \(m_{\rho }, m^{\prime }_{\rho ^{\prime }},m^{\prime \prime }_{\rho ''}\) and \({m^{\prime \prime \prime }}_{{\rho ^{\prime \prime \prime }}}\) with \(m_{\rho } \trianglelefteq m^{\prime \prime }_{\rho ''}\) and \(m^{\prime }_{\rho ^{\prime }} \trianglelefteq m^{\prime \prime \prime }_{{\rho ^{\prime \prime \prime }}}\), we always find that \(A_{T}(m_{\rho },m^{\prime }_{\rho ^{\prime }}) \trianglelefteq A_{T}(m^{\prime \prime }_{\rho ''},m^{\prime \prime \prime }_{{\rho ^{\prime \prime \prime }}})\) and \(A_{S}(m_{\rho },m^{\prime }_{\rho ^{\prime }}) \trianglelefteq A_{S}(m^{\prime \prime }_{\rho ''},m^{\prime \prime \prime }_{{\rho ^{\prime \prime \prime }}})\). Indeed, let us remind that T-norms and T-conorms are monotone, and as shown in Fig. 4: \(\forall \rho \forall k \in \mathbb {N}^*, CR_{\rho } \trianglelefteq CR_{\rho +k}\) and \(CW_{\rho +k} \trianglelefteq CW_{\rho }\) and \(CW_{\rho } \trianglelefteq CR_{\rho }\). So the defined M-norm and M-conorm are monotone.

Table 4 Aggregation of all combinations of \(\mathcal {M}^{AR}\) with \(A_{T}\)
Table 5 Aggregation of all combinations of \(\mathcal {M}^{AR}\) with \(A_{S}\)

1.3 Associativity of \(A_{T}\) and \(A_{S}\)

The demonstration of these properties requires a case by case verification of all possible combinations of modifiers. The number of combination to consider is reduced as \(A_{T}\) and \(A_{S}\) are commutative. Let us begin by the proof of the associativity of the M-norm \(A_{T}\), that is \(\forall m_{\rho }, m^{\prime }_{\rho ^{\prime }} m^{\prime \prime }_{\rho ''} \in \mathcal {M}^{AR}\):

\(A_{T}(m_{\rho },A_{T}(m^{\prime }_{\rho ^{\prime }},m^{\prime \prime }_{\rho ''})) = A_{T}(A_{T}(m_{\rho },m^{\prime }_{\rho {\prime }}),m^{\prime \prime }_{\rho ''})\)

Case 1. \(A_{T}(CC,A_{T}(CC,CC)) = A_{T}(A_{T}(CC,CC),CC)\)

Proof

\(A_{T}(CC,A_{T}(CC,CC)) = A_{T}(CC,CC) = CC\)

\(A_{T}(A_{T}(CC,CC),CC) = A_{T}(CC,CC) = CC\)

Case 2.

\(A_{T}(CW_{\alpha },A_{T}(CW_{\beta },CW_{\gamma })) = A_{T}(A_{T}(CW_{\alpha },CW_{\beta }),CW_{\gamma })\)

Proof

\(A_{T}(CW_{\alpha },A_{T}(CW_{\beta },CW_{\gamma })) = \)

\(A_{T}(CW_{\alpha },CW_{\lnot T(\lnot \beta , \lnot \gamma )}) = CW_{\lnot T (\lnot \alpha ,\lnot \lnot T(\lnot \beta , \lnot \gamma ))} = CW_{S(\alpha ,S(\beta ,\gamma ))}\) with \(S\) the dual conorm of \(T\).

\(A_{T}(A_{T}(CW_{\alpha },CW_{\beta }),CW_{\gamma }) = A_{T}(CW_{\lnot T(\lnot \alpha , \lnot \beta )},CW_{\gamma }) = CW_{\lnot T (\lnot \lnot T(\lnot \alpha , \lnot \beta ),\lnot \gamma )} = CW_{S(S(\alpha ,\beta ),\gamma )}\)

T-conorms are associative so \(CW_{S(\alpha ,S(\beta ,\gamma ))} = CW_{S(S(\alpha ,\beta ),\gamma )}\)

Case 3. \(A_{T}(CW_{\alpha },A_{T}(CC,CC)) = A_{T}(A_{T}(CW_{\alpha },CC),CC)\)

Proof

\(A_{T}(CW_{\alpha },A_{T}(CC,CC)) = A_{T}(CW_{\alpha },CC) = CW_{\alpha }\)

\(A_{T}(A_{T}(CW_{\alpha },CC),CC) = A_{T}(CW_{\alpha },CC) = CW_{\alpha }\)

Case 4.

\(A_{T}(CW_{\alpha },A_{T}(CW_{\beta },CC)) = A_{T}(A_{T}(CW_{\alpha },CW_{\beta }),CC)\)

Proof

\(A_{T}(CW_{\alpha },A_{T}(CW_{\beta },CC)) = A_{T}(CW_{\alpha },CW_{\beta }) = CW_{\lnot T (\lnot \alpha , \lnot \beta ))}\)

\(A_{T}(A_{T}(CW_{\alpha },CW_{\beta }),CC) = A_{T}(CW_{\lnot T (\lnot \alpha , \lnot \beta ))},CC) = CW_{\lnot T (\lnot \alpha , \lnot \beta ))}\)

Case 5.

\(A_{T}(CR_{\alpha },A_{T}(CR_{\beta },CR_{\gamma })) = A_{T}(A_{T}(CR_{\alpha },CR_{\beta }),CR_{\gamma })\)

Proof

\(A_{T}(CR_{\alpha },A_{T}(CR_{\beta },CR_{\gamma })) = A_{T}(CR_{\alpha },CR_{T(\beta ,\gamma )}) = CR_{T(\alpha ,T(\beta ,\gamma ))}\)

\(A_{T}(A_{T}(CR_{\alpha },CR_{\beta }),CR_{\gamma }) = A_{T}(CR_{T(\alpha ,\beta )},CR_{\gamma }) = CR_{T(T(\alpha ,\beta ),\gamma )}\)

T-norms are associative so \(CR_{T(\alpha ,T(\beta ,\gamma ))} = CR_{T(T(\alpha ,\beta ),\gamma )}\)

Case 6. \(A_{T}(CR_{\alpha },A_{T}(CC,CC)) = A_{T}(A_{T}(CR_{\alpha },CC),CC)\)

Proof

\(A_{T}(CR_{\alpha },A_{T}(CC,CC)) = A_{T}(CR_{\alpha },CC) = CC\)

\(A_{T}(A_{T}(CR_{\alpha },CC),CC) = A_{T}(CC,CC) = CC\)

Case 7. \(A_{T}(CR_{\alpha },A_{T}(CR_{\beta },CC)) = A_{T}(A_{T}(CR_{\alpha },CR_{\beta }),CC)\)

Proof

\(A_{T}(CR_{\alpha },A_{T}(CR_{\beta },CC)) = A_{T}(CR_{\alpha },CC) = CC\)

\(A_{T}(A_{T}(CR_{\alpha },CR_{\beta }),CC) = A_{T}(CR_{T(\alpha ,\beta )},CC) = CC\)

Case 8.

\(A_{T}(CW_{\alpha },A_{T}(CR_{\beta },CC)) = A_{T}(A_{T}(CW_{\alpha },CR_{\beta }),CC)\)

Proof

\(A_{T}(CW_{\alpha },A_{T}(CR_{\beta },CC)) = A_{T}(CW_{\alpha },CC) = CW_{\alpha }\)

\(A_{T}(A_{T}(CW_{\alpha },CR_{\beta }),CC) = A_{T}(CW_{\alpha },CC) = CW_{\alpha }\)

Case 9.

\(A_{T}(CW_{\alpha },A_{T}(CW_{\beta },CR_{\gamma })) = A_{T}(A_{T}(CW_{\alpha },CW_{\beta }),CR_{\gamma })\)

Proof

\(A_{T}(CW_{\alpha },A_{T}(CW_{\beta },CR_{\gamma })) = A_{T}(CW_{\alpha },CW_{\beta }) = CW_{\lnot T(\lnot \alpha , \lnot \beta )}\)

\(A_{T}(A_{T}(CW_{\alpha },CW_{\beta }),CR_{\gamma }) = A_{T}(CW_{\lnot T(\lnot \alpha , \lnot \beta )},CR_{\gamma }) = CW_{\lnot T(\lnot \alpha , \lnot \beta )}\)

Case 10.

\(A_{T}(CW_{\alpha },A_{T}(CR_{\beta },CR_{\gamma })) = A_{T}(A_{T}(CW_{\alpha },CR_{\beta }),CR_{\gamma })\)

Proof

\(A_{T}(CW_{\alpha },A_{T}(CR_{\beta },CR_{\gamma })) = A_{T}(CW_{\alpha },CR_{T(\beta ,\gamma )}) = CW_{\alpha }\)

\(A_{T}(A_{T}(CW_{\alpha },CR_{\beta }),CR_{\gamma }) = A_{T}(CW_{\alpha },CR_{\gamma }) = CW_{\alpha }\)

Let us proof that the M-conorm \(A_{S}\) is associative, that is \(\forall m_{\rho }, m^{\prime }_{\rho ^{\prime }}, m^{\prime \prime }_{\rho ''} \in \mathcal {M}^{AR}\):

\(A_{S}(m_{\rho },A_{S}(m^{\prime }_{\rho ^{\prime }},m^{\prime \prime }_{\rho ''})) = A_{S}(A_{S}(m_{\rho },m^{\prime }_{\rho ^{\prime }}),m^{\prime \prime }_{\rho ''})\)

Case 1. \(A_{S}(CC,A_{S}(CC,CC)) = A_{S}(A_{S}(CC,CC),CC)\)

Proof

\(A_{S}(CC,A_{S}(CC,CC)) = A_{S}(CC,CC) = CC\)

\(A_{S}(A_{S}(CC,CC),CC) = A_{S}(CC,CC) = CC\)

Case 2.

\(A_{S}(CW_{\alpha },A_{S}(CW_{\beta },CW_{\gamma })) = A_{S}(A_{S}(CW_{\alpha },CW_{\beta }),CW_{\gamma })\)

Proof

\(A_{S}(CW_{\alpha },A_{S}(CW_{\beta },CW_{\gamma })) \!=\! A_{S}(CW_{\alpha },CW_{\lnot S(\lnot \beta , \lnot \gamma )}) \!=\! CW_{\lnot S (\lnot \alpha ,\lnot \lnot S(\lnot \beta , \lnot \gamma ))} = CW_{T(\alpha ,T(\beta ,\gamma ))}\) with \(T\) the dual norm of \(S\).

\(A_{S}(A_{S}(CW_{\alpha },CW_{\beta }),CW_{\gamma }) = A_{S}(CW_{\lnot S(\lnot \alpha , \lnot \beta )},CW_{\gamma }) = CW_{\lnot S (\lnot \lnot S(\lnot \alpha , \lnot \beta ),\lnot \gamma )} = CW_{T(T(\alpha ,\beta ),\gamma )}\)

T-norms are associative so \(CW_{T(\alpha ,T(\beta ,\gamma ))} = CW_{T(T(\alpha ,\beta ),\gamma )}\)

Case 3. \(A_{S}(CW_{\alpha },A_{S}(CC,CC)) = A_{S}(A_{S}(CW_{\alpha },CC),CC)\)

Proof

\(A_{S}(CW_{\alpha },A_{S}(CC,CC)) = A_{S}(CW_{\alpha },CC) = CC\)

\(A_{S}(A_{S}(CW_{\alpha },CC),CC) = A_{T}(CC,CC) = CC\)

Case 4.

\(A_{S}(CW_{\alpha },A_{S}(CW_{\beta },CC)) = A_{S}(A_{S}(CW_{\alpha },CW_{\beta }),CC)\)

Proof

\(A_{S}(CW_{\alpha },A_{S}(CW_{\beta },CC)) = A_{S}(CW_{\alpha },CC) = CC\)

\(A_{S}(A_{S}(CW_{\alpha },CW_{\beta }),CC) = A_{S}(CW_{\lnot S (\lnot \alpha , \lnot \beta ))},CC) = CC\)

Case 5.

\(A_{S}(CR_{\alpha },A_{S}(CR_{\beta },CR_{\gamma })) = A_{S}(A_{S}(CR_{\alpha },CR_{\beta }),CR_{\gamma })\)

Proof

\(A_{S}(CR_{\alpha },A_{S}(CR_{\beta },CR_{\gamma })) = A_{S}(CR_{\alpha },CR_{S(\beta ,\gamma )}) = CR_{S(\alpha ,S(\beta ,\gamma ))}\)

\(A_{S}(A_{S}(CR_{\alpha },CR_{\beta }),CR_{\gamma }) = A_{S}(CR_{S(\alpha ,\beta )},CR_{\gamma }) = CR_{S(S(\alpha ,\beta ),\gamma )}\)

T-conorms are associative so \(CR_{S(\alpha ,S(\beta ,\gamma ))} = CR_{S(S(\alpha ,\beta ),\gamma )}\)

Case 6. \(A_{S}(CR_{\alpha },A_{S}(CC,CC)) = A_{S}(A_{S}(CR_{\alpha },CC),CC)\)

Proof

\(A_{S}(CR_{\alpha },A_{S}(CC,CC)) = A_{s}(CR_{\alpha },CC) = CR_{\alpha }\)

\(A_{S}(A_{S}(CR_{\alpha },CC),CC) = A_{S}(CR_{\alpha },CC) = CR_{\alpha }\)

Case 7. \(A_{S}(CR_{\alpha },A_{S}(CR_{\beta },CC)) = A_{S}(A_{S}(CR_{\alpha },CR_{\beta }),CC)\)

Proof

\(A_{S}(CR_{\alpha },A_{S}(CR_{\beta },CC)) = A_{T}(CR_{\alpha },CR_{\beta }) = CR_{S(\alpha ,\beta )}\)

\(A_{S}(A_{S}(CR_{\alpha },CR_{\beta }),CC) = A_{S}(CR_{S(\alpha ,\beta )},CC) = CR_{S(\alpha ,\beta )}\)

Case 8. \(A_{S}(CW_{\alpha },A_{S}(CR_{\beta },CC)) = A_{S}(A_{S}(CW_{\alpha },CR_{\beta }),CC)\)

Proof

\(A_{S}(CW_{\alpha },A_{S}(CR_{\beta },CC)) = A_{S}(CW_{\alpha },CR_{\beta }) = CR_{\beta }\)

\(A_{S}(A_{S}(CW_{\alpha },CR_{\beta }),CC) = A_{S}(CR_{\beta },CC) = CR_{\beta }\)

Case 9.

\(A_{S}(CW_{\alpha },A_{S}(CW_{\beta },CR_{\gamma })) = A_{S}(A_{S}(CW_{\alpha },CW_{\beta }),CR_{\gamma })\)

Proof

\(A_{S}(CW_{\alpha },A_{S}(CW_{\beta },CR_{\gamma })) = A_{S}(CW_{\alpha },CR_{\gamma }) = CR_{\gamma }\)

\(A_{S}(A_{S}(CW_{\alpha },CW_{\beta }),CR_{\gamma }) = A_{S}(CW_{\lnot S(\lnot \alpha , \lnot \beta )},CR_{\gamma }) = CR_{\gamma }\)

Case 10.

\(A_{S}(CW_{\alpha },A_{S}(CR_{\beta },CR_{\gamma })) = A_{S}(A_{S}(CW_{\alpha },CR_{\beta }),CR_{\gamma })\)

Proof

\(A_{S}(CW_{\alpha },A_{S}(CR_{\beta },CR_{\gamma })) = A_{S}(CW_{\alpha },CR_{S(\beta ,\gamma )}) = CR_{S(\beta ,\gamma )}\)

\(A_{S}(A_{S}(CW_{\alpha },CR_{\beta }),CR_{\gamma }) = A_{S}(CR_{\beta },CR_{\gamma }) = CR_{S(\beta ,\gamma )}\)

1.4 Neutral elements of \(A_{T}\) and \(A_{S}\)

Property 2

\(CR_{M-1}\) is the neutral element of \(A_{T}\).

Proof

Saying that \(CR_{M-1}\) is the neutral element of \(A_{T}\) means that \(A_{T}(m_{\rho },CR_{M-1})=m_{\rho }\) for all \(m_{\rho } \in \mathcal {M}^{AR}\). The definition of \(A_{T}\) includes three cases, let us check this equality case by case.

  • Aggregation with \(CC: A_{T}(CC,CR_{M-1})=CC\).

  • Aggregation with \(CR_{\rho }\): \(A_{T}(CR_{\rho },CR_{M-1})=CR_{\gamma }\) with \(\tau _{\gamma } = T(\tau _{\rho },\tau _{M-1}). \tau _{M-1}\) is the neutral element of every T-norm so \(\tau _{\gamma }=\tau _{\rho }\) and thus \(A_{T}(CR_{\rho },CR_{M-1})=CR_{\rho }\).

  • Aggregation with \(CW_{\rho }: A_{T}(CW_{\rho },CR_{M-1})= CW_{\rho }\).

Property 3

\(CW_{M-1}\) is the neutral element of \(A_{S}\).

Proof

Saying that \(CW_{M-1}\) is the neutral element of \(A_{S}\) means that \(A_{S}(m_{\rho },CW_{M-1})=m_{\rho }\) for all \(m_{\rho } \in \mathcal {M}^{RA}\). The definition of \(A_{S}\) includes three cases, let us check this equality case by case.

  • Aggregation with \(CC: A_{S}(CC,CW_{M-1})=CC\).

  • Aggregation with \(CR_{\rho }: A_{S}(CR_{\rho },CW_{M-1})=CR_{\rho }\).

  • Aggregation with \(CW_{\rho }: A_{S}(CW_{\rho },CW_{M-1})= CW_{\gamma }\) with \(\tau _{\gamma } = \lnot S(\lnot \tau _{\rho }, \lnot \tau _{M-1}) = T(\tau _{\rho },\tau _{M-1})\) with \(T\) the dual T-norm of \(S. \tau _{M-1}\) is the neutral element of every T-norm so \(\tau _{\gamma }=\tau _{\rho }\) and thus \(A_{S}(CW_{\rho },CW_{M-1})=CW_{\rho }\).

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Bel Hadj Kacem, S., Borgi, A. & Tagina, M. Extended symbolic approximate reasoning based on linguistic modifiers. Knowl Inf Syst 42, 633–661 (2015). https://doi.org/10.1007/s10115-014-0730-6

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  • DOI: https://doi.org/10.1007/s10115-014-0730-6

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