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.
Similar content being viewed by others
Notes
Denoted mathematically by “\(X \in _{\alpha } A\)”: the object \(X\) belongs with a degree \(\alpha \) to the multi-set \(A\).
With \(M\) a positive integer not null, which represents the number of truth-degrees in the scale \(\mathcal {L}_{M}\).
Also denoted: If “\(X\) is \(A\)” is \(\tau _{\alpha }\)-true, then “\(Y\) is \(B\)” is \(\tau _{\beta }\)-true.
The set \(\mathbb {N}^*\) refers to the set of all natural numbers excluding zero.
Such a modifier \(m\) can be a modification operator or the composition of several operators.
LCM is the least common multiple.
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 \({M^{\prime }}\).
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.
The stability of aggregators is shown farther.
same mode and same nature.
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 \(\mathcal {M}^{AR}\).
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 \(\lnot \circ T\).
\(T(\tau _{\gamma },\tau _{0}) = \tau _{0}\) for any T-norm \(T\).
Similarly \(S(\tau _{\alpha },\tau _{\beta }) = \lnot T(\lnot \tau _{\alpha },\lnot \tau _{\beta })\).
For any T-norm T, \(T(\tau _{\alpha },\tau _{\beta }) \le \min (\tau _{\alpha },\tau _{\beta })\).
For any T-conorm S, \(S(\tau _{\alpha },\tau _{\beta }) \ge \max (\tau _{\alpha },\tau _{\beta })\).
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 \(A_{\vee }\)).
References
Akdag H (1992) Une approche logique du raisonnement incertain. Ph.D. thesis. University of Paris VI
Akdag H, Glas MD, Pacholczyk D (1992) A qualitative theory of uncertainty. Fundam Inf 17(4):333–362
Akdag H, Mokhtari M(1996) Approximative conjunctions processing by multi-valued logic. In: Proceedings of IEEE international symposium on multiple-valued logic, Spain, pp 130–135
Akdag H, Truck I, Borgi A, Mellouli N (2001) Linguistic modifiers in a symbolic framework. Int J Uncertain Fuzziness Knowl Based Syst 9(Supplement):49–61
Baldwin J, Pilsworth B (1980) Axiomatic approach to implication for approximate reasoning with fuzzy logic. Fuzzy Sets Syst 3(2):193–219
Bartusek T, Navara M (2001) Conjunctions of many-valued criteria. In: Proceedings of the international conference uncertainty modelling. Bratislava, Slovakia, pp 67–77
Bedregal B, Santos H, Callejas-Bedregal R (2006) T-norms on bounded lattices: t-norm morphisms and operators. In: IEEE international conference on fuzzy systems, pp 22–28
Borgi A, Kacem SBH, Ghédira K (2008) Approximate reasoning in a symbolic multi-valued framework. In: Lee RY, Kim HK (eds) Computer and information science. Studies in computational intelligence. Springer, Berlin, pp 203–217
Bosc P, HadjAli A, Pivert O, Smits G (2010) Trimming plethoric answers to fuzzy queries: an approach based on predicate correlation. In: International conference on information processing and management of uncertainty in knowledge-based systems. IPMU’10 Dortmund, Germany, pp 595–604
Bouchon-Meunier B, Valverde L (1999) A fuzzy approach to analogical reasoning. Softw Comput 3:141–147
Chen TY (2012) A signed-distance-based approach to importance assessment and multi-criteria group decision analysis based on interval type-2 fuzzy set. Knowl Inf Syst. doi:10.1007/s10115-012-0497-6
Chung HT, Schwartz DG (1995) A resolution-based system for symbolic approximate reasoning. Int J Approx Reason 13(3):201–246
Ciliz MK (2005) Rule base reduction for knowledge-based fuzzy controllers with application to a vacuum cleaner. Expert Syst Appl 28(1):175–184
Cornelis C, Kerre EE (2001) Inclusion-based approximate reasoning. In: International conference on computational science (2), Lecture notes in computer science, vol. 2074. Springer, Berlin, pp 221–230
Dombi J (1982) A general class of fuzzy operators, the DeMorgan class of fuzzy operators and fuzziness measures induced by fuzzy operators. Fuzzy Sets Syst 8(2):149–163
Dombi J, Vas Z (1983) Basic theoretical treatment of fuzzy connectives. Acta Cybernetica 6(2):191–201
Dubois D, Prade H (1985) A review of fuzzy set aggregation connectives. Fuzzy Sets Syst 36:85–121
Dubois D, Prade H (2003) Fuzzy set and possibility theory-based methods in artificial intelligence. Artif Intell 148:1–9
El-Sayed M, Pacholczyk D (2003) Towards a symbolic interpretation of approximate reasoning. Electr Notes Theor Comput Sci 82(4):1–12
Fodor J, Roubens M (1994) Fuzzy preference modelling and multicriteria decision support. Kluwer Academic Publishers, Dordrecht
Fukami S, Mizumoto M, Tanaka K (1980) Some considerations of fuzzy conditional inference. Fuzzy Sets Syst 4(3):243–273
Gacgne L (1997) Elements de Logique floue. Editions Hermes
Genno H, Fujiwara Y, Yoneda H, Fukushima K (1990) Human sensory perception oriented image processing in color copy system. In: International conference on fuzzy logic and neural networks. Iizuka, Japan, pp 423–427
Ginsberg ML (1988) Multivalued logics: a uniform approach to reasoning in artificial intelligence. Comput Intell 4(3):265–316
Glas MD (1989) Knowledge representation in a fuzzy setting. Report 89–48. LAFORIA. University of Paris VI
Grabisch M, Marichal JL, Mesiar R, Pap E (2011) Aggregation functions: construction methods, conjunctive, disjunctive and mixed classes. Inf Sci 181(1):23–43
Hohle U (1978) Probabilistic uniformization of fuzzy topologies. Fuzzy Sets Syst 1:311–332
Kacem SBH, Borgi A, Ghédira K (2008) Generalized modus ponens based on linguistic modifiers in a symbolic multi-valued framework. In: Proceeding of the 38th IEEE international symposium on multiple-valued logic. Dallas, USA, pp 150–155
Kacem SBH, Borgi A, Tagina M (2009) On some properties of generalized symbolic modifiers and their role in symbolic approximate reasoning. In: Huang DS, Jo KH, Lee HH, Kang HJ, Bevilacqua V (eds) ICIC (2), Lecture Notes in Computer Science, vol 5755. Springer, Berlin, pp 190–208
Khoukhi F (1996) Approche logico-symbolique dans le traitement des connaissances incertaines et imprécises dans les systèmes à base de connaissances. Ph.D. thesis, Université de Reims, France
Klement EP, Mesiar R, Pap E (2004) Triangular norms. Position paper i: basic analytical and algebraic properties. Fuzzy Sets Syst 143(1):5–26
Lascio HD, Gisolfi A, Cortés U (1999) Linguistic hedges and the generalized modus ponens. Int J Intell Syst 14:981–993
Menger K (1942) Statistical metrics. Proc Natl Acad Sci USA 28(12):535–537
Mikut R, Jkel J, Grll L (2005) Interpretability issues in data-based learning of fuzzy systems. Fuzzy Sets Syst 150(2):179–197
Mizumoto M (1981) Fuzzy sets and their operations i–ii. Inf Control 48(1):30–48, 50(2):160–174
Pacholczyk D (1992) Contribution au traitement logico-symbolique de la connaissance. Ph.D. thesis, University of Paris VI
Rojas K, Gmez D, Rodrguez J, Montero J (2012) Some properties of consistency in the families of aggregation operators. In: Melo-Pinto P, Couto P, Serdio C, Fodor J, De Baets B (eds) Eurofuse 2011, Advances in intelligent and soft computing, vol 107. Springer, Berlin, pp 169–176
Schwartz DG (1991) A system for reasoning with imprecise linguistic information. Int J Approx Reasoning 5(5):463–488
Schweizer B, Sklar A (1960) Statistical metrics spaces. Pacific J Math 10(1):314–334
Schweizer B, Sklar A (1961) Associative functions and statistical triangle inequalities. Publicationes Mathematicae Debrecen 8:169–186
Truck I, Akdag H (2006) Manipulation of qualitative degrees to handle uncertainty : Formal models and applications. Knowl Inf Syst 9(4):385–411
Truck I, Borgi A, Akdag H (2002) Generalized modifiers as an interval scale: towards adaptive colorimetric alterations. In: Garijo FJ, Santos JCR, Toro M (eds.) IBERAMIA, Lecture notes in computer science, vol 2527. Springer, Berlin, pp 111–120
Weber S (1983) A general concept of fuzzy connectives, negations and implications based on t-norms and t-conorms. Fuzzy Sets Syst 11:103–113
Xia M, Xu Z, Zhu B (2012) Some issues on intuitionistic fuzzy aggregation operators based on Archimedean t-conorm and t-norm. Knowl-Based Syst 31:78–88
Yager RR (1980) On a general class of fuzzy connectives. Fuzzy Sets Syst 4(3):235–242
Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353
Zadeh LA(1975) The concept of a linguistic variable and its application to approximate reasoning: i–iii. Inf Sci 8:199–249, 8:301–357, 9:43–80
Zadeh LA (1979) A theory of approximate reasoning. Mach Intell 9:149–194
Author information
Authors and Affiliations
Corresponding author
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.
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 \(A_{S}(CW_{\rho },CW_{M-1})=CW_{\rho }\).
Rights and permissions
About this article
Cite this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10115-014-0730-6