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Numerical solution of fuzzy relational equations based on smooth fuzzy norms

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Abstract

In this paper, we study and formulate a BP learning algorithm for fuzzy relational neural networks based on smooth fuzzy norms for functions approximation. To elaborate the model behavior more, we have used different fuzzy norms led to a new pair of fuzzy norms. An important practical case in fuzzy relational equations (FREs) is the identification problem which is studied in this work. In this work we employ a neuro-based approach to numerically solve the set of FREs and focus on generalized neurons that use smooth s-norms and t-norms as fuzzy compositional operators.

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Authors and Affiliations

  1. Department of Electrical Engineering, Amirkabir University of Technology (AUT), Hafez Ave., P.O. Box 15914, Tehran, Iran

    Arya Aghili Ashtiani & Mohammad Bagher Menhaj

Authors
  1. Arya Aghili Ashtiani
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  2. Mohammad Bagher Menhaj
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Correspondence to Arya Aghili Ashtiani.

Appendix

Appendix

For the probabilistic sum we can define:

$$ \begin{array}{l} p^{1} : = s\left( {a,0} \right) = a \hfill \\ p^{2} : = s\left( {a,b} \right) = a + b - ab \hfill \\ p^{3} : = s\left( {a,b,c} \right) = s\left( {a,s\left( {b,c} \right)} \right) \hfill \\ \qquad = a + b + c - \left( {ab + ac + bc} \right) + abc \hfill \\ p^{n} : = s\left( {a_{1} ,a_{2} , \ldots ,a_{n} } \right) \hfill \\ \end{array} $$

Now the above functions can be expressed according to the following scheme:

$$ p_{{}}^{n} = P_{1}^{n} \left( {\left[ {a_{1} ,a_{2} , \ldots ,a_{n} } \right]^{T} } \right) $$
$$ P_{{k_{0} }}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right): = \sum\limits_{{k = k_{0} }}^{n} {N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)} ,\;N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right): = \left( { - 1} \right)^{k + 1} M_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right),\;M_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right): = \frac{1}{{k!\left( {n - k} \right)!}}L_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) $$
$$ L_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right): = \left| \begin{gathered} \hfill \\ \;\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \;}}^{{k\;{\text{vectors}}}}\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} \;}}^{{\left( {n - k} \right){\text{vectors}}}}\; \hfill \\ \hfill \\ \end{gathered} \right|_{ + } ;\;\;\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } = \left[ \begin{gathered} \lambda_{1} \hfill \\ \vdots \hfill \\ \lambda_{n} \hfill \\ \end{gathered} \right],\;\;\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} : = \left[ \begin{gathered} 1 \hfill \\ \vdots \hfill \\ 1 \hfill \\ \end{gathered} \right] $$
$$ \Rightarrow \;\;\;\;\;N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) = \frac{{\left( { - 1} \right)^{k + 1} }}{{k!\left( {n - k} \right)!}}\left| \begin{gathered} \hfill \\ \;\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \;}}^{{k\;{\text{vectors}}}}\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u} \;}}^{{\left( {n - k} \right){\text{vectors}}}}\; \hfill \\ \hfill \\ \end{gathered} \right|_{ + } $$

In the next step we consider the derivatives as follows:

$$ \begin{gathered} \frac{{\partial p^{1} }}{\partial a} = 1,\;\frac{{\partial p^{2} }}{\partial a} = 1 - b,\;\frac{{\partial p^{3} }}{\partial a} = 1 - \left( {b + c} \right) + bc, \hfill \\ \frac{{\partial p^{4} }}{\partial a} = 1 - \left( {b + c + d} \right) + \left( {cd + bd + bc} \right) - bcd, \hfill \\ \end{gathered} $$

and so on.

In \( P_{1}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right), \) the dimension \( n \) is indeed the length of the vector \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \) and can be obtained by the argument \( \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right), \) hence we can omit the superscript for the sake of brevity. Also by omitting the subscript \( k_{0} , \) we mean the common value \( k_{0} = 1. \) Therefore the following notations are equal:

$$ P_{1}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) = P_{1}^{{}} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) = P\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) $$

Now, we look for the derivatives:

$$ \begin{aligned} Q\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right): & = \frac{{\partial P\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }} = \frac{\partial }{{\partial \lambda_{i} }}\left( {\sum\limits_{k = 1}^{n} {N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)} } \right) = \sum\limits_{k = 1}^{n} {\frac{{\partial N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }}} \\ & = \sum\limits_{k = 1}^{n} {\frac{{\partial N_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial M_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}\frac{{\partial M_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }}} = \sum\limits_{k = 1}^{n} {\frac{{\left( { - 1} \right)^{k + 1} }}{{k!\left( {n - k} \right)!}}\frac{{\partial L_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }}} \\ \end{aligned} $$
$$ \frac{{\partial L_{k}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }} = \frac{\partial }{{\partial \lambda_{i} }}\left| \begin{gathered} \hfill \\ \;\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \;}}^{{k\;{\text{vectors}}}}\overbrace {{\;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} \; \cdots \;\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{1} \;}}^{{\left( {n - k} \right){\text{vectors}}}}\; \hfill \\ \hfill \\ \end{gathered} \right|_{ + } = \frac{\partial }{{\partial \lambda_{i} }}\left| {\begin{array}{*{20}c} {\lambda_{1} } & \ldots & {\lambda_{1} } \\ \vdots & {} & \vdots \\ {\lambda_{i} } & \cdots & {\lambda_{i} } \\ \vdots & {} & \vdots \\ {\lambda_{n} } & \cdots & {\lambda_{n} } \\ \end{array} \;\;\;\begin{array}{*{20}c} 1 & \ldots & 1 \\ \vdots & {} & \vdots \\ 1 & \cdots & 1 \\ \vdots & {} & \vdots \\ 1 & \cdots & 1 \\ \end{array} } \right|_{ + } = k\;L_{k - 1}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) $$

In the above equation, \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime} \) is a \( \left( {n - 1} \right) \times 1 \) vector which is obtained from \( \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } \) by omitting the \( i \)’th element.

$$ \begin{gathered} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime} = \left[ {\begin{array}{*{20}c} {\lambda_{1} } & \cdots & {\lambda_{i - 1} } & {\lambda_{i + 1} } & \cdots & {\lambda_{n} } \\ \end{array} } \right]^{T} \hfill \\ \left\{ \begin{gathered} \frac{{\partial P\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }} = \sum\limits_{k = 1}^{n} {\frac{{\left( { - 1} \right)^{k + 1} k}}{{k!\left( {n - k} \right)!}}L_{k - 1}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right)} \hfill \\ M_{k - 1}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) = \frac{1}{{\left( {k - 1} \right)!\left( {n - k} \right)!}}L_{k - 1}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) \hfill \\ \end{gathered} \right. \Rightarrow \frac{{\partial P\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)}}{{\partial \lambda_{i} }} = \sum\limits_{k = 1}^{n} {\left( { - 1} \right)^{k + 1} M_{k - 1}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right)} \hfill \\ \end{gathered} $$

To obtain a more compact form we can write:

$$ \begin{aligned} & \left\{ {\begin{array}{*{20}c} {P_{1}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right) = \sum\limits_{k = 1}^{n} {N_{k}^{n} = \sum\limits_{k = 1}^{n} {\left( { - 1} \right)^{k + 1} } } M_{k} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)} \\ {Q\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{'} } \right) = \sum\limits_{k = 1}^{n} {\left( { - 1} \right)^{k + 1} M_{k - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{'} } \right)} } \\ \end{array} } \right. \\ & \Rightarrow Q\left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) = \sum\limits_{k = 0}^{n - 1} {\left( { - 1} \right)^{k} M_{k}^{{}} } \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) = - \sum\limits_{k = 0}^{n - 1} {\left( { - 1} \right)^{k + 1} M_{k}^{{}} } \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) = - \sum\limits_{k = 0}^{n - 1} {N_{k}^{{}} } \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) = - P_{0}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right) \hfill \\ & \Rightarrow \boxed{\frac{\partial }{{\partial \lambda_{i} }}\left( {P_{1}^{n} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda } } \right)} \right) = - P_{0}^{n - 1} \left( {\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\lambda }^{\prime}} \right)} \hfill \\ \end{aligned} $$
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Aghili Ashtiani, A., Menhaj, M.B. Numerical solution of fuzzy relational equations based on smooth fuzzy norms. Soft Comput 14, 545–557 (2010). https://doi.org/10.1007/s00500-009-0425-1

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