Universal approximation theorem for uninorm-based fuzzy systems modeling. (Q1414810)

The fundamental result of this study concerns the universal approximation property of fuzzy systems endowed with arbitrary uninorms. Uninorms are two-argument functions \(U:[0,1]^2\to [0,1]\) satisfying the following properties: (a) identity \(U(g,x)=x,(b)\) commutativity \(U(x,y)=U(y,x)\), (c) associativity \(U(x,U (y,z))= U(U(x, y),z)\), (d) monotonicity \(U(x,y)\leq U(x',y')\) for all \(x\leq x'\) and \(y\leq y'\). For \(g=0\) the uninorm becomes a certain s-norm while for \(g=1\) the definition coincides with the one for any t-norm. Furthermore the authors introduce a relevancy transformation \(h:[0,1]^2\to[0,1]\) such that (a) \(h(1,x)=x\); \(h(0,x)= g\); (b) if \(y<y'\) then \(h(x,y)\leq h(x,y')\), and (c) if \(x<x'\) then \(h(x,y)\leq h(x',y)\) for \(y\leq g\) and \(h(x,y)\geq h(x',y)\) for \(y\geq g\). The property of universal approximation is derived for the Mamdani type of model with the fuzzy relation of the fuzzy system being computed in the following form \[ R(x,y)=U \biggl( h\bigl(A_1(x), B_1(y) \bigr), \dots,h \bigl(A_N(x), B_N(y)\bigr)\biggr). \] Here the model is based on a collection of \(N\)-rules of the form ``if \(A_i\) then \(B_i\)'' involving fuzzy sets \(A_i\) and \(B_i\) defined in the input and output space, respectively.