An Analysis of Ruspini Partitions in Gödel Logic

Abstract: By a Ruspini partition we mean a finite family of fuzzy sets ${f_1, \ldots, f_n}$, $f_i : [0,1] \to [0,1]$, such that $\sum_{i=1}^n f_i(x)=1$ for all $x \in [0,1]$, where $[0,1]$ denotes the real unit interval. We analyze such partitions in the language of G\"odel logic. Our first main result identifies the precise degree to which the Ruspini condition is expressible in this language, and yields inter alia a constructive procedure to axiomatize a given Ruspini partition by a theory in G\"odel logic. Our second main result extends this analysis to Ruspini partitions fulfilling the natural additional condition that each $f_i$ has at most one left and one right neighbour, meaning that $\min_{x \in [0,1]}{{f_{i_1}(x),f_{i_2}(x),f_{i_3}(x)}}=0$ holds for $i_1\neq i_2\neq i_3$.