An Efficient Computational Framework for Discrete Fuzzy Numbers Based on Total Orders
Arnau Mir, Alejandro Mus, Juan Vicente Riera
TL;DR
The work tackles the expensive task of ranking and unranking discrete fuzzy numbers under total (admissible) orders by introducing two exact, memory-efficient algorithms for computing pos and pos^{-1}. These algorithms operate on the interval lattice of $L_n$ and rely on combinatorial counts of generalized discrete fuzzy numbers to achieve a deterministic time complexity of $O(n^{2} m \log n)$ with constant extra memory, avoiding full enumeration of $\left| \mathcal{D}_1^{L_n\rightarrow Y_m} \right|$. The main contributions are the ranking/unranking mechanisms, their complexity analysis, and empirical validation showing linear scaling in $m$ for fixed $n$, which enables scalable aggregation and implication operations on dfns. The results have practical impact for efficient fuzzy reasoning with linguistic information and open avenues for extending the framework to discrete Z-numbers, with reproducible implementations available on GitHub.
Abstract
Discrete fuzzy numbers, and in particular those defined over a finite chain $L_n = \{0, \ldots, n\}$, have been effectively employed to represent linguistic information within the framework of fuzzy systems. Research on total (admissible) orderings of such types of fuzzy subsets, and specifically those belonging to the set $\mathcal{D}_1^{L_n\rightarrow Y_m}$ consisting of discrete fuzzy numbers $A$ whose support is a closed subinterval of the finite chain $L_n = \{0, 1, \ldots, n\}$ and whose membership values $A(x)$, for $x \in L_n$, belong to the set $Y_m = \{ 0 = y_1 < y_2 < \cdots < y_{m-1} < y_m = 1 \}$, has facilitated the development of new methods for constructing logical connectives, based on a bijective function, called $\textit{pos function}$, that determines the position of each $A \in \mathcal{D}_1^{L_n\rightarrow Y_m}$. For this reason, in this work we revisit the problem by introducing algorithms that exploit the combinatorial structure of total (admissible) orders to compute the $\textit{pos}$ function and its inverse with exactness. The proposed approach achieves a complexity of $\mathcal{O}(n^{2} m \log n)$, which is quadratic in the size of the underlying chain ($n$) and linear in the number of membership levels ($m$). The key point is that the dominant factor is $m$, ensuring scalability with respect to the granularity of membership values. The results demonstrate that this formulation substantially reduces computational cost and enables the efficient implementation of algebraic operations -- such as aggregation and implication -- on the set of discrete fuzzy numbers.