Recent years have witnessed a widespread in the use of interval type-2 fuzzy logic systems (IT2 FLSs) in real-world applications. It has been shown recently that interval type-2 fuzzy sets (IT2 FSs) are more general than interval-valued fuzzy sets (IV FSs) [1]. Hence, there is a need to explore the capabilities of the more general forms of IT2 FSs (beyond IV FSs) and the applications areas they will be more suitable for. In addition, there is a need to develop the theory of the general forms of IT2 FLSs (gfIT2 FLSs), which employ IT2 FSs that are not equivalent to IV FSs and can have nonconvex secondary membership functions (MFs). Although these systems could be considered within the scope of general type-2 FLSs (GT2 FLSs), the practical implementation of GT2 FLSs has traditionally required the secondary MFs to be convex and normal type-1 fuzzy sets (T1 FSs). In addition, the type-reduction operation still presents a challenge for GT2 FLSs because of its computational complexity. In this paper, we present a complete framework for a type-2 FLS that uses the most recent perception of IT2 FSs (the so called general forms of interval type-2 fuzzy sets, gfIT2 FSs), whose secondary grades can be nonconvex T1 FSs. This framework includes new equations for the meet and join operations of gfIT2 FSs, as well as a new type reduction procedure for the type-2 FLS involving gfIT2 FSs. In addition, we present the type-2 FLS operation for singleton and nonsingleton fuzzification. We will introduce the various operations employed within a gfIT2 FLSs, from fuzzification (including singleton and nonsingleton) to inference, type-reduction, and defuzzification. We will also present two examples in which these gfIT2 FSs arise naturally when modeling sonar sensors input noise and the antecedents/consequents from a survey including different users.

中文翻译：

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基于区间二类模糊集一般形式的模糊逻辑系统

近年来，在实际应用中广泛使用区间类型 2 模糊逻辑系统 (IT2 FLS)。最近已经表明，区间类型 2 模糊集 (IT2 FS) 比区间值模糊集 (IV FS) [1] 更通用。因此，有必要探索更一般形式的 IT2 FS（超越 IV FS）的能力以及它们更适合的应用领域。此外，有必要发展 IT2 FLSs (gfIT2 FLSs) 的一般形式的理论，它使用不等价于 IV FSs 的 IT2 FSs，并且可以具有非凸的二级隶属函数 (MFs)。虽然这些系统可以考虑在一般类型 2 FLS (GT2 FLS) 的范围内，GT2 FLS 的实际实现传统上要求辅助 MF 是凸的和正常的类型 1 模糊集（T1 FS）。此外，由于其计算复杂性，类型归约操作仍然对 GT2 FLS 提出了挑战。在本文中，我们提出了一个完整的 2 类 FLS 框架，该框架使用 IT2 FS（所谓的区间 2 类模糊集的一般形式，gfIT2 FS）的最新感知，其二级等级可以是非凸的 T1 FS . 该框架包括 gfIT2 FS 的会合和连接操作的新方程，以及涉及 gfIT2 FS 的类型 2 FLS 的新类型归约程序。此外，我们还介绍了单例和非单例模糊化的类型 2 FLS 操作。我们将介绍 gfIT2 FLS 中使用的各种操作，从模糊化（包括单例和非单例）到推理、类型缩减和去模糊化。我们还将展示两个示例，其中这些 gfIT2 FS 在对声纳传感器输入噪声和包括不同用户的调查中的前因/后果进行建模时自然出现。