非可换逻辑证明论与模糊推理算法研究

Non-classical logic, one of the most important foundation theory of intelligent information processing technology, has received extensive attention from research scholars in recent years. Taking into account this situation, this project will focus on the research in the two new directions such as the proof theory of non-commutative logics and fuzzy reasoning algorithms. As for the proof theory of non-commutative logics, we will conduct the study on the essential characteristics of the left and right residual operations induced by the pseudo-uninorms and intend to build the Hilbert system for the non-commutative logic psUL based on pseudo-uninorms as well as the proof of its soundness and completeness. Concurrently, we will do the research on the logic rules characterizing the continuity of the continuous uninorms and intend to construct the Hypersequent calculus for fuzzy logic BUL based on continuous uninorms. Also the logic rules for the axioms of the residual implications induced by pseudo-uninorms will be investigated to establish the the Hypersequent calculus for the non-commutative logic psUL. In regard to the fuzzy reasoning algorithms, respectively combining the idea of the full implication triple I method with the fuzzy logic UL and non-commutative logic psUL, we will obtain the triple I algorithm for fuzzy reasoning based on the fuzzy logic UL and the left R-type (right R-type) triple I algorithm for fuzzy reasoning based on the logic psUL, which are incorporated into the framework of fuzzy logics, and simultaneously provide the strict logical demonstration of the new fuzzy reasoning algorithms. Given these facts, this project will further deepen the research into fuzzy logic theory and have important significance for the practical applications of fuzzy reasoning, fuzzy control and decision etc.

非经典逻辑是智能信息处理技术的重要基础理论之一，近年受到研究学者的广泛关注。本项目将围绕两个新方向开展研究：非可换逻辑证明论、模糊推理算法。在非可换逻辑证明论方向上，研究伪一致范数诱导的左、右蕴涵的本质特性，拟构建基于伪一致范数非可换逻辑psUL的希尔伯特公理系统，证明其可靠性与完备性；探寻刻画连续一致范数"连续性"的逻辑规则，拟构造基于连续一致范数模糊逻辑BUL超串演算系统；构造左、右蕴涵公理的逻辑规则，拟创建非可换逻辑psUL的超串演算系统。在模糊推理算法方向上，将全蕴涵三I算法思想分别与模糊逻辑UL、非可换逻辑psUL结合起来，给出基于模糊逻辑UL的模糊推理三I算法、基于非可换逻辑psUL的模糊推理左R-型（右R-型）三I算法，并将其纳入模糊逻辑框架，给出模糊推理算法的严格逻辑论证。该项目研究进一步深化模糊逻辑理论研究成果，在模糊推理、模糊控制与决策等实际应用中有重要意义。

本项目研究非可换逻辑证明论与模糊推理算法及其应用。在非可换逻辑证明论方向上，研究伪一致范数诱导的左、右剩余的本质特性，构建基于伪一致范数非可换逻辑的希尔伯特公理系统，证明其可靠性与完备性；构造基于连续一致范数模糊逻辑超串演算系统；研究逻辑代数的滤子及导子理论。在模糊推理算法方向上，给出基于伪一致范数与区间值模糊集的模糊推理三I算法与反向三I算法，并给出基于一簇区间值三角范数的模糊推理算法，并将其纳入模糊逻辑框架，给出模糊推理算法的严格逻辑论证。进一步，将模糊推理算法分别与图像分类与倒立摆结合起来，给出新型的图像分类方法与倒立摆控制方法；将区间值模糊集及直觉模糊集与多属性决策相结合，给出多属性决策新算法。该项目研究不仅深化模糊逻辑理论研究成果，而且将理论成果成功应用到图像处理与多属性决策中，表明模糊推理在实际应用中有重要意义。