几类重要实代数簇的等式理论的完备性、可计算性及其在智能计算中的应用

The project studies fuzzy logical systems UL、IUL and HpsUL* for their completeness, computability and applications to intelligent computing, its major topics and significance are as follows: theoretically, in order to obtain profound results on these systems we explore the open problem in the fuzzy logical community about the standard completeness of IUL with the help of theories and techniques of model theory and proof theory; with the help of theories and skills on decision problems in equational logic and equational theory we study another open problem in the fuzzy logical community about the computability of UL, IUL, and HpsUL* system and hope obtain computer algorithms to determine properties of the truth function in these systems; practically, we will attempt to use UL, IUL, and HpsUL* to formalize fuzzy inference models and algorithms based on uninorms which may lay firm mathematical foundations for these fuzzy inference models, and with the help of these systems we will attempt to interpret mathematically the combining functions of certainty factors in expert system MYCIN and PROSPECTOR and attempt to expand their applications, which lay firm theoretical foundations for study the properties of information aggregation based on uninorms.

本项目研究模糊逻辑系统UL、IUL和HpsUL*等的完备性和可计算性及其在智能计算中的应用，其主要内容和意义如下：理论方面，利用模型论和证明论的理论和技巧围绕IUL的标准完备性这一模糊逻辑界的公开难题开展研究，以期获得关于这些系统的深刻结果；利用等式逻辑和等式理论关于判定问题的理论和技巧研究UL、IUL和HpsUL*等系统的可计算性这一公开难题，以期得到计算机算法来判断这些系统中真值函数的性质；应用方面，基于UL、IUL和HpsUL*等形式化基于一致模的模糊推理模型和算法，为这类模糊推理模型建立数学基础，同时基于这些系统对专家系统MYCIN和PROSPECTOR中的确定性因子的组合函数给出数学上的解释和应用上的拓展，为基于一致模的信息聚合算子的性质研究提供理论基础。

本项目研究模糊逻辑系统UL、IUL和HpsUL*等的完备性、可计算性以及在智能计算中的应用。主要成果如下：解决了IUL 的标准完备性这一模糊逻辑界长达十年的公开难题；基于该成果进一步解决了HpsUL*的完备性，这一问题由Metcalfe, Olivetti, Gabbay 和Tsinakis等人2009年在其专著专著中提出；在UL、IUL和HpsUL*等的判定性方面，通过添加一个简单的公理到UL，IUL和HpsUL *，给出了有限UL和IUL代数的公理刻画，从而对相应的有限代数类有了深入的认识，为进一步研究这些系统的判定问题和可计算性问题奠定了基础；在基于UL、IUL和HpsUL*等的应用基础研究方面，项目组也取得一系列初步成果。