柔性区间逻辑及推理研究
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摘要
本文的研究工作源于国家自然科学基金项目(60273087)“经验知识推理理论研究”与北京市自然科学基金项目(4032009)“不精确推理理论研究”。
已经发展十分完善的经典数理逻辑是刚性逻辑，只能解决确定性问题。研究具有矛盾和不确定性问题的各种非经典逻辑和现代逻辑是人工智能的主要发展方向之一。何华灿教授所提出的泛逻辑学旨在研究逻辑的一般规律，重点在研究具有矛盾和不确定性的各种柔性推理过程和演化过程。
本文将泛逻辑学的思想引入到区间逻辑中，实现了区间逻辑的值域和运算模型的柔性化，并对柔性区间推理进行了研究，主要创新点如下：
1．根据泛逻辑学原理，把广义相关性引入到区间逻辑中，定义了区间逻辑的补、与、或、平均、等价和组合运算模型簇。重点论证了柔性区间与、柔性区间或和柔性区间平均运算簇的性质。
2．提出了四种柔性区间蕴涵运算模型簇，重点证明了第一种柔性区间蕴涵的边界条件、单调性和伴随性，证明了柔性区间与、柔性区间或和柔性区间蕴涵可构成剩余格，且其具有良好的代数性质。
3．提出了柔性区间推理模型，证明了该运算模型满足推理规则的基本要求。
4．将柔性区间逻辑应用到粗糙逻辑研究中，定义了粗糙与、或、补和蕴涵运算模型，证明了粗糙蕴涵和粗糙与、或、补可构成FI代数。
上述研究成果为进一步证明柔性区间逻辑推理的有效性和可靠性，为最终建立柔性区间逻辑形式演绎系统奠定了理论基础。
This dissertation comes from the National Nature Science Foundation of China "Research on Theories of Experience Knowledge Reasoning" (No. 60273087) and Beijing Nature Science Foundation of China "Research on Theories of imprecise reasoning" (No.4032009), and it is part of the research field of the Artificial Intelligence.
The well-developed classical mathematical logic is rather rigid and it can only solve certainty problems. Studies of various non-classical logic and modern logic, which contain contradiction and uncertainties, have been the mainstream topics for further development of AI. In the study of general laws of logics, Professor Huacan He proposed a new flexible logic which was called Universal Logics. Various flexible-reasoning processes and evolution processes were comprehensively investigated, containing contradiction and uncertainties.
In this dissertation, the idea of Universal Logics is introduced into the Interval Logic. The flexibility of the valued field and operations models of the Interval Logic has been realized. And Flexible Interval-reasoning process is deeply studied. The innovation and main results are summarized as follows:
1. Based on Universal Logics principles, the generalized correlativity is introduced into the Interval Logic. The operation model clusters of the Interval Logic are defined, including the Flexible Interval Complement, the Flexible Interval Intersection, the Flexible Interval Union, the Flexible Interval Implication, the Flexible Interval Equivalence, the Flexible Interval Average and the Flexible Interval Combination; Moreover, properties of the Flexible Interval Intersection, the Flexible Interval Union, the Flexible Interval Implication and the Flexible Interval Average are investigated.
2. Four operation model clusters of the Flexible Interval Implication are proposed. The boundary condition and monotonicity of the first operation model clusters of the Flexible Interval Implication are investigated; The Flexible Interval-intersection and the Flexible Interval-implication are proved to be an adjoint pair; Furthermore, the Flexible Interval-intersection, the Flexible Interval Union and the Flexible Interval-implication are proved to constitute complete residuated lattice, and have the well-known algebraic properties.
3. The models of the Flexible Interval-reasoning are proposed, and found to
已经发展十分完善的经典数理逻辑是刚性逻辑，只能解决确定性问题。研究具有矛盾和不确定性问题的各种非经典逻辑和现代逻辑是人工智能的主要发展方向之一。何华灿教授所提出的泛逻辑学旨在研究逻辑的一般规律，重点在研究具有矛盾和不确定性的各种柔性推理过程和演化过程。
本文将泛逻辑学的思想引入到区间逻辑中，实现了区间逻辑的值域和运算模型的柔性化，并对柔性区间推理进行了研究，主要创新点如下：
1．根据泛逻辑学原理，把广义相关性引入到区间逻辑中，定义了区间逻辑的补、与、或、平均、等价和组合运算模型簇。重点论证了柔性区间与、柔性区间或和柔性区间平均运算簇的性质。
2．提出了四种柔性区间蕴涵运算模型簇，重点证明了第一种柔性区间蕴涵的边界条件、单调性和伴随性，证明了柔性区间与、柔性区间或和柔性区间蕴涵可构成剩余格，且其具有良好的代数性质。
3．提出了柔性区间推理模型，证明了该运算模型满足推理规则的基本要求。
4．将柔性区间逻辑应用到粗糙逻辑研究中，定义了粗糙与、或、补和蕴涵运算模型，证明了粗糙蕴涵和粗糙与、或、补可构成FI代数。
上述研究成果为进一步证明柔性区间逻辑推理的有效性和可靠性，为最终建立柔性区间逻辑形式演绎系统奠定了理论基础。
This dissertation comes from the National Nature Science Foundation of China "Research on Theories of Experience Knowledge Reasoning" (No. 60273087) and Beijing Nature Science Foundation of China "Research on Theories of imprecise reasoning" (No.4032009), and it is part of the research field of the Artificial Intelligence.
The well-developed classical mathematical logic is rather rigid and it can only solve certainty problems. Studies of various non-classical logic and modern logic, which contain contradiction and uncertainties, have been the mainstream topics for further development of AI. In the study of general laws of logics, Professor Huacan He proposed a new flexible logic which was called Universal Logics. Various flexible-reasoning processes and evolution processes were comprehensively investigated, containing contradiction and uncertainties.
In this dissertation, the idea of Universal Logics is introduced into the Interval Logic. The flexibility of the valued field and operations models of the Interval Logic has been realized. And Flexible Interval-reasoning process is deeply studied. The innovation and main results are summarized as follows:
1. Based on Universal Logics principles, the generalized correlativity is introduced into the Interval Logic. The operation model clusters of the Interval Logic are defined, including the Flexible Interval Complement, the Flexible Interval Intersection, the Flexible Interval Union, the Flexible Interval Implication, the Flexible Interval Equivalence, the Flexible Interval Average and the Flexible Interval Combination; Moreover, properties of the Flexible Interval Intersection, the Flexible Interval Union, the Flexible Interval Implication and the Flexible Interval Average are investigated.
2. Four operation model clusters of the Flexible Interval Implication are proposed. The boundary condition and monotonicity of the first operation model clusters of the Flexible Interval Implication are investigated; The Flexible Interval-intersection and the Flexible Interval-implication are proved to be an adjoint pair; Furthermore, the Flexible Interval-intersection, the Flexible Interval Union and the Flexible Interval-implication are proved to constitute complete residuated lattice, and have the well-known algebraic properties.
3. The models of the Flexible Interval-reasoning are proposed, and found to
引文
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