对称逻辑公式在经典逻辑度量空间中的分布

摘要

The concept of symmetric Boolean functions treated in cryptology is transplanted into quantitative logic, and the concepts of symmetric logic formulas and pseudo-symmetric logic formulas are introduced. It is pointed out that logic fornulas in two valued logic are closely related to Boolean functions while they have crucial differences. It is proved that the ratio of the number of symmetric formulas with n atoms over the number of all formulas with n atoms converges to zero when n tends to infinite. It is proved that the set of truth degrees of synmmetric logic formulas is dense in [0,1] .It is proved from the viewpoint of topology that the set consisting of all symmetric logic formulas is a nowhere dense set in the classical logic metric space.%将密码学中对称布尔函数的概念引入到计量逻辑学理论之中,定义了对称逻辑公式和准对称逻辑公式.指出二值逻辑公式与布尔函数既密切相关,又有重要区别.证明了n元对称公式占全体n元逻辑公式的比例随n的增大而趋向于零,然而全体对称公式的真度之集却在[0,1]中稠密.最后从拓扑学的观点证明了全体对称公式之集在经典逻辑度量空间中无处稠密.