布尔函数的计量化及其在多值逻辑系统中的应用研究

In quantitative logic, truth degree of formulas and similarity degree between formulas are systematically introduced through establishing graded versions of basic logical notions. Meanwhile the pseudo-distance between formulas is constructed and theory of the logic metric space is established correspondingly. Based on this thought, the following contents are mainly studied in this project. (1)The thought of graded versions is applied to the research of Boolean functions, quantitative properties of linear Boolean functions, symmetric Boolean functions as well as avalanche Boolean functions and their distribution in the classical logic metric space are researched. (2)The Shannon expansion of Boolean functions is applied to many-valued logic systems. The expansions of many-valued functions which are induced by logical formulas are gained. The normal forms and construction methods are researched. (3)The definitions of linear many-valued functions, symmetric many-valued functions, avalanche many-valued functions and their distributions in many-valued logic metric spaces are studied. These results will further enrich the research of quantitative logic. Meanwhile the construction methods of these special Boolean functions are gained from the perspective of quantitative logic.

计量逻辑学从基本概念的程度化入手，系统地引入了公式的真度理论和公式间的相似度理论，定义了公式间的伪距离，最终建立起了逻辑度量空间理论. 基于这种思想，本项目主要研究以下内容：(1)将程度化思想引入到布尔函数的研究之中，研究线性、对称和雪崩布尔函数的计量性质及其在经典逻辑度量空间中的分布情况. (2)将布尔函数Shannon展开式的巧妙思想应用到多值逻辑系统之中，给出由逻辑公式导出的多值函数的展开式.基于多值函数的展开式，给出多值函数的构造方法和范式表示. (3)在多值逻辑系统中，给出线性、对称和雪崩多值函数的定义，研究这几类特殊的多值函数的计量性质及其在多值逻辑度量空间中的分布情况. 这将进一步丰富计量逻辑学的研究内容，并能从计量逻辑学角度出发，给出这几类特殊的布尔函数的构造方法.

计量逻辑学从基本概念的程度化入手, 系统地引入了公式的真度理论和公式间的相似度理论, 定义了公式间的伪距离, 最终建立起了逻辑度量空间理论. 布尔函数理论既是经典逻辑度量空间中真度理论的基础, 又是密码学中常用的基本工具, 可见计量逻辑学与密码学之间存在着紧密的联系. 基于这种思想, 本项目从计量逻辑学的角度研究了密码学中的雪崩布尔函数, 给出了雪崩布尔函数新的构造方法, 并改进了其个数的下界. 其次, 本项目将布尔函数Shannon展开式的巧妙思想应用到多值逻辑系统之中, 给出了由逻辑公式导出的多值函数的展开式. 基于多值函数的展开式, 给出多值逻辑系统中逻辑公式的析取范式和合取范式表示, 并研究逻辑公式的计数问题. 最后, 在逻辑度量空间中给出了反射变换和仿射变换的概念, 研究了在这两类变换下逻辑公式的计量性质.