命题逻辑公式

The soundness and completeness theorem of this method were proved at last. This part consists of the following points: 1. The properties of the indecomposable extremely simple form of LP were discussed, the theorem shown that every logical formula of LP was equals of a generalized conjunction normal form. 2. The method for judging a logical formula is a indecomposable extremely simple form or not was discussed, the necessary and sufficient conditions of a logical formula with not more than three implicative connective being a indecomposable extremely simple form was given. 3. The α-resolution fields of the indecomposable extremely simple form of LP were discussed, some elementary results were obtained. 4. The automated reasoning method based on the resolution principle of classical logic was discussed, the author given a method to describe the set of clauses by using a matrix, and translated the resolution of two clauses to column operations of this matrix.

二、关于格值命题逻辑系统LP中α-自动推理方法的研究 1、对LP中不可分极简式的性质进行了研究，证明了LP中任何一个格值逻辑公式等值于一个广义合取范式； 2、对LP中不可分极简式的判定方法进行了研究，讨论了含有不超过3个蕴涵运算的格值逻辑式为不可分极简式的充分必要条件； 3、对LP中不可分极简式的α-归结域进行了研究，得到了一些基本结果； 4、对经典逻辑中基于归结原理的自动推理方法进行了研究，给出了用符号矩阵表示子句集合的一种方法，把不同的归结过程转换成矩阵的列变换。

Based on paper [1] and [2], this paper gives two generating algorithms to calculate and output the special conjunctive normal forms and special disjunctive forms of given propositional formulas automatically and formally.

摘要在文[1]和文[2]的基础上，给出了命题逻辑中任一命题公式的主析取范式和主合取范式的自动生成算法，并实现了多个命题公式主范式的同时形式化输出。

Using the infinite product of unevenly distributed probability space with potential of 3, the concept of formula truth degree was introduced into the Gdel 3-valued propositional logic system. Moreover, it was proved that in the 3-value (1/4, 1/2, 1/4), measure the set of truth of degree of propositions was dense in [0, 1], and a general expression of truth degree was obtained, providing a basis for further developing approximate reasoning of 3-valued propositional logic system.

利用势为3的非均匀概率空间的无穷乘积，在Gdel三值命题逻辑系统中引入公式的真度概念，在三值逻辑(四分之一，二分之一，四分之一)测度下证明G3中全体公式的真度值之集在[0， 1]上是稠密的，并给出公式真度的表达通式，为进一步在三值命题逻辑系统中展开近似推理奠定基础。

Finally,a simplified case of BOFL,i.e.Boolean Operator Propositional Logicestablished on a Boolean algebra,is further discussed.A complete algorithmfor finding the true level and false level of a formula in BOPL,which subsumes the re-lated work by Wang H.in the propositional logic,is also provided.

对布尔算子模糊逻辑的简化情形一布尔算子命题逻辑作了进一步讨论，放宽了对真值域的要求，将布尔算子命题逻辑建立在布尔代数上，并推广了命题逻辑中的王浩算法，给出了一个完备的求给定公式恒真水平和恒假水平的机械推导算法。

Furthermore, we discuss the truth degree of the propositional formula in two-valued propositional logic system.

并对命题公式的蕴涵问题进行等价描述；对二值命题逻辑系统的程度化研究中命题公式的真度问题进行讨论。

Having combination of the theory of truth degree in metrology of logic with the theory of ∑-α-tautologies in fuzzy logic, which have been introduced by professor G.J.Wang, the theory of ∑Γ-truth degree was introduced, and its properties which induce a pseudometric on F were investigated. The results gained have complemented and enhanced the original theory, and the work delivers a new frame for fuzzy reasoning.

将模糊命题逻辑中的∑-α-重言式理论与计量逻辑学中的真度理论相结合，在经典二值命题演算系统L中引入了公式相对于有限理论的∑Γ-真度理论，较为详细地讨论了它们的性质，并利用∑Γ-真度的性质在公式集F上引入了ρΓ-伪距离，对原有的理论进行了加强和补充，为在模糊命题逻辑系统的有限理论中讨论结论的程度化问题奠定了基础。

Part Three The study of Lattice-valued Modal First-order Logic System and Its Resolution Principle In this part, we introduced quantifiers and predicate into LMP, put up lattice-valued modal first-order logic system LMF, and gave its semantic interpretation and syntax structure, proved soundness theorem and consistence theorem. Moreover, in order to judge the satisfiability of formula, defined Skolem standard type and H-interpretation. Based on these work, made a primary discussion of a - resolution principle based on LMF.

第三部分：关于格值模态一阶逻辑系统及其归结原理的研究第n页西南交通大学博士研究生学位论文这一部分主要是在格值模态命题逻辑系统LMP队）中引进量词和谓词，建立格值模态一阶逻辑系统LMF（广并给出其语又解释和语法结构，证明了系统的可靠性和协调性；另外，为了判断公式的可满足性，定义了格值模态一阶公式的 Skolem标准型和体解释；在此基础上，对基于系统LMF的a一归结原理进行了初步探讨。

Finally, the given symbolic logic formula can prove the rationality of component model and validity of framework extension.

在分析了框架元模型组成元素的基础上，应用面向领域的特征模型的方法，对构件模型组织结构进行统一的抽象和描述，并用命题逻辑公式来表示构件之间的逻辑关系和语义约束；用UniCon语言给出了扩展点及其依赖关系的表示方式；最后通过给定的命题逻辑规则公式来验证构件模型的合理性、框架扩展的正确性。

Analysis is conducted on model checking for three valued logic formulae of modal transition system using existing model checking techniques. We reduce the three valued logic model checking problem for modal transition system to the two valued model checking problem under Kripke structure. This reduction is linear in the number of states, size of transition relations, and number of atomic propositional formulae in the new formed model compared with those in the original model. It does not increase the comple...

分析了现有的模型检验技术应用于模态转移系统的三值逻辑公式的模型检验中存在的问题·提出了把模态转移系统转换成Kripke结构的算法以及三值逻辑公式转换成2个二值逻辑的算法，经过转换后可用现有的模型检验技术进行模型检验·用该算法转换后，状态数、转移数和原子命题数目与原模型呈线性关系，没有增加模型检验的复杂度

The theory of ∑-α-tautologies of fuzzy propositional logic was combined with the theory of truth degree in metrology of logic introduced by professor G.J.Wang, and the theory of ∑Γ－fuzzy truth degrees of formula relative to the finite theory in propositional logic system n was introduced.

将模糊命题逻辑系统中的∑-理论与计量逻辑学中的真度理论相结合，在n-值Lukasievicz模糊命题逻辑系统Ln中引入了公式相对于有限理论的∑Γ－模糊真度理论，讨论了其中的主要性质。

They weren't aggressive, but I yelled and threw a rock in their direction to get them off the trail and away from me, just in case.

他们没有侵略性，但我大喊，并在他们的方向扔石头让他们过的线索，远离我，以防万一。

In slot 2 in your bag put wrapping paper, quantity does not matter in this case.

在你的书包里槽2把包装纸、数量无关紧要。