LOGICAL OPERATORS - The Ma讲义sters College ppt课件
4归纳逻辑中文课件(卡尔纳普)℉
归纳逻辑与演绎逻辑的比较
卡尔纳普认为，演绎逻辑与归纳逻辑所 共同面临的最基本的任务有这样三项：由 前提(或证据)出发寻找相应的结论(或假说)， 给定一定的前提(或证据)以及一定的结论 (或假说)，检查该结论能否由该前提所推出， 该假说能否由该证据所证实；任给由一前 提推出一结论或由一证据证实一假说的证 明，检查此证明是否正确。
论，我们可以建立起研究归纳前提(证 据)对结论(假说)的证实度的归纳逻辑， 这是归纳逻辑的真正研究对象，正如 演绎逻辑以逻辑蕴涵为研究对象一样。 这也就是归纳推理、归纳逻辑的合理 性之所在，这样的归纳逻辑理论是具 有实用意义的。
能建立目的较为有限制的归纳机器
卡尔纳普认为，我们能够建立目的较 为有限制的归纳机器，即可按程序计 算给定的归纳前提(证据)对给定的结论 (假说)的证实度的机器，这是卡尔纳普 的设想，其实现的必要条件是建立起 他所主张的归纳逻辑理论。
第二节 归纳逻辑及其与演绎逻辑的比较
卡尔纳普将归纳逻辑理解为研究包括 归纳推理在内的一切非演绎、非论证 性推理前提对结论的证实度的理论， 这一思想是针对波普否认归纳推理、 归纳逻辑的合理性、必要性以及证实 的可能性而提出的。
波普在这方面的主张
有限不能证明无限，过去不能证明将来， 归纳推理无论’以确定的抑或概率的命题 为结论，其合理性都是不能得到证明的， 同理，科学理论不可能通过证实来予以检 验而具有意义；另一方面，从科学考察的 实际来看，归纳推理并无用处，科学方法 根本不具有归纳的性质，科学理论不是通 过归纳推理由经验材料建立起来的，而是 靠想象、猜测、灵机一动产生出来，再经 由“证伪”来予以检验。波普由此认为， 归纳逻辑既无必要建立，也不可能建立起 来。
第一节 生平及关于归纳逻辑的基本思想
逻辑学的英文翻译PPT
C.The intensions and extensions. 概念的内涵和外延 The intension is the essential character reflected by the concept,the extension is the objects that possess the essential character reflected by the concept. 概念的内涵是指概念所反映的事物的本质属性，反映的事物质的 方面，回答的事物“是什么”的问题；概念的外延是指具有概念所反映的本质属 性的那些事物，反映事物量的方面，回答事物“有哪些”或“哪些是该事物”的 问题。
charactering judgrment infer wether another charactering judgrment is true.性质推理是指由一种已知的性质判断推断出另一种性质判断的真假的思维 形式。
B.演进过程 入oros logic 逻辑 理则学
严 复
牟宗三
3 The logical implication 逻辑含义
A 本原、本体、原初、发端。 Principle、Noumenon、Primary、inception （四词都有“原始”的意思） B一般规律、原理和规则。 The general pattern、axiom and regulation (四词都有“规范”的意思） C尺度、关系、比例、比率。 measurement、relationship、proportion、ratio （ 四词都含“数量关系”的意思) D 思维形式 和 思维规律 Thinking form and Thinking regular
B.表达形式：The representation of concept are languages、words and
数理逻辑(Mathematical Logic)
数理逻辑（MathematicalLogic）数理逻辑(Mathematical logic)是用数学方法研究诸如推理的有效性、证明的真实性、数学的真理性和计算的可行性等这类现象中的逻辑问题的一门学问。
其研究对象是对证明和计算这两个直观概念进行符号化以后的形式系统。
数理逻辑是数学基础的一个不可缺少的组成部分。
数理逻辑的研究范围是逻辑中可被数学模式化的部分。
以前称为符号逻辑（相对于哲学逻辑），又称元数学，后者的使用现已局限于证明论的某些方面。
历史背景“数理逻辑”的名称由皮亚诺首先给出，又称为符号逻辑。
数理逻辑在本质上依然是亚里士多德的逻辑学，但从记号学的观点来讲，它是用抽象代数来记述的。
某些哲学倾向浓厚的数学家对用符号或代数方法来处理形式逻辑作过一些尝试，比如说莱布尼兹和朗伯（Johann Heinrich Lambert）；但他们的工作鲜为人知，后继无人。
直到19世纪中叶，乔治·布尔和其后的奥古斯都·德·摩根才提出了一种处理逻辑问题的系统性的数学方法（当然不是定量性的）。
亚里士多德以来的传统逻辑得到改革和完成，由此也得到了研究数学基本概念的合适工具。
虽然这并不意味着1900年至1925年间的有关数学基础的争论已有了定论，但这“新”逻辑在很大程度上澄清了有关数学的哲学问题。
在整个20世纪里，逻辑中的大量工作已经集中于逻辑系统的形式化以及在研究逻辑系统的完全性和协调性的问题上。
本身这种逻辑系统的形式化的研究就是采用数学逻辑的方法.传统的逻辑研究（参见逻辑论题列表）较偏重于“论证的形式”，而当代数理逻辑的态度也许可以被总结为对于内容的组合研究。
它同时包括“语法”（例如，从一形式语言把一个文字串传送给一编译器程序，从而转写为机器指令）和“语义”（在模型论中构造特定模型或全部模型的集合）。
数理逻辑的重要著作有戈特洛布·弗雷格（Gottlob Frege）的《概念文字》（Begriffsschrift）、伯特兰·罗素的《数学原理》（Principia Mathematica）等。
英语词组
5.2 Boolean Algebra and Logic Circuit布尔代数与逻辑电路Boolean Algebra 布尔代数Logic Circuit 逻辑电路Logic Gate 逻辑门mathematical logic数理逻辑logic variable 逻辑变量logic operation逻辑运算expression 表达式AND operation 与运算AND gate 与门function table 真值表，函数表，功能表OR operation 或运算OR gate或门NOT operation 非运算complement 补码，反码Inversion 取反，反向Inverter 反相器De Morgan’s Theorem 摩根定理identity 恒等式composite operations 复合运算NAND 与非NOR 或非integrated circuit 集成电路logic flow diagram 逻辑流程图Flip-Flop（FF）触发器memory characteristic 记忆特性SET 置位CLEAR 复位symmetrical 对称的illegal 非法的internal内部的circuitry 电路, 线路memory element 存储元件, 记忆元件burglar alarm 防盗报警器deactivate 使无效，使不活动photocell 光电池，光电管illuminate 照明，照亮saturated 饱和的activate 使动作，使活动synchronous 同步的sequential 顺序的synchronous sequential system同步时序系统synchronize v.同步master clock 主时钟periodic 周期的pulse 脉冲periodic pulse 周期脉冲square wave 方波duty cycle 占空比interval 时间间隔，间隔rising edge 上升沿falling edge 下降沿positive-going edge 正向沿negative-going edge 负向沿clocked flip-flop 时钟触发器frequency 频率propagation delays 传播延迟trigger 触发waveforms 波形trigger input 触发脉冲输入control input 控制输入edge-triggered 边沿触发triangle 三角形Chapter 4 Electric Circuit Theory电路理论field 场circuit 电路potential 电位potential difference 电位差electric charge电荷mass 质量particle 粒子electric field 电场electron 电子potential energy 势能，位能electric field strength 电场强度scalar 标量displacement转移vector向量, 矢量integrate积分product 乘积current电流frame of reference参照物detector探测器, 检测器magnetic force磁力magnetic field 磁场intensity 强度magnet磁体, 磁铁equation等式，方程式accelerate 加速transmission传送, 传输voltage电压electromagnetic theory 电磁理论generator 发电机battery 电池motor 电机transformer 变压器coordinate 坐标lumped集总的distributed分布式的element 元件conduction current传导电流displacement current 位移电流crystal lattice 晶格charge carrier电荷载体, 载荷子conductor导体copper铜aluminium铝random 随机velocity 速度thermal agitation 热搅动, 剧烈振荡net 净的load 负载electro-chemical 电化学的thermo-electric 热电的photo-electric光电的circuit analysis 电路分析voltage source电压源current source电流源terminal 端子、终端terminal voltage 端电压function 函数temperature 温度pressure压力power 功率passive element 无源元件电阻resistanceOhm’s law欧姆定律active element有源元件电阻器resistor reciprocal倒数conductance 电导，导率inductance电感inductor电感器capacitance 电容capacitor电容器electrostatic 静电的step 阶跃Chapter 5 Electronic Technology 电子技术5.1 Signal Processing信号处理电子技术Electronic Technology 信号处理Signal Processing物理变量physical variable 无线电广播radio broadcast物理量physical quantity 数字的、数值的numerical股票市场stock market 电气系统electrical system机械系统mechanical system 液压系统hydraulic system模拟信号Analog Signal 数字信号Digital Signal热电偶thermocouple 接点，节点junction温差temperature difference 离散的discrete自动调温器，温度调节装置thermostat 预设值，预定值preset value开关switch 电子零件electronic part 接通,打开，开启turn on关闭，关掉，断开turn off 输入信号input signal传感器transducer 电气激励electrical excitation 音量volume显示display 指针needle 仪表meter 声能sound energy扩音器, 喇叭loudspeaker 十进制decimal system 位置计数法positional notation乘multiply 计算calculate 算术算法arithmetic 加法addition乘法multiplication 小数点decimal point 算术运算arithmetic operation数字digit 整数integer 基数base 根, 基数radix 晶体管transistor五进制quinary system 十二进制duodecimal system 二进制binary system类比，相似analogy 继电器relay 数字计算机digital computer双稳态bistable state。
《计算机科学导论》(第三版)弗罗赞(翻译)
Chapter 1 练习复习题1.定义一个基于图灵模型的计算机。
答：Turing proposed that all kinds of computation could be performed by a special kind of a machine. He based the model on the actions that people perform when involved in computation. He abstracted these actions into a model for a computational machine that has really changed the world. 图灵模型假设各种各样的运算都能够通过一种特殊的机器来完成，图灵机的模型是基于各种运算过程的。
图灵模型把运算的过程从计算机器中分离开来，这确实改变了整个世界。
2.定义一个基于冯·诺伊曼模型的计算机。
答：The von Neumann Model defines the components of a computer, which are memory, the arithmetic logic unit (ALU), the control unit and the input/output subsystems.冯·诺伊曼模型定义了计算机的组成，它包括存储器、算术逻辑单元、控制单元和输入/输出系统。
3.在基于图灵模型的计算机中，程序的作用是什么？答：Based on the Turing model a program is a set of instruction that tells the computer what to do.基于图灵模型的计算机中程序是一系列的指令，这些指令告诉计算机怎样进行运算。
4.在基于冯·诺伊曼模型的计算机中，程序的作用是什么？答：The von Neumann model states that the program must be stored in the memory. The memory of modern computers hosts both programs and their corresponding data.冯·诺伊曼模型的计算机中，程序必须被保存在存储器中，存储程序模型的计算机包括了程序以及程序处理的数据。
Logical NAND and NOR Operations Using Algorithmic Selfassembly of DNA Molecules
P h ysics Procedia 33 ( 2012 )954 – 9611875-3892 © 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Committee.doi: 10.1016/j.phpro.2012.05.160012 International Conference on Medical Physics and Biomedical EngineeringLogical NAND and NOR Operations Using Algorithmic Self-assembly of DNA MoleculesYanfeng Wang a , Junwei Sun a ,Guangzhao Cui b , Xuncai Zhang b , Yan Zheng baDepartment of Control Science and Engineering,Huazhong University of Science and Technology,Wuhan, 430074, Chinayanfengwang@bHenan Key Lab of Information-based Electrical Appliances,Zhengzhou University of Light Industry,Zhengzhou, 450002, Chinacgzh@AbstractDNA self-assembly is the most advanced and versatile system that has been experimentally demonstrated for programmable construction of patterned systems on the molecular scale. It has been demonstrated that the simple binary arithmetic and logical operations can be computed by the process of self assembly of DNA tiles. Here we report a one-dimensional algorithmic self-assembly of DNA triple-crossover molecules that can be used to execute five steps of a logical NAND and NOR operations on a string of binary bits. To achieve this, abstract tiles were translated into DNA tiles based on triple-crossover motifs. Serving as input for the computation, long single stranded DNA molecules were used to nucleate growth of tiles into algorithmic crystals. Our method shows that engineered DNA self-assembly can be treated as a bottom-up design techniques, and can be capable of designing DNA computer organization and architecture.© 2011 Published by Elsevier Ltd. Selection and/or peer-review under responsibility of [name organizer]Keywords:DNA Self-assembly; DNA Tile; Logical NAND Operation; Logical NOR Operation.1.IntroductionDNA self-assembly is a methodology for the construction of molecular scale structures. In this method, artificially synthesized single stranded DNA self-assemble into DNA crossover molecules (DNA tiles). These DNA tiles have sticky ends that preferentially match the sticky ends of certain other DNA tiles, facilitating the further assembly into tiling lattices. DNA self-assembly, using only a small number of component tiles, can provide arbitrarily complex assemblies. In those decades, the potential of DNA self-assembly has been widely developed in the fields of molecular computing, biophysics, nanotechnology and so on. It has important significance especially for the development of DNA nanotechnology. The rapid emergence of DNA nanotechnology in recent years has aroused much excitement among scientistsAvailable online at © 2012 Published by Elsevier B.V. Selection and/or peer review under responsibility of ICMPBE International Committee.Open access under CC BY -NC-ND license.Open access under CC BY -NC-ND license.Y anfeng Wang et al. / P hysics Procedia 33 ( 2012 )954 – 961 955 due to DNA self-assembly, as the most advanced and versatile system, which has been experimentally demonstrated for programmable construction of patterned systems on the molecular scale, and further-scale conventional silicon-based integrated circuits.In traditional silicon-based digital computer, the hardware circuits consist of the different kinds of logical gates circuits, and digital logic circuit is the basis for digital computer architecture. In a sensethat molecular logic gates built by DNA tiles could be considered as the basic structure of the DNA computer. In this contribution, we will focus on construction of theoretical models on the logical NANDand NOR operations based on the algorithmic of DNA self-assembly.The paper is organized as follows. Section II briefly reviews the Boolean logic, followed by, in SectionIII, the introduction of programming self-assembly of DNA tiles. In Section IV, we describe in detail the theoretical model of the common molecular logical NAND and NOR operations. Finally, we conclude in Section V with a brief discussion.2.Boolean logicBoolean logic, developed by George Boole, is often used to refine the determination of system status orto set or clear specific bits. Boolean logic is simply a way of comparing individual bits. It uses what are called operators to determine how the bits are compared. Boolean logic is always used for circuits designin electrical engineering, in which 0 and 1 may represent the two different states of one bit in a digital circuit, typically high and low voltage. Circuits are described by expressions containing variables, andtwo such expressions are equal for all values of the variables, if and only if, the corresponding circuits have the same input-output behavior. Furthermore, every possible input-output behavior can be modeledby a suitable Boolean expression.2.1.Logical NAND OperationLogical NAND, for Not And, sometimes denoted by a symoperation that is equivalent to the negation of the conjunction operation, expressed in ordinary language AND, it is one of the two functionally complete binary operators of propositional logic. The NAND operation is of particular significance to computer designers since any Boolean expression can be realizedby an expression using only the NAND operation.The logical NAND operation implements logical conjunction, which behaves according to the truth table (see Table I). The NAND operation with inputs x1 and x2 and output y implements the logical expression y=¬(x1x2).956Y anfeng Wang et al. / P hysics Procedia 33 ( 2012 )954 – 9612.2.Logical NOR OperationThe logical NOR operation or Inclusive-NOR operation is a combination of the digital logical OR operation with that of an inverter or NOT operation connected together in series. The NOR (Not-OR) operation has an output thatany of its inputs are at logic level of the OR operation. The NOR operation is of particular significance to computer designers since any Boolean expression can be also realized by an expression using only the NOR operation.The logical NOR operation implements logical disjunction, which behaves according to the truth table (see Table II). The NOR operation with inputs x1, x2 and output y implements the logical expression y=¬(x1x2).3.Programming self-assembly of dna tilesWe now focus on a common self-assembly approach: Computation by self-assembly. In this case, self-assembly is the spontaneous self-ordering of substructures into superstructures driven by annealing of Watson-Crick base-pairing DNA sequences. Computation by self-assembly entails the building up of superstructures from starting units such that the assembly process itself performs the actual computation.Adleman made use of a simple form of computation by self-assembly in his original experiment [1]: instead of blindly generating all possible sequences of vertices; The oligonucleotide sequences and the logic of Watson-Crick complementarity guide the self-assembly processes so that only valid paths are generated. Winfree [2] generalized this approach to two-dimensional self-assembly processes and showed that computation by self-assembly is Turing-universal. Jonoska et al. proposed the use of self-assembled DNA nanostructures to solve NP-complete problems [3].Programming self-assembly of DNA tiles amounts to the design of the pads of the DNA tiles. The use of pads relation of tiles in the final assembly to be intimately controlled, thus the only large-scale superstructures formed during assembly are those that encode valid mappings of input to output. Consequently, the approach for executing Boolean logical operations by using algorithmic of DNA self-assembly essentially contains the following steps. (i) mixing the input oligonucleotides to form the DNA tiles, (ii) allowing the tiles to self-assemble into superstructures, (iii) ligating strands that have been co-localized, and (iv) then performing a single separation to identify the correct output.4.Theoretical model of the common molecular logical operationsA variety of different DNA tile types have been used in previous assemblies, including DNA double-crossover molecules [4], DNA triple-crossover molecules [5], and parallelograms produced from HollidayY anfeng Wang et al. / P hysics Procedia 33 ( 2012 ) 954 – 961 957junction analogues [6]. Mao et al. had made the first experimental demonstrations of computation usingDNA tile assembly [7]. It demonstrated a 1-layer, linear assembly of triple-crossover molecules (TX tiles) that executed a bit-wise cumulative XOR computation. In this computation, n bits are input and n bits are output, where the i th output is the XOR of the first i input bits. This is the computation occurring when one determines the output bits of a full-carry binary adder circuit. The success of this experiment indicates that it is possible to use DNA components to do algorithmic assembly. Here we use the following modified triple-crossover molecules, whose structure is illustrated in Figure 1.Figure 1. The modified triple-crossover molecular structureThe molecule contains three DNA single strands (shown in red, green, and purple) that self-assemble through Watson-Crick base pairing to produce three double helices in a roughly planar arrangement, and they have three sticky ends at the three corners, each double helix is connected to adjacent double helical domains at two points in which their strands cross over between them. Notice a single DNA strand (colored red) passing from the bottom left to the top right of the tile.10i i y x 11i i y x 10i i y x 01i i y x Figure 2. Component tiles for performing logical NAND operation958 Y anfeng Wang et al. / P hysics Procedia 33 ( 2012 ) 954 – 961Figure 4. Self-assembly process of logical NAND operationThe ends of the central double helix are closed by hairpin loops, but the other helices can terminate insticky ends containing informa-tion that directs the assembly of the tiles. 4.1.Logical NAND OperationComponent Tiles for Logical NAND Operation: The tiles illustrated in Figure 1 are represented schematically as in Figure 2, which are used to perform the logical NAND operation. The three helices are depicted as connected rectangular forms terminating in sticky ends, which are represented as geometrical shapes, which are drawn flush. The three helical domains are drawn as rectangles, flanked by sticky ends shown as geometrical triangle shapes. The value of each tile is in the upper-right rectangle. The meaning of each sticky end is also indicated. The figure shows thetwox tiles (blue), the four ytiles (yellow), and the initialization corner tiles,c 1and c 2 (green).Based on the logical NAND operation, we propose a gener- al schema for the computation of the modified TX tiles. The tiles code the truth table of the NAND operation (Table I). The NAND consists ofa series of Boolean inputs x 1, x 2x n and a series of Boolean outputs y 1, y 2 y n , where x 1= y 1, and for i >1, y 1=1 y i =¬(x i y i-1). The value of any y i in these calculations also reports the even or odd parity of the first i values of x . Thus, two different kinds of input xtiles are needed, one whose value is 0 and a secondd by the0i y 10i i y x 01i i y x 0i y Figure 3. Component tiles for performing logical NOR operationY anfeng Wang et al. / P hysics Procedia 33 ( 2012 )954 – 961 959Figure 5. Self-assembly process of logical NOR operationThe x tiles are shown in blue in Figure 2, their value (0 or 1) is shown in their upper-left rectangle, the upper-left sticky end reports this value, and the upper-right and lower-left sticky ends provide the meansof connecting successive x tiles. These sticky ends are shown as geometrically complementary, as they would be for a general set of 16 parallel calculations. However, to demonstrate the efficacy of the procedure, the calculations performed here are definite five-bit calculations for which the order of the input x tiles is specified exactly by a series of different sticky ends.Like x tiles, two values of y tiles are necessary, again representing 0 and 1. However, there are two ways to get each of these results: the value of a y tile can be 1 either because both inputs are 0 or becausethe value of one input is 0 and the other is 1, or vice versa. Likewise, the value of a y tile can be 0 because both are 1. Thus, four different y tiles are needed according to the requirement.The y tiles are shown in yellow in Figure 2. The tile values again are displayed on the upper-right domain, and reported by the sticky end on the right of the upper domain. The two inputs are derived fromthe sticky ends on the left (y i-1) and right (x i) of the bottom domain. Note that the same sticky end in the input domain represents a given input, independence of the other end. For example, the right-side sticky end x i =1 has the same shape (sticky end) regardless of whether the left-side sticky end represents y i-1=0 (leading to a tile value of 1) or y i-1=0 (leading to a tile value of 0). For simplicity, they are labeled withthe values, as shown in Figure 2. There are only two different left sticky ends in the input (bottom) domain, and similarly only two different right sticky ends. Consequ-ently, both sticky ends on each tile must pair correctly for the proper tile to be inserted in the assembly.Self-assembly Process of Logical NAND Operation: According to component tiles, one NAND related self-assembly have been performed, and illustrated in Figure of the x and y tiles, two corner tiles, c1and c2(green in Figure 2) which are used to initialize the two values of x1 and y1, and to connect the input to the output. The values of each tile are the same as in Figure 2, and the sticky ends are the same, although the labels have been omitted for clarity. The tiles are shown assembled to perform NAND calculation. We note the complementarity of sticky ended association at each molecular interface, the sequences are complementary, but not self-complementary. The operations are designed to proceed from lower left to upper right, because the x1 and c1 and c2 tiles have longer sticky ends than the y tiles. After the first y tile has been added, x1=0, for both arrays, so y1=1. In the next step, the array in Figure 2 contains x2=0: because x2=0, y1=1, y2=¬(x2y1)=1. Because x3=1, y2=1, y3=¬(x3y2)=0. x4=1, y3=0, y4=¬(x4y3)=1. At last, x5=1, y4=1, y5=¬(x5y4)=0, as shown.960Y anfeng Wang et al. / P hysics Procedia 33 ( 2012 )954 – 961After five transition steps, the self-assembly is completed. The self-assembly in Figure 4 has the inputs x1=x2=0and x3=x4=x5=1. These correspond to output values of y1=y2=y4=1and y3=y5=0.Note that for i > 1, the sticky ends on the bottom domain of each y i tile complement those on the y i-1 tile on its left and the tile x i on its right. The (yellow) output layer then assemble specifically starting from the bottom left using the inputs from the blue layer. The tiles were designed such that an output reporter strand ran through all the n tiles of the assembly by bridges across the adjoining pads in input and output tiles. This reporter strand was pasted together from the short DNA sequences within the tiles using ligation enzyme. When the solution was warmed, this output strand was isolated and identified. The output data was read by experimentally determining the sequence of cut sites. In principle, the output could be used for subsequent computations.4.2.Logical NOR OperationComponent Tiles for Logical NOR Operation: The schema for the computation of the modified TX tiles. The tiles code the truth table of the NOR operation (Table II). The NOR consists of a series ofBoolean inputs x1, x2x n and a series of Boolean outputs y1, y2 y n, where x1= y1, and for i>1,y1=1, and for i > 1, y i=¬(x i y i-1). Other is the same as the logical NAND operation.The x tiles (blue) and the initialization corner tiles, c1 and c2 (green) are shown in Figure 3. Their specific structures are the same as the previous tiles of logical NAND operation. The y tiles are shown in yellow in Figure 3. The tiles are different from the logical NAND tiles. The tile values again are displayed on the upper-right domain, and this value is reported by the sticky end on the right of the upper domain. The two inputs are derived from the sticky ends on the left (y i-1) and right (x i) of the bottom domain. We note that the same sticky end in the input domain represents a given input value, independent of the other end. For example, the right-side sticky end x i=0 has the same shape (sticky end) regardless of whether the left-side sticky end represents y i-1=0 (leading to a tile value of 1) or y i-1=1 (leading to a tile value of 0). Both sticky ends on each tile also comply with the rules that correct tiles are participating with partially correct tiles.Self-assembly Process of Logical NOR Operation: Basing on the component tiles, we also have performed one NOR related self-assembly, illustrated in Figure 5. The one-dimensional template is also arbitrary, and consists of the x and y tiles, two corner tiles, 1c and 2c(green in Figure 2).The NOR operation are also same with the NAND operation which proceeds from lower left to upper right, because the x1 and c1 and c2 tiles have longer sticky ends than the y tiles. After the firsty tile has been added, x1=1, for both arrays, so y1=0. In the next step, the array in Figure 5 contains x2=0: because x2=0, y1=0, y2 should be 1. Because x3=0, y2=1, y3 should be 0. x4=1, y3=0, y4 should be0. At last, x5=0, y4=0, y5 should be 1. The self-assembly in Figure 5 has the inputs x1 = x4=1 and x2 =x3= x5=0. These correspond to output values of y1 = y3= y4=0 and y2= y5=1.4.3.Solution ExtractionOnce the self-assembly has occurred, it is necessary to extract the solution. For this purpose, each molecular tile contains a reporter strand, which traverses the tile in a diagonal pathway, the reporter strand is illustrated as a thick red strand in the tile, which is an x-type tile. Following self-assembly, the reporter strands are ligated to each other to produce a long reporter strand that contains the inputs and outputs of the calculation. The ligated long reporter strand in the vicinity of the corner of the assembly is shown as a thick black strand on the molecular diagram in Figure 6. The three TX tiles join diagonally.For extracting the solutions, the tiles were first assembled individually from their component strands by cooling slowly from 90 to room temperature, as done in [7]. The output in this case was read byY anfeng Wang et al. / P hysics Procedia 33 ( 2012 )954 – 961 961 determining which of two possible cut sites (endonuclease cleavage sites) were present at each position inthe tile assembly. This was executed by first isolating the reporter strand, then digesting separate aliquotswith each endonuclease separately and the two together, finally these samples were examined by gel electrophoresis and the output values were displayed as banding patterns on the gel. Another method for output is the use of atomic force microscope (AFM) observable patterning. The patterning was made by designing the tiles computing a bit 1 to have a stem loop protruding from the top of the tile, The sequenceof this molecular patterning was clearly viewable under appropriate AFM imaging conditions.After ligation of the tiling assembly (this joins together each tiles segments of the reporter strands), the reporter strand provides an encoding of the output of the tiling assembly computation (and typically alsothe inputs). Note this input/output can occur in parallel for multiple distinct tiling assemblies. Finally, the tiling assembly is disassembled by denaturing (e.g.,via heating) and the resulting ssDNA Reporter Strands provide the result (these may be used as scaffold strands for later cycles of assembly computation, or the readout may be by PCR, restriction cutting, sequencing, or DNA expression chips).5.DisscussionThe algorithmic molecular assembly described here demonstrates a non-trivial Boolean logic done by selfassembly. DNA self-assembly technology has been applied not only in DNA computation, but also in nanomaterial area. However, there are many difficulties in applying DNA self-assembly to Boolean logic. Firstly, the design of DNA codewords is the most difficult problem, as it decides whether the experimentcan be implemented. Secondly, there are two significant keys to effect on the yield of selfassembly structure: the assembled angle between DNA single-strands and the connections among motifs.Despite the engineering challenges met in laboratory investigations, DNA self-assembly technology shows encouraging theoretical advances and has already led to new computing paradigms which certainly enriched our understanding of the nature of computation. Moreover, the proposed method might be usedfor the Bottom-Up design of DNA computer architecture with the further advances in biological techniques.AcknowledgementThe work for this paper was supported by the National Science Foundation of China (Grant No. 60573190, 60773122, 60970084), Basic and Frontier Technology Research Program of Henan Province (Grant No. 082300413203, 092300410166), and Innovation Scientists and Technicians Troop Construction Projects of Henan Province (Grant No. 0941********).References[1] L. Science, vol. 266, no. 5187, pp. 10211024, 1994.[2] 1st DIMACS Workshop on DNA Based Computers, R. J. Lipton and E. B. Baum, Eds., vol. 27, Princeton, 1996, pp. 199221.[3] Proceedings of 1997 IEEE International Conference on Evolutionary Computation, vol. 13, no. 16, 1997, pp. 261266.[4] -assembly of two-Nature, vol. 394, no. 6693, pp. 539544, 1998.[5] T. H. LaBean, H. Yan, J. Kopatsch, F. Liu, E. Winfree, J. H. Reif, and N. C. Seeself-J.Am. Chem. Soc., vol. 122, no. 9, pp. 18481860, 2000.[6] C. Mao, W. Sun, and N. C. S DNA holliday junction arrays visualized by atomic forceJ. Am. Chem. Soc., vol. 121, no. 23, pp. 54375443, 1999.[7] -assembly of DNA triple-Nature, vol. 407, no. 6803, pp. 493496, 2000.。
ChLogic离散数学英文实用PPT课件
• Algorithm (Algorithmic Analysis, Algorithmic Thinking) • Universal, Existential Quantification, Rule of Inference • Probability, Statistics, Event, Independent, Dependent • Operation, Operator, Product, Sum, Division, Integer, Real Number,
• Steps to convert an English sentence to a statement in propositional logic • Identify propositions and using propositional variables • Determine appropriate logical operations
• “If I go to iShow (爱秀) or to CHIC (东京秀客 ), t hen I will no t go sho pping at Golden Ea gle .” • p: I go to iShow • q: I go to CHIC • r: I go shopping at GE
Solution: Let p be A is a knight, and q be B is a
kniቤተ መጻሕፍቲ ባይዱht
• If A is a knight, then p True. Since knights tell
the truth, then
If p or q then not r.
常用离散数学名词中英文对照
常用离散数学名词中英文对照集合：set元素：element严格定义:well defined成员:member外延原理:principle of extension泛集(全集):universal set空集:empty set(null set)子集:subset文氏图:venn diagram并:union交:intersection相对补集:relative complement绝对补集:absolute complement补集:complement对偶性:duality幂等律:idempotent laws组合律:associative laws交换律:commutative laws分配律:distributive laws同一律:identity laws对合律:involution laws求补律:complement laws对偶原理:principle of duality有限集:finite set计算原理:counting principle类:class幂集:power set子类:subclass子集合:subcollection命题:proposition命题计算:proposition calculus语句:statement复合:compound子语句:substatement合取:conjunction析取:disjuction否定:negation真值表:truth table重言式:tautology矛盾:contradiction逻辑等价:logical equivalence命题代数:algebra of propositions 逻辑蕴涵:logicalimplication关系:relation有序对:ordered pair划分:parti-on偏序:partial order整除性:divisibility常规序:usual order上确界:supremum下确界:infimum上(下)界:upper(lower) bound乘积集:product set笛卡儿积:cartesian product笛卡儿平面:cartesian plane二元关系:binary relation定义域:domain值域:range相等:equality恒等关系:identity relation全关系:universal ralation空关系:empty ralation图解:graph坐标图:coordinate diagram关系矩阵:matrix of the relation 连矢图:arrow diagram有向图:directed graph逆关系:inverse relation转置:transpose复合:composition自反:reflexive对称的:symmetric反对称的:anti-symmetric可递的:transitive等价关系:equivalence relation半序关系:partial ordering relation 函数:function映射:mapping变换:transformation像点:image象:image自变量:independent variable因变量:dependent variable函数图象:graph of a function合成函数:composition function可逆函数:invertible function一一对应:one to one correspondence 内射:injective满射:surjective双射:bijective基数度:cardinality基数:cardinal number图论:graph theory多重图:multigraphy顶点:vertix(point,node)无序对:unordered pair边:edge相邻的adjacent端点:endpoint多重边:multiple edge环:loop子图:subgraph生成子图:generated subgraph平凡图:trivial graph入射:incident孤立点:isolated vertex连通性:connectivity通路:walk长度:length简单通路:chain(trail)圈:path回路:cycle连通的:connected连通分支:connected component距离:distance欧拉图:eulerian graph欧拉链路:eulerian trail哈密顿图:hamilton graph哈密顿回路:hamilton cycle货郎行程问题:traveling salesman完全图:complete graph正则图:regular graph偶图:bipartive graph树图:tree graph加权图:labeled graph同构图:isomorphic graph同构:isomorphism同胚的:homeomorphic平面图:planar graph着色问题:colortion区域:region地图:map非平面图:nonplanargraph着色图:colored graphs顶点着色:vertex coloring色散:chromatic number四色原理:four color theorem对偶地图:dual map退化树:degenerate tree生成树:spanning tree有根树:rooted tree根:root水平(深度):level(depth)叶子:leaf分支:branch有序有根树:ordered rooted tree二元运算符:binary operational symbol半群:semigroup单位元素:identity element右(左)单位元素:right(left) identity左(右)消去律:left(right) cancellation law) 逆:inverse并列:juxtaposition有限群:finite group正规子群:normal subgroup非平凡子群:nontrivial subgroup循环群:cyclic group环:ring整环:integral domain域:field交换环:commutative ring加性环:additive group汇合:meet格:lattice有界格:bounded lattice分配格:distributeve lattice补格:complemented lattice表示定理:representation theorem。
《电子技术数字基础 Digital Fundamentals》双语课件PPT-第03章 Logic Gates
29
3-5 The NOR Gate
The NOR gate produces a LOW output when any of the inputs is HIGH, and HIGH only when all of its input are LOW. For a 2-input NOR gate, the output X is LOW if either A or B is HIGH, and HIGH if both A and B are LOW.
4
1-3-1 NOT Logic Operation
Function （逻辑功能）: changes one logic level to the opposite level. Changes a 1 to a 0 and a 0 to a 1. Logic symbol
•
E
•
A Y
5
1-3 Basic Logic Operations
X AB
NAND gates can be used in combination to perform the AND，OR and inverter operation.
28
3-5 The NOR Gate （或非门）
The term NOR is a contraction of NOT-OR and implies an OR function with a complemented (inverted) output.
XA
The complement variable can be read as “A bar” or “not A” （A反）. If A=0, A=1
14
高级数理逻辑第5讲
4.5 一阶谓词语义系统 4.5.1 什么是形式系统语义抽象公理系统或者形式系统，具有较高的抽象性。
因此，已经脱离了任何一个具体的系统，但是我们可以对形式系统作出各种解释。
通过这种解释将形式系统对应到各种具体的系统中取。
例如可以将一阶谓词逻辑系统，解释到平面几何系统中。
怎样将形式系统解释成具体系统呢？我们先看下面的例子：如果我们要知道)),1((x f xP ∀的具体的真值=1，我们至少要知道以下事情： 1、 x 在什么范围之内，x 范围是实数。
2、 f 是什么？ (X+1)3、 P 是什么？P 代表的是大于＝04、 a=？a=15、 x=?，x ＝5，－4例如，我们可以作出以下解释： 1、解释1：● x 在实数中取值 ● P 表示等于0●),(a x f 表示x-a● a=5因此，公式解释为05==-x 。
令x=5, 则1))),(((=x a f P v s(x) ->5s(f(a,x) ->I(f)(I(a),s(x)) 令x=6，则0))),(((=x a f P u 2、解释2：● x 在实数中取值 ● P 表示大于等于0●),(a x f 表示2)(a x -因此，公式解释为0)(2>=-a x 。
这个公式不必对a 和x 作出具体解释，就可以确定公式的真值。
即对于任何实数x ，和赋值映射v ，1)))(((2=-a x P v 。
由上面的例子可以看出，要对形式系统作出解释，我们要了解以下问题：✓x取值于哪里？即规定讨论问题的领域。
✓给出谓词的含义和谓词的真值✓给出函数的解释✓给出变量和常量的值✓根据连接词的赋值规则，赋值这就是我们要研究的语义系统－指称语义的主要内容。
现代逻辑语义学理论的创始人是美籍波兰逻辑学家、哲学家A. Tarski，其奠基性文章是他在1933年发表的《形式语言中的真实概念》。
后来被称为模型论—标准语义学理论。
进一步的发展由维特根斯坦最早提出设想，卡尔纳普最早把它展开为系统。
Logical Operator
Logical Operator/p/1938383351执行指定的逻辑运算输入逻辑和位操作Logical Operator块执行在其输入端的指定的逻辑运算。
如果是非零输入值是TRUE（1），如果它是零，FALSE（0）。
您选择的布尔操作与连接输入Operator参数列表.如果您选择rectangular为Icon shape 属性，块更新，以显示所选择的操作符的名称。
支持的操作在下面给出。
如果您选择distinctive为Icon shape，块的外观表明其功能。
Simulink软件会显示一个独特的形状为选定的操作符，符合IEEE标准图形符号逻辑函数：指定的数量的输入端口为Number of input ports参数. 指定的输出类型Output data type参数.如果TRU E输出值是1，如果FALSE为0。
输出的大小取决于输入矢量的大小和选定的运算符：•如果块有一个以上的输入，任何非标量输入必须具有相同的维。
例如，如果任何的输入是一个2×2阵列的，所有其他的非标量输入也必须是2×2阵列。
标量输入被扩展为具有相同的尺寸的非标量输入。
如果块有一个以上的输入，输出作为输入（标量扩展后）具有相同的尺寸和每个输出元件是施加到相应的输入元件的指定的逻辑运算的结果。
例如，如果指定的操作是与AND输入的是2×2阵列，输出的是一个2×2阵列的其顶部左元素是应用的结果，与输入到左上角元素等。
•对于一个单一的矢量输入端，块适用于操作（除NOT运算符）的向量的所有元素。
输出始终是一个标量。
•NOT运算符只接受一个输入端，它可以是一个标量或向量。
如果输入是一个矢量，输出是相同的大小，元素的向量包含的逻辑互补的输入向量当配置为一个多输入的XOR门，此块执行加法模2的操作，由IEEE标准逻辑单元的授权。
Logical Operator块接受任何Simulink数字数据类型的支持，包括定点数据类型的实际信号。
离散数学中英文名词对照表
离散数学中英文名词对照表离散数学中英文名词对照表外文中文AAbel category Abel 范畴Abel group (commutative group) Abel 群(交换群)Abel semigroup Abel 半群accessibility relation 可达关系action 作用addition principle 加法原理adequate set of connectives 联结词的功能完备（全）集adjacent 相邻（邻接）adjacent matrix 邻接矩阵adjugate 伴随adjunction 接合affine plane 仿射平面algebraic closed field 代数闭域algebraic element 代数元素algebraic extension 代数扩域（代数扩张）almost equivalent 几乎相等的alternating group 三次交代群annihilator 零化子antecedent 前件anti symmetry 反对称性anti-isomorphism 反同构arboricity 荫度arc set 弧集arity 元数arrangement problem 布置问题associate 相伴元associative algebra 结合代数associator 结合子asymmetric 不对称的(非对称的)atom 原子atomic formula 原子公式augmenting digeon hole principle 加强的鸽子笼原理augmenting path 可增路automorphism 自同构automorphism group of graph 图的自同构群auxiliary symbol 辅助符号axiom of choice 选择公理axiom of equality 相等公理axiom of extensionality 外延公式axiom of infinity 无穷公理axiom of pairs 配对公理axiom of regularity 正则公理axiom of replacement for the formula Ф关于公式Ф的替换公式axiom of the empty set 空集存在公理axiom of union 并集公理Bbalanced imcomplete block design 平衡不完全区组设计barber paradox 理发师悖论base 基Bell number Bell 数Bernoulli number Bernoulli 数Berry paradox Berry 悖论bijective 双射bi-mdule 双模binary relation 二元关系binary symmetric channel 二进制对称信道binomial coefficient 二项式系数binomial theorem 二项式定理binomial transform 二项式变换bipartite graph 二分图block 块block 块图（区组）block code 分组码block design 区组设计Bondy theorem Bondy 定理Boole algebra Boole 代数Boole function Boole 函数Boole homomorophism Boole 同态Boole lattice Boole 格bound occurrence 约束出现bound variable 约束变量bounded lattice 有界格bridge 桥Bruijn theorem Bruijn 定理Burali-Forti paradox Burali-Forti 悖论Burnside lemma Burnside 引理Ccage 笼canonical epimorphism 标准满态射Cantor conjecture Cantor 猜想Cantor diagonal method Cantor 对角线法Cantor paradox Cantor 悖论cardinal number 基数Cartesion product of graph 图的笛卡儿积Catalan number Catalan 数category 范畴Cayley graph Cayley 图Cayley theorem Cayley 定理center 中心characteristic function 特征函数characteristic of ring 环的特征characteristic polynomial 特征多项式check digits 校验位Chinese postman problem 中国邮递员问题chromatic number 色数chromatic polynomial 色多项式circuit 回路circulant graph 循环图circumference 周长class 类classical completeness 古典完全的classical consistent 古典相容的clique 团clique number 团数closed term 闭项closure 闭包closure of graph 图的闭包code 码code element 码元code length 码长code rate 码率code word 码字coefficient 系数coimage 上象co-kernal 上核coloring 着色coloring problem 着色问题combination number 组合数combination with repetation 可重组合common factor 公因子commutative diagram 交换图commutative ring 交换环commutative seimgroup 交换半群complement 补图(子图的余) complement element 补元complemented lattice 有补格complete bipartite graph 完全二分图complete graph 完全图complete k-partite graph 完全k-分图complete lattice 完全格composite 复合composite operation 复合运算composition (molecular proposition) 复合（分子）命题composition of graph (lexicographic product)图的合成（字典积）concatenation (juxtaposition) 邻接运算concatenation graph 连通图congruence relation 同余关系conjunctive normal form 正则合取范式connected component 连通分支connective 连接的connectivity 连通度consequence 推论（后承）consistent (non-contradiction) 相容性（无矛盾性）continuum 连续统contraction of graph 图的收缩contradiction 矛盾式（永假式）contravariant functor 反变函子coproduct 上积corank 余秩correct error 纠正错误corresponding universal map 对应的通用映射countably infinite set 可列无限集（可列集）covariant functor (共变)函子covering 覆盖covering number 覆盖数Coxeter graph Coxeter 图crossing number of graph 图的叉数cuset 陪集cotree 余树cut edge 割边cut vertex 割点cycle 圈cycle basis 圈基cycle matrix 圈矩阵cycle rank 圈秩cycle space 圈空间cycle vector 圈向量cyclic group 循环群cyclic index 循环（轮转）指标cyclic monoid 循环单元半群cyclic permutation 圆圈排列cyclic semigroup 循环半群DDe Morgan law De Morgan 律decision procedure 判决过程decoding table 译码表deduction theorem 演绎定理degree 次数,次(度)degree sequence 次(度)序列derivation algebra 微分代数Descartes product Descartes 积designated truth value 特指真值detect errer 检验错误deterministic 确定的diagonal functor 对角线函子diameter 直径digraph 有向图dilemma 二难推理direct consequence 直接推论（直接后承）direct limit 正向极限direct sum 直和directed by inclution 被包含关系定向discrete Fourier transform 离散 Fourier 变换disjunctive normal form 正则析取范式disjunctive syllogism 选言三段论distance 距离distance transitive graph 距离传递图distinguished element 特异元distributive lattice 分配格divisibility 整除division subring 子除环divison ring 除环divisor (factor) 因子domain 定义域Driac condition Dirac 条件dual category 对偶范畴dual form 对偶式dual graph 对偶图dual principle 对偶原则(对偶原理) dual statement 对偶命题dummy variable 哑变量（哑变元）Eeccentricity 离心率edge chromatic number 边色数edge coloring 边着色edge connectivity 边连通度edge covering 边覆盖edge covering number 边覆盖数edge cut 边割集edge set 边集edge-independence number 边独立数eigenvalue of graph 图的特征值elementary divisor ideal 初等因子理想elementary product 初等积elementary sum 初等和empty graph 空图empty relation 空关系empty set 空集endomorphism 自同态endpoint 端点enumeration function 计数函数epimorphism 满态射equipotent 等势equivalent category 等价范畴equivalent class 等价类equivalent matrix 等价矩阵equivalent object 等价对象equivalent relation 等价关系error function 错误函数error pattern 错误模式Euclid algorithm 欧几里德算法Euclid domain 欧氏整环Euler characteristic Euler 特征Euler function Euler 函数Euler graph Euler 图Euler number Euler 数Euler polyhedron formula Euler 多面体公式Euler tour Euler 闭迹Euler trail Euler 迹existential generalization 存在推广规则existential quantifier 存在量词existential specification 存在特指规则extended Fibonacci number 广义 Fibonacci 数extended Lucas number 广义Lucas 数extension 扩充（扩张）extension field 扩域extension graph 扩图exterior algebra 外代数Fface 面factor 因子factorable 可因子化的factorization 因子分解faithful (full) functor 忠实（完满）函子Ferrers graph Ferrers 图Fibonacci number Fibonacci 数field 域filter 滤子finite extension 有限扩域finite field (Galois field ) 有限域（Galois 域）finite dimensional associative division algebra有限维结合可除代数finite set 有限（穷）集finitely generated module 有限生成模first order theory with equality 带符号的一阶系统five-color theorem 五色定理five-time-repetition 五倍重复码fixed point 不动点forest 森林forgetful functor 忘却函子four-color theorem(conjecture) 四色定理（猜想）F-reduced product F-归纳积free element 自由元free monoid 自由单元半群free occurrence 自由出现free R-module 自由R-模free variable 自由变元free-?-algebra 自由?代数function scheme 映射格式GGalileo paradox Galileo 悖论Gauss coefficient Gauss 系数GBN (G?del-Bernays-von Neumann system)GBN系统generalized petersen graph 广义petersen 图generating function 生成函数generating procedure 生成过程generator 生成子（生成元）generator matrix 生成矩阵genus 亏格girth （腰）围长G?del completeness theorem G?del 完全性定理golden section number 黄金分割数（黄金分割率）graceful graph 优美图graceful tree conjecture 优美树猜想graph 图graph of first class for edge coloring 第一类边色图graph of second class for edge coloring 第二类边色图graph rank 图秩graph sequence 图序列greatest common factor 最大公因子greatest element 最大元（素）Grelling paradox Grelling 悖论Gr?tzsch graph Gr?tzsch 图group 群group code 群码group of graph 图的群HHajós conjecture Hajós 猜想Hamilton cycle Hamilton 圈Hamilton graph Hamilton 图Hamilton path Hamilton 路Harary graph Harary 图Hasse graph Hasse 图Heawood graph Heawood 图Herschel graph Herschel 图hom functor hom 函子homemorphism 图的同胚homomorphism 同态（同态映射）homomorphism of graph 图的同态hyperoctahedron 超八面体图hypothelical syllogism 假言三段论hypothese (premise) 假设(前提)Iideal 理想identity 单位元identity natural transformation 恒等自然变换imbedding 嵌入immediate predcessor 直接先行immediate successor 直接后继incident 关联incident axiom 关联公理incident matrix 关联矩阵inclusion and exclusion principle 包含与排斥原理inclusion relation 包含关系indegree 入次（入度）independent 独立的independent number 独立数independent set 独立集independent transcendental element 独立超越元素index 指数individual variable 个体变元induced subgraph 导出子图infinite extension 无限扩域infinite group 无限群infinite set 无限（穷）集initial endpoint 始端initial object 初始对象injection 单射injection functor 单射函子injective (one to one mapping) 单射（内射）inner face 内面inner neighbour set 内（入）邻集integral domain 整环integral subdomain 子整环internal direct sum 内直和intersection 交集intersection of graph 图的交intersection operation 交运算interval 区间invariant factor 不变因子invariant factor ideal 不变因子理想inverse limit 逆向极限inverse morphism 逆态射inverse natural transformation 逆自然变换inverse operation 逆运算inverse relation 逆关系inversion 反演isomorphic category 同构范畴isomorphism 同构态射isomorphism of graph 图的同构join of graph 图的联JJordan algebra Jordan 代数Jordan product (anti-commutator) Jordan乘积（反交换子）Jordan sieve formula Jordan 筛法公式j-skew j-斜元juxtaposition 邻接乘法Kk-chromatic graph k-色图k-connected graph k-连通图k-critical graph k-色临界图k-edge chromatic graph k-边色图k-edge-connected graph k-边连通图k-edge-critical graph k-边临界图kernel 核Kirkman schoolgirl problem Kirkman 女生问题Kuratowski theorem Kuratowski 定理Llabeled graph 有标号图Lah number Lah 数Latin rectangle Latin 矩形Latin square Latin 方lattice 格lattice homomorphism 格同态law 规律leader cuset 陪集头least element 最小元least upper bound 上确界（最小上界）left (right) identity 左（右）单位元left (right) invertible element 左（右）可逆元left (right) module 左（右）模left (right) zero 左（右）零元left (right) zero divisor 左（右）零因子left adjoint functor 左伴随函子left cancellable 左可消的left coset 左陪集length 长度Lie algebra Lie 代数line- group 图的线群logically equivanlent 逻辑等价logically implies 逻辑蕴涵logically valid 逻辑有效的（普效的）loop 环Lucas number Lucas 数Mmagic 幻方many valued proposition logic 多值命题逻辑matching 匹配mathematical structure 数学结构matrix representation 矩阵表示maximal element 极大元maximal ideal 极大理想maximal outerplanar graph 极大外平面图maximal planar graph 极大平面图maximum matching 最大匹配maxterm 极大项（基本析取式）maxterm normal form(conjunctive normal form) 极大项范式（合取范式）McGee graph McGee 图meet 交Menger theorem Menger 定理Meredith graph Meredith 图message word 信息字mini term 极小项minimal κ-connected graph 极小κ-连通图minimal polynomial 极小多项式Minimanoff paradox Minimanoff 悖论minimum distance 最小距离Minkowski sum Minkowski 和minterm (fundamental conjunctive form) 极小项（基本合取式）minterm normal form(disjunctive normal form)极小项范式（析取范式）M?bius function M?bius 函数M?bius ladder M?bius 梯M?bius transform (inversion) M?bius 变换（反演）modal logic 模态逻辑model 模型module homomorphism 模同态(R-同态)modus ponens 分离规则modus tollens 否定后件式module isomorphism 模同构monic morphism 单同态monoid 单元半群monomorphism 单态射morphism (arrow) 态射（箭）M?bius function M?bius 函数M?bius ladder M?bius 梯M?bius transform (inversion) M?bius 变换（反演）multigraph 多重图multinomial coefficient 多项式系数multinomial expansion theorem 多项式展开定理multiple-error-correcting code 纠多错码multiplication principle 乘法原理mutually orthogonal Latin square 相互正交拉丁方Nn-ary operation n-元运算n-ary product n-元积natural deduction system 自然推理系统natural isomorphism 自然同构natural transformation 自然变换neighbour set 邻集next state 下一个状态next state transition function 状态转移函数non-associative algebra 非结合代数non-standard logic 非标准逻辑Norlund formula Norlund 公式normal form 正规形normal model 标准模型normal subgroup (invariant subgroup) 正规子群（不变子群）n-relation n-元关系null object 零对象nullary operation 零元运算Oobject 对象orbit 轨道order 阶order ideal 阶理想Ore condition Ore 条件orientation 定向orthogonal Latin square 正交拉丁方orthogonal layout 正交表outarc 出弧outdegree 出次(出度)outer face 外面outer neighbour 外（出）邻集outerneighbour set 出(外)邻集outerplanar graph 外平面图Ppancycle graph 泛圈图parallelism 平行parallelism class 平行类parity-check code 奇偶校验码parity-check equation 奇偶校验方程parity-check machine 奇偶校验器parity-check matrix 奇偶校验矩阵partial function 偏函数partial ordering (partial relation) 偏序关系partial order relation 偏序关系partial order set (poset) 偏序集partition 划分,分划,分拆partition number of integer 整数的分拆数partition number of set 集合的划分数Pascal formula Pascal 公式path 路perfect code 完全码perfect t-error-correcting code 完全纠-错码perfect graph 完美图permutation 排列（置换）permutation group 置换群permutation with repetation 可重排列Petersen graph Petersen 图p-graph p-图Pierce arrow Pierce 箭pigeonhole principle 鸽子笼原理planar graph （可）平面图plane graph 平面图Pólya theorem Pólya 定理polynomail 多项式polynomial code 多项式码polynomial representation 多项式表示法polynomial ring 多项式环possible world 可能世界power functor 幂函子power of graph 图的幂power set 幂集predicate 谓词prenex normal form 前束范式pre-ordered set 拟序集primary cycle module 准素循环模prime field 素域prime to each other 互素primitive connective 初始联结词primitive element 本原元primitive polynomial 本原多项式principal ideal 主理想principal ideal domain 主理想整环principal of duality 对偶原理principal of redundancy 冗余性原则product 积product category 积范畴product-sum form 积和式proof (deduction) 证明（演绎）proper coloring 正常着色proper factor 真正因子proper filter 真滤子proper subgroup 真子群properly inclusive relation 真包含关系proposition 命题propositional constant 命题常量propositional formula(well-formed formula,wff)命题形式（合式公式）propositional function 命题函数propositional variable 命题变量pullback 拉回(回拖) pushout 推出Qquantification theory 量词理论quantifier 量词quasi order relation 拟序关系quaternion 四元数quotient (difference) algebra 商（差）代数quotient algebra 商代数quotient field (field of fraction) 商域（分式域）quotient group 商群quotient module 商模quotient ring (difference ring , residue ring) 商环（差环，同余类环）quotient set 商集RRamsey graph Ramsey 图Ramsey number Ramsey 数Ramsey theorem Ramsey 定理range 值域rank 秩reconstruction conjecture 重构猜想redundant digits 冗余位reflexive 自反的regular graph 正则图regular representation 正则表示relation matrix 关系矩阵replacement theorem 替换定理representation 表示representation functor 可表示函子restricted proposition form 受限命题形式restriction 限制retraction 收缩Richard paradox Richard 悖论right adjoint functor 右伴随函子right cancellable 右可消的right factor 右因子right zero divison 右零因子ring 环ring of endomorphism 自同态环ring with unity element 有单元的环R-linear independence R-线性无关root field 根域rule of inference 推理规则Russell paradox Russell 悖论Ssatisfiable 可满足的saturated 饱和的scope 辖域section 截口self-complement graph 自补图semantical completeness 语义完全的（弱完全的）semantical consistent 语义相容semigroup 半群separable element 可分元separable extension 可分扩域sequent 矢列式sequential 序列的Sheffer stroke Sheffer 竖（谢弗竖）simple algebraic extension 单代数扩域simple extension 单扩域simple graph 简单图simple proposition (atomic proposition) 简单（原子）命题simple transcental extension 单超越扩域simplication 简化规则slope 斜率small category 小范畴smallest element 最小元（素）Socrates argument Socrates 论断（苏格拉底论断）soundness (validity) theorem 可靠性（有效性）定理spanning subgraph 生成子图spanning tree 生成树spectra of graph 图的谱spetral radius 谱半径splitting field 分裂域standard model 标准模型standard monomil 标准单项式Steiner triple Steiner 三元系大集Stirling number Stirling 数Stirling transform Stirling 变换subalgebra 子代数subcategory 子范畴subdirect product 子直积subdivison of graph 图的细分subfield 子域subformula 子公式subdivision of graph 图的细分subgraph 子图subgroup 子群sub-module 子模subrelation 子关系subring 子环sub-semigroup 子半群subset 子集substitution theorem 代入定理substraction 差集substraction operation 差运算succedent 后件surjection (surjective) 满射switching-network 开关网络Sylvester formula Sylvester公式symmetric 对称的symmetric difference 对称差symmetric graph 对称图symmetric group 对称群syndrome 校验子syntactical completeness 语法完全的（强完全的）Syntactical consistent 语法相容system ?3 , ?n , ??0 , ??系统?3 , ?n , ??0 , ??system L 公理系统Lsystem ?公理系统?system L1 公理系统 L1system L2 公理系统 L2system L3 公理系统 L3system L4 公理系统 L4system L5 公理系统 L5system L6 公理系统 L6system ?n 公理系统?nsystem of modal prepositional logic 模态命题逻辑系统system Pm 系统 Pmsystem S1 公理系统 S1system T (system M) 公理系统 T(系统M)Ttautology 重言式（永真公式）technique of truth table 真值表技术term 项terminal endpoint 终端terminal object 终结对象t-error-correcing BCH code 纠 t -错BCH码theorem (provable formal) 定理（可证公式）thickess 厚度timed sequence 时间序列torsion 扭元torsion module 扭模total chromatic number 全色数total chromatic number conjecture 全色数猜想total coloring 全着色total graph 全图total matrix ring 全方阵环total order set 全序集total permutation 全排列total relation 全关系tournament 竞赛图trace (trail) 迹tranformation group 变换群transcendental element 超越元素transitive 传递的tranverse design 横截设计traveling saleman problem 旅行商问题tree 树triple system 三元系triple-repetition code 三倍重复码trivial graph 平凡图trivial subgroup 平凡子群true in an interpretation 解释真truth table 真值表truth value function 真值函数Turán graph Turán 图Turán theorem Turán 定理Tutte graph Tutte 图Tutte theorem Tutte 定理Tutte-coxeter graph Tutte-coxeter 图UUlam conjecture Ulam 猜想ultrafilter 超滤子ultrapower 超幂ultraproduct 超积unary operation 一元运算unary relation 一元关系underlying graph 基础图undesignated truth value 非特指值undirected graph无向图union 并(并集)union of graph 图的并union operation 并运算unique factorization 唯一分解unique factorization domain (Gauss domain) 唯一分解整域unique k-colorable graph 唯一k着色unit ideal 单位理想unity element 单元universal 全集universal algebra 泛代数（Ω代数）universal closure 全称闭包universal construction 通用结构universal enveloping algebra 通用包络代数universal generalization 全称推广规则universal quantifier 全称量词universal specification 全称特指规则universal upper bound 泛上界unlabeled graph 无标号图untorsion 无扭模upper (lower) bound 上（下）界useful equivalent 常用等值式useless code 废码字Vvalence 价valuation 赋值Vandermonde formula Vandermonde 公式variery 簇Venn graph Venn 图vertex cover 点覆盖vertex set 点割集vertex transitive graph 点传递图Vizing theorem Vizing 定理Wwalk 通道weakly antisymmetric 弱反对称的weight 重（权）weighted form for Burnside lemma 带权形式的Burnside引理well-formed formula (wff) 合式公式(wff)word 字Zzero divison 零因子zero element (universal lower bound) 零元（泛下界）ZFC (Zermelo-Fraenkel-Cohen) system ZFC系统form)normal(Skolemformnormalprenex-存在正则前束范式(Skolem 正则范式)3-value proposition logic 三值命题逻辑。
逻辑导论课件全.ppt.Convertor
《逻辑导论》第一章绪论一、"逻辑"的含义1、"逻辑"的由来"逻辑"一词由英文"Logic"音译而来。
"Logic"源于古希腊语"λγο"（逻各斯），原意是言辞、思想、理性、秩序、尺度、规律等。
直到近代，才用来表示逻辑学这门学科。
2、"逻辑"的含义（1）客观事物的规律（2）某种理论、观点或主张（3）思维的形式规律或规则（4）逻辑学二、逻辑学的起源--源于思想的自我反思逻辑学是一门历史悠久的学科，至今已有两千多年的历史。
这门学科是由古希腊思想家亚里士多德（Aristotle 公元前384--322年）创立的。
公元前6世纪-4世纪是思想争论的时代。
逻辑学是为了满足这个时期的政治与法庭的辩论、数学的证明、伦理学和形而上学的论证等思想争论的需要发展起来的，是在这个时代的思想的自我反思基础之上发展而来的。
三、逻辑学的对象、问题、任务逻辑学以推论作为研究对象。
逻辑学的研究问题：推论的必然性即有效性问题。
逻辑学的研究任务：解决判定问题：研究有效推论的判定程序与方法。
解决推导问题：发现有效推论的规则与方法。
四、逻辑学的性质和作用在联合国教科文组织颁布的学科分类中，逻辑学被列为七大基础学科之一。
逻辑学是一门基础性学科。
逻辑学是一门工具性学科。
逻辑学是一门应用广泛性的学科。
逻辑学具有重要的理论意义和实践意义逻辑学是思想分析与批判的工具。
逻辑学是思想推论与论证的工具。
逻辑学在理论研究和社会实践中具有广泛的应用。
第二章命题逻辑关于命题联结词的规律一、命题命题：有所陈述的语句。
命题结构：命题可以分解为联结词和被联结的部分。
联结词称为逻辑常项，它起联结作用并反映被联结的部分之间的逻辑关系。
命题形式：命题结构或命题内在逻辑关系的表达式。
由变项和逻辑常项组成。
命题的真假：二、命题联结词与复合命题命题联结词：联结命题的联结词。
算术逻辑单元英文-PPT
Carry propagate function Pi and carry generate function Gi: Gi=Xi·Yi carry generate function Pi=Xi+Yi carry propagate function
算术逻辑单元英文
5、1 The von Neumann puter model
The vast majority of puter systems used today are constructed on the van Neumann puter model、
A puter is viewed as a stored program puter、
Gi:when Xi and Yi are“1”,no matter whether there is loworder carry bit,the current carry bit is generated、
Pi:when either of Xi and Yi is 1,if there exist low-order carry bit, then Ci-1 is propagated to high-order carry bit
An arithmetic unit (ALU) is the heart of the CPU
The ALU usually has a binary adder
The performance of the ALU is mainly determined by its adder、
We need to design a fast adder to get rid of the excessive carry-propagation time of the ripplecarry adder、
几个逻辑相关的英语单词
几个逻辑相关的英语单词逻辑 logic数理逻辑 mathematical logic模型论 model theory集合论 set theory递归论 recursion theory证明论 proof theory非标准分析 nonstandard analysis反推数学 reverse mathematics元数学 metamathematics二阶算术的子系统 subsystems of the second-order arithmetic 直觉主义 intuitionism构造性数学 constructive mathematics语言 language元语言 metalanguage元定理 metatheorem公理 axiom定理 theorem命题 proposition命题演算 propositional calculus 谓词演算 predicate calculus合取 conjunction析取 disjunction非,否定 negation量词 quantifier全称量词 universal quantifier 存在量词 existential quantifier 关系 relation函数 function常量 constant变元,变量 variable项 term公式 formula原子公式 atomic formula句子,命题 sentence永真命题 tautology前束标准型 prenex normal form 理论 theory可满足的 satisfiable和谐性,相容性 consistency句法 syntax语义 semantics可靠性定理 soundness theorem完备性定理 completeness theorem 紧致性定理 compactness theorem可公理化 axiomatizable有限可公理化 finitely axiomatizable 同构 isomorphism同态 homomorphism初等等价 elementary equivalent 初等嵌入 elementary embedding 初等子模型 elementary submodel 初等扩张 elementary extension图象 diagram正图象 positive diagram初等图象 elementary diagram模型 model可数模型 countable model不可数模型 uncountable model原子模型 atomic model素模型 prime model齐性模型 homogeneous model万有模型 universal model饱和模型 saturated model特殊模型 special model递归饱和模型 recursively saturated model 布尔值模型 boolean-valued model格值模型 lattice-valued model超滤 ultrafilter超积 ultraproduct超幂 ultrapower模型完备 model complete子模型完备 submodel complete量词消去 quantifier elimination稳定性理论 stable theory集,集合 set子集 subset幂集 power set空集 empty set有限集 finite set无限集 infinite set可数集 countable set不可数集 uncountable set 有限集 finite set无限集 infinite set序数 ordinal极限序数 limit ordinal后继序数 successor ordinal 基数 cardinal大基数 large cardinal可测基数 measurable cardinal正则基数 regular cardinal奇异基数 singular cardinal不可达基数 inaccessible力迫法 forcing连续统假设 Continuum Hypothesis 选择公理 Axiom of Choice决定性公理 Axiom of Determinacy 归纳法 induction超限归纳法 transfinite induction 超限递归 transfinite recursion递归 recursion原始递归 primitive recursive递归函数 recursive function递归可枚举 recursively enumerable递归可判定 recursively decidable 递归不可分 recursively inseparable 递归集 recursive set算术集 arithmetical set解析集 analytic set单纯集 simple set创造集 creative set多一归约 many-one reducible一一归约 one-one reducible图灵归约 Turing reducible不可解度 degree of unsolvability 图灵度 Turing degree一阶逻辑 first-order logic二阶逻辑 second-order logic高阶逻辑 higher-order logic非古典逻辑 non-classical logic 无穷逻辑 infinitary logic古典逻辑 classical logic直觉主义逻辑 intuitionistic logic 模态逻辑 modal logic多值逻辑 many-valued logic。
数学专业英语2.11数理逻辑入门分析
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We can denote the statement “x is greater than 3” by P(x), where P denotes the predicate “is greater than 3” and x is the variable.
把语句“x大于3”记为P(x), 其中P表示谓词“大于3”， 而x是变量。
2.11 数理逻辑入门 Elementary Mathematical Logic
New Words & Expressions:
assert 断言，主张
predicate 谓词
conjunction 合取
quantifier 量词
connective 连词
quantification 量词化
disjunction 析取
许多数学语句认为，性质对论域这个特殊领域内的变 量的所有值都成立。
Such a statement is expressed using a universal quantification. 这样的语句可用全称量词化表示。
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The universal quantification of a propositional function is the proposition that assert that P(x) is true for all values of x in the universe of discourse. The universe of discourse specifies the possible values of the variable x.
命题函数的全称量词化是一个命题，认为P(x)对论域 中x的所有值P(x)都是真的。论域指定变量x的可能取 值.
数字电子技术加英文注释
2.1 概述
二、逻辑电平 Vcc
Logic Levels
The voltages used to represent a 1 and a 0 are called logic levels.
Vo
实际开关为晶体二极管、三极
VI
管以及场效应管等电子器件
S
VI控制开关S的断、通情况。 S断开，VO为高电平；S接通，VO为低电平。
④74LS：低功耗肖特基系列；74LS系列成为功耗延迟积较小的系列。74LS系列 产品具有最佳的综合性能，是TTL集成电路的主流，是应用最广的系列。
⑤74AS：先进肖特基系列；
④74ALS：先进低功耗肖特基系列。
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2.4 TTL集成门电路
74LS系列常用芯片
VCC 3A 3B 3Y 4A 4B 4Y
Implement inverter with BJT
+VCC Rc iC
Rb b
c uo
ui
iB
利用二极管的压降为0.7Ve，保证
输入电压在1V以下时，开关电路
可三靠极地管截开止关。 电路
输入为低，输出为高； A
输入为高，输出为低。
0
1
A(V) Y(V) <0.8 5 >2 0.2
YA
Y
A
1
Y
1
0
②外加反向电压，二极管截止。
门坎电压Uth 硅 PN 结伏安特性
iD(mA)
0.7V
uD(V)
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2.2 半导体二极管和三极管的开关特性
二、二极管开关特性
D uI
Vcc R
uo
利用二极管的单向导电性，相当 于一个受外加电压极性控制的开关。
马斯科莱尔高微课件第1章
第1章 偏好和选择
• 问题： 能否对“弱公理”进行强化，得到“行为 能被偏好理性化”的必要及充分条件？
第1章 偏好和选择
（注：代表一个偏好关系的效用函数如果存在 ，就不是唯一的)
第1章 偏好和选择
• 理性假设与效用函数ห้องสมุดไป่ตู้
命题1.B.2：只有当偏好关系 是理性的时，它才可以用一个效用函数来代表。
第1章 偏好和选择
2、选择规则
（1）选择结构: A. 预算集“集类”；B.选择规则
（2）显示偏好弱公理 （3）显示偏好关系
第1章 偏好和选择
• “理性假设”存在的问题：
恰可感觉到的差异； 构架问题; 集体选择的非传递性（孔多塞悖论）； 爱好变化。
第1章 偏好和选择
• “严格偏好关系”与“无差异关系”的性质:
－命题1.B.1 （第8页）
（back）
第1章 偏好和选择
• 效用函数的定义
定义1.B.2：若对于所有x, y X ， x y u x u y 则函数u : X R为“代表偏好关系 ”的一个效用函数。
*
第1章 偏好和选择
• 满足“弱公理”的选择行为并不一定可以被视 作是由理性偏好关系导出。
定义1.D.1：给定一个选择结构 B，C ，如果对于所有的B B, 有 C B C * B, , 也就是说，如果 导出选择结构 B，C ,我们称理性偏好关系 “理性化”与B相关的C 。
定义1.C.1：若下述性质成立，则称选择结构 B，C 满足“显示偏好弱公理”： 若对于某一B B, 且x, y B, 我们有x C B , 则对于任意B B, 且x, y B, y C B , 我们必有x C B 。