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1、Requirements: 掌握逻辑代数的基本概念，学会用逻辑函数描述逻 辑问题 掌握逻辑代数的公理、基本定理和重要规则 学会使用卡诺图进行化简P由于充满理性的人脑用逻辑去思考，那么，如果能 用数学来表征逻辑，我们也就可以用数学来描述大 脑是如何工作的。Chapter 2 Combinational Logic CircuitDate1religionphilosophyScienceUnknown worldChapter 2 Com. Logic CircuitlLevels of cognitionDate2讨论 2: 学习金字塔学习金字塔是美国缅因州国家训练实验室的研究成果，它用数字形
2、式形象显示了：采用不同的学习方式，学习者在两周以后还能记住内容（平均学习保持率）的多少。它是一种现代学习方式的理论。最早它是由美国学者、著名的学习专家爱德加戴尔1946年首先发现并提出的。 Chapter 2 Com. Logic Circuit (cont.)Date3Chapter 2 Com. Logic Circuit (cont.)你是如何学习的？大学里的学习方式与中学有何不同？你认真考虑过如何有效地进行学习吗？举一个你“学有所悟”或“学有所成”的例子与大家分享。Date4Assignments:2-3、2-4、2-5、2-6、2-26、2-27、2-28、2-29、2-30、2-3
3、4.Chapter 2 Com. Logic Circuit (cont.)Date5l Part 1 Gate Circuits and Boolean Equations l Binary Logic and Gates l Boolean Algebra l Standard Formsl Part 2 Circuit Optimization l Two-Level Optimization l Map Manipulation l Practical Optimization (Espresso) l Multi-Level Circuit Optimizationl Part 3
4、Additional Gates and Circuits l Other Gate Types l Exclusive-OR Operator and Gates l High-Impedance OutputsOverviewDate6lBinary variables denoted by A, B, y, z, or X1, RESET, START_IT, or ADD1, others take on one of two values.Binary Logic and GateslLogical operators operate on binary values and bin
5、ary variables.lBasic logical operators are the logic functions AND, OR and NOT.lLogic gates implement logic functions.lBoolean Algebra: a useful mathematical system for specifying and transforming logic functions. We study Boolean algebra as a foundation for designing and analyzing digital systems!D
6、ate7l Logic gates have special symbols:Logic Gate Symbols and Behaviorl Waveform behavior in timeDate8l In actual physical gates, if one or more input changes causes the output to change, the output change does not occur instantaneously.Gate DelaytGtGInputOutputTime (ns)001100.511.5tG = 0.3 nsl The
7、delay between an input change(s) and the resulting output change is the gate delay denoted by tG:Date9l Truth tables are unique; expressions and logic diagrams are not. But they describe the same thing! This gives flexibility in implementing functions.Logic Expressions and DiagramsXYFZLogic DiagramT
8、ruth Table11 1 111 1 011 0 111 0 000 1 100 1 010 0 100 0 0X Y ZZ YX F+=EquationF=X+Y ZDate10Logic Function Implementationl Using Switches l For inputs: l logic 1 is switch closed l logic 0 is switch openl For outputs: l logic 1 is light on l logic 0 is light off.l NOT uses a switch such that: l logi
9、c 1 is switch open l logic 0 is switch closedSwitches in series = ANDSwitches in parallel = ORNormally-closed switch = NOTDate11l (A1) if X1, then X0,;(A1) if X0, then X1; l (A2) if X0, then X1,(A2) if X1, then X0; l (A3) 000,(A3) 111; l (A4) 111,(A4) 000; l (A5) 01100,(A5) 10011.Boolean Algebral An
10、 algebraic structure, L=K, +, , -, 0, 1 defined on a set of logical variables, constant 0 and 1, and three binary operators (denoted +, and ) that satisfies the following axioms:布尔 1854, Date12Some Theoremsl Single-Variable Theoremsl All of the theorems can be proved using perfect induction.Date13So
11、me Theorems (cont.)l Two- and Three-Variable Theoremsl The order of evaluation, Parentheses, NOT, AND, OR l T8 is not true in the ordinary algebra l T9 and T10 are used extensively in the minimization of logic functions.Date14Some Theorems (cont.)l n-Variable Theoremsl All of the theorems can be pro
12、ved using finite induction.l In all of the theorems, it is possible to replace each variable with an arbitrary logic expression.Date15Principle of Dualityl Any theorem or identity in switching algebra remains true if 0 and 1 are swapped and and + are swapped throughout .l The dual of a logic express
13、ion formally defined:If F(X1,X2,Xn, +, , ) is a fully parenthesized logic expression involving the variables X1,X2,Xn and the operators +, , and , then the dual of F, written FD, is the same expression with + and swapped:l Duality is important because it doubles the usefulness of everything that we
14、learn about Boolean algebra and manipulation of Boolean functions.l Complementing functionsDate16Expression Simplificationl Simplify an expression to contain the smallest number of literals (complemented and uncomplemented variables)l Problem: How do you know whether you have obtained the simplest e
15、xpression?Date17Canonical Formsl It is useful to specify Boolean functions in a form that:l Allows comparison for equality.l Has a correspondence to the truth tables.l Canonical Forms in common usage:l Sum of Minterms (SOM)l Product of Maxterms (POM)l Some definitions:l A literal is a variable or th
16、e complement of a variable. Examples: X, Y, X,Y.l A product term is a single literal or a logical product of two or more literals. Examples: Z, WXY, XYZ, WYZ.Date18Canonical Forms (cont.)l A sum-of-products expression is a logical sum of product terms. Example: Z + WXY + XYZ + WYZ. l A sum term is a single literal or a logical sum of two or more literals. Examples: Z, W + X + Y, X
