diff --git a/Srt/lec1b.srt b/Srt/lec1b.srt index 635f294..58ebc41 100644 --- a/Srt/lec1b.srt +++ b/Srt/lec1b.srt @@ -1,5994 +1,5972 @@ 1 -00:00:00,000 --> 00:00:00,000 +00:00:01,840 --> 00:00:02,800 哈尔滨工业大学 IBM技术中心 倾情制作 2 -00:00:00,000 --> 00:00:00,000 +00:00:02,800 --> 00:00:04,800 压制&&特效：（Tacitus） 字幕&&时间轴：曹竞帆（ChingfanTsou） 3 -00:00:00,000 --> 00:00:00,000 +00:00:04,800 --> 00:00:06,800 特别感谢：裘宗燕 4 -00:00:01,840 --> 00:00:14,160 +00:00:06,820 --> 00:00:08,860 [音乐] [MUSIC PLAYING BY J.S. BACH] 5 +00:00:08,920 --> 00:00:13,080 +过程及计算过程： +代换模型 + +6 00:00:14,160 --> 00:00:14,700 教授 大家好 PROFESSOR: Hi. -6 +7 00:00:15,960 --> 00:00:17,820 大家都已经知道 程序员的工作 You've seen that the job of a programmer -7 +8 00:00:19,060 --> 00:00:22,080 就是设计出能够达成特定目标的程序 is to design processes that accomplish particular goals, -8 +9 00:00:23,400 --> 00:00:25,200 比如求出一个数的平方根 such as finding the square roots of numbers, -9 +10 00:00:25,940 --> 00:00:27,720 或者其它一些你想要做的事 or other sorts of things you might want to do. -10 +11 00:00:28,460 --> 00:00:30,140 目前为止我们还没介绍别的什么东西 We haven't introduced anything else yet. -11 +12 00:00:31,760 --> 00:00:32,370 当然了 Of course, -12 +13 00:00:32,380 --> 00:00:35,310 程序员完成工作的方式就是构造“咒语” the way in which a programmer does this is by constructing spells, -13 +14 00:00:36,080 --> 00:00:39,960 通过过程和表达式构造出来的“咒语” which are constructed out of procedures and expressions. -14 +15 00:00:40,700 --> 00:00:45,380 这些“咒语”一定程度上指明了 And that these spells are somehow direct a process -15 +16 00:00:45,480 --> 00:00:47,680 程序员想要的达成目标的方式 to accomplish the goal that was intended by the programmer. -16 +17 00:00:48,840 --> 00:00:50,660 程序员为了使整个过程变得高效 In order for the programmer to do this effectively, -17 +18 00:00:50,940 --> 00:00:52,660 必须理解他写下的代码 he has to understand the relationship -18 +19 00:00:52,880 --> 00:00:54,730 这些“咒语” between the particular things that he writes, -19 +20 00:00:54,800 --> 00:00:55,940 和他想要控制的程序的行为 these particular spells, -20 +21 00:00:56,200 --> 00:00:59,020 之间的联系 and the behavior of the process that he's attempting to control. -21 +22 00:01:01,300 --> 00:01:03,180 所以我们在这门课中要做的就是试图 So what we're doing this lecture is attempt to -22 +23 00:01:03,180 --> 00:01:06,000 以尽可能清晰的方式建立这样的联系 establish that connection in as clear a way as possible. -23 +24 00:01:07,260 --> 00:01:09,330 我们尤其要理解 What we will particularly do is understand -24 +25 00:01:09,330 --> 00:01:13,390 特定过程和表达式的模式 how particular patterns of procedures and expressions -25 +26 00:01:14,280 --> 00:01:16,210 将会导致怎样的特定程序执行模式 cause particular patterns of execution, -26 +27 00:01:16,460 --> 00:01:19,000 和计算过程的特定的行为 particular behaviors from the processes. -27 +28 00:01:22,100 --> 00:01:23,230 让我们开始吧 Let's get down to that. -28 +29 00:01:23,860 --> 00:01:25,770 我将会从一个非常简单的程序开始 I'm going to start with a very simple program. -29 +30 00:01:27,920 --> 00:01:31,000 这是一个计算两个数的平方和的程序 This is a program to compute the sum of the squares of two numbers. -30 +31 00:01:33,320 --> 00:01:44,600 我们将x和y的平方和定义为 And we'll define the sum of the squares of x and y to be -31 +32 00:01:45,700 --> 00:01:49,720 x的平方 the sum of the square of x -32 +33 00:01:49,720 --> 00:01:50,780 我这么写 -- I'm going to write it that way-- -33 +34 00:01:51,440 --> 00:01:53,630 -和y的平方的和 +加上y的平方的和 and the square of y, -34 +35 00:01:56,260 --> 00:02:11,000 -x的平方是指x和x的乘积 +这里的x的平方是指x和x的乘积 where the square of x is the product of x and x. -35 +36 00:02:13,840 --> 00:02:16,080 现在 设想这么一件事 Now,supposing I were to say something to this, -36 +37 00:02:16,780 --> 00:02:19,970 比如 在这个程序这么定义好之后 like,to the system after having defined these things, -37 +38 00:02:20,180 --> 00:02:24,300 跟据定义的形式 求3和4的平方和 of the form,the sum of the squares of 3 and 4, -38 +39 00:02:25,240 --> 00:02:28,560 我希望得到25 I am hoping that I will get out a 25. -39 +40 00:02:29,120 --> 00:02:30,610 因为3的平方是9 Because the square of 3 is 9, -40 +41 00:02:30,610 --> 00:02:31,880 4的平方是16 and the square of 4 is 16, -41 +42 00:02:32,580 --> 00:02:33,970 它们的和是25 and 25 is the sum of those. -42 +43 00:02:34,980 --> 00:02:35,960 但这都是怎么回事呢 But how does that happen? -43 +44 00:02:36,380 --> 00:02:39,660 如果我们要理解程序的执行以及它的控制方法 If we're going to understand processes and how we control them, -44 +45 00:02:39,660 --> 00:02:45,060 那么我们就得把过程的机制 then we have to have a mapping from the mechanisms of this procedure -45 +46 00:02:45,680 --> 00:02:48,260 与过程执行时所产生的行为对应起来 into the way in which these processes behave. -46 +47 00:02:49,160 --> 00:02:50,720 我们将要得到的是一个正式的 What we're going to have is a formal, -47 +48 00:02:50,960 --> 00:02:53,140 或者说半正式的机械的模型 or semi-formal mechanical model, -48 +49 00:02:53,820 --> 00:02:57,280 你要理解计算机如何事实上 大体上实现这个机制 whereby you understand how a machine could,in fact,in principle,do this. -49 +50 00:02:57,600 --> 00:03:00,060 不论实际的计算机究竟是怎么实现的 Whether or not the actual machine really does -50 +51 00:03:00,080 --> 00:03:02,720 我此刻要告诉你们的都与实际的实现完全没有关系 what I'm about to tell you is completely irrelevant at this moment. -51 +52 00:03:03,490 --> 00:03:04,920 实际上 这是一个工程模型 In fact,this is an engineering model -52 +53 00:03:05,440 --> 00:03:07,630 就好像电阻 in the same way that,electrical resistor, -53 +54 00:03:07,840 --> 00:03:09,810 我们写出一个模型V=IR we write down a model v equals i r, -54 +55 00:03:10,360 --> 00:03:11,410 这基本上是正确的 it's approximately true. -55 +56 00:03:12,020 --> 00:03:13,060 但不是完全正确的 It's not really true. -56 +57 00:03:13,540 --> 00:03:15,450 当电流通过电阻时 电阻会增加 If I put up current through the resistor it goes boom. -57 +58 00:03:16,560 --> 00:03:19,820 所以电压并不总与电流成线性关系 So the voltage is not always proportional to the current, -58 +59 00:03:20,120 --> 00:03:23,220 但是对于部分情况来说这个模型是适用的 but for some purposes the model is appropriate. -59 +60 00:03:23,860 --> 00:03:26,260 比如我们将要介绍的这个模型 In particular,the model we're going to describe right now, -60 +61 00:03:26,260 --> 00:03:27,850 我把这个模型称作代换模型 which I call the substitution model, -61 +62 00:03:28,180 --> 00:03:30,280 这是我们能接触到的最简单的模型 is the simplest model that we have -62 +63 00:03:30,500 --> 00:03:33,020 它可以帮助我们理解过程和程序执行的原理 for understanding how procedures work and how processes work. -63 +64 00:03:33,770 --> 00:03:35,470 以及过程如何使得程序执行 How procedures yield processes. -64 +65 00:03:36,000 --> 00:03:38,110 代换模型 对接下来几天 And that substitution model will be accurate -65 +66 00:03:38,240 --> 00:03:40,770 我们将要接触到的东西都是适用的 for most of the things we'll be dealing with in the next few days. -66 +67 00:03:41,580 --> 00:03:45,490 但最终 幻想总会破灭 But eventually,it will become impossible to sustain the illusion -67 +68 00:03:45,500 --> 00:03:46,250 如果你认为计算机实际就是这么工作的 that that's the way the machine works, -68 +69 00:03:46,490 --> 00:03:49,600 并且我们将要学习其它的更特殊的模型 and we'll go to other more specific and particular models -69 +70 00:03:49,920 --> 00:03:51,020 届时会讨论更多的细节 that will show more detail. -70 +71 00:03:53,200 --> 00:03:57,640 好了 接下来 我们来看下 OK,well,the first thing,of course,is we say, -71 +72 00:03:57,640 --> 00:03:58,780 黑板上的这些内容 what are the things we have here? -72 +73 00:03:58,780 --> 00:04:00,030 这里已经有了一些神秘的符号 We have some cryptic symbols. -73 +74 00:04:00,900 --> 00:04:03,770 这些神秘的符号是由几块组成的 And these cryptic symbols are made out of pieces. -74 +75 00:04:04,090 --> 00:04:05,380 还有几种表达式 There are kinds of expressions. -75 +76 00:04:05,720 --> 00:04:08,100 现在我们看一看都有哪些类型的表达式 So let's write down here the kinds of expressions there are. -76 +77 00:04:17,580 --> 00:04:18,180 有 And we have-- -77 +78 00:04:18,260 --> 00:04:20,180 首先有数字 and so far I see things like numbers. -78 +79 00:04:25,060 --> 00:04:27,550 有像那样的符号 I see things like symbols like that. -79 +80 00:04:31,790 --> 00:04:34,940 我们之前见过Lambda表达式 We have seen things before like lambda expressions, -80 -00:04:34,940 --> 00:04:35,710 -当然了没出现在黑板上 -but they're not here. - 81 -00:04:35,870 --> 00:04:36,720 -先撇开不谈 -I'm going to leave them out. +00:04:34,940 --> 00:04:36,760 +当然了没出现在黑板上 先撇开不谈 +but they're not here.I'm going to leave them out. 82 -00:04:36,720 --> 00:04:37,680 -Lambda表达式 -Lambda expressions, +00:04:36,800 --> 00:04:39,260 +Lambda表达式 稍后再讨论 +Lambda expressions,we'll worry about them later. 83 -00:04:38,190 --> 00:04:39,290 -稍后再讨论 -we'll worry about them later. - -84 00:04:44,500 --> 00:04:45,640 还有定义 Things like definitions. -85 +84 00:04:51,550 --> 00:04:52,660 还有条件表达式 Things like conditionals. -86 +85 00:04:58,040 --> 00:05:00,050 最后还有组合式 And finally,things like combinations. -87 +86 00:05:06,660 --> 00:05:09,050 这些表达式 These kinds of expressions are -88 +87 00:05:10,090 --> 00:05:14,090 我们稍后再讨论 它们是特殊形式 -- I'll worry about later-- these are special forms. -89 +88 00:05:14,740 --> 00:05:16,920 它们有一些专门的规则 There are particular rules for each of these. -90 -00:05:17,320 --> 00:05:18,430 -然而我要告诉你们的是 -I'm going to tell you,however, - -91 -00:05:18,430 --> 00:05:20,370 -处理通常情况的规则 -the rules for doing a general case. +89 +00:05:17,320 --> 00:05:20,420 +然而我要告诉你们的是 处理通常情况的规则 +I'm going to tell you,however,the rules for doing a general case. -92 +90 00:05:20,750 --> 00:05:22,370 通常是怎样对一个组合式求值的 How does one evaluate a combination? -93 +91 00:05:23,200 --> 00:05:24,370 因为 实际上 在黑板的那边 Because,in fact,over here, -94 +92 00:05:24,760 --> 00:05:27,800 全都是一些组合式以及一些符号和数字 all I really have are combinations and some symbols and numbers. -95 +93 00:05:28,990 --> 00:05:30,610 而简单的 比如一个数字 And the simple things like a number,well, -96 +94 00:05:30,610 --> 00:05:32,090 求值的结果就是这个数字所代表的数值 it will evaluate to itself. -97 +95 00:05:33,000 --> 00:05:34,680 在我将要介绍的模型中 In the model I will have for you, -98 +96 00:05:34,700 --> 00:05:35,860 符号的概念将不会出现 the symbols will disappear. -99 +97 00:05:36,810 --> 00:05:39,450 只有当你需要了解 They won't be there at the time when you need them, -100 +98 00:05:39,450 --> 00:05:40,970 当你需要真正理解它的时候才讨论 when you need to get at them. -101 +99 00:05:41,420 --> 00:05:43,530 所以我只需要向大家讲解 So the only thing I really have to explain to you is, -102 +100 00:05:43,530 --> 00:05:44,850 如何对组合式求值 how do we evaluate combinations? -103 +101 00:05:47,950 --> 00:05:49,010 好吧 继续 OK,let's see -104 +102 00:05:50,010 --> 00:05:52,540 先看下第一张幻灯片 So first I want to get the first slide. -105 +103 00:05:53,310 --> 00:05:58,440 这是对一个表达式求值的规则 Here is the rule for evaluating an application. -106 +104 00:06:01,040 --> 00:06:06,140 我们看到这条规则指出 What we have is a rule that says, -107 +105 00:06:06,140 --> 00:06:07,050 要对一个组合式求值 to evaluate a combination, -108 +106 00:06:07,050 --> 00:06:08,900 主要有三个部分 there are two parts,three parts to the rule. -109 +107 00:06:09,540 --> 00:06:11,790 组合式有若干个部分 The combination has several parts. -110 +108 00:06:12,020 --> 00:06:15,090 有运算符和运算对象 It has operators and it has operands. -111 +109 00:06:16,420 --> 00:06:19,060 运算符返回一个过程 The operator returns into a procedure. -112 +110 00:06:19,810 --> 00:06:20,910 也就是说如果我们对运算符求值 If we evaluate the operator, -113 +111 00:06:20,930 --> 00:06:21,890 结果是一个过程 we will get a procedure. -114 +112 00:06:22,100 --> 00:06:23,200 比如你们也看到了 And you saw,for example, -115 +113 00:06:23,200 --> 00:06:24,980 我向计算机输入一个“+” how I'll type "+" at the machine -116 +114 00:06:24,980 --> 00:06:27,170 然后产生了一些复合的过程以及其它的一些东西 and out came compound procedure something or other. -117 +115 00:06:28,520 --> 00:06:30,260 而对运算对象求值就会得到参数 And the operands produce arguments. -118 +116 00:06:31,620 --> 00:06:35,250 一旦我们完成了对运算符的求值得到了一个过程 Once we've gotten the operator evaluated to get a procedure, -119 +117 00:06:35,250 --> 00:06:37,540 并且对运算对象也完成求值得到了参数 and the argument is evaluated to get argument -120 +118 00:06:37,550 --> 00:06:39,090 也就是运算对象作为参数的值 -- the operand's value to get arguments-- -121 +119 00:06:39,090 --> 00:06:41,170 我们通过复制过程体 we apply the procedure to these arguments -122 +120 00:06:41,880 --> 00:06:43,950 来将其应用到这些参数上 by copying the body of the procedure, -123 +121 00:06:44,240 --> 00:06:46,940 用术语来说 过程体指的就是定义这个过程的表达式 which is the expression that the procedure is defined in terms of. -124 +122 00:06:47,220 --> 00:06:48,570 接下来要做的是 What is it supposed to do? -125 +123 00:06:49,280 --> 00:06:53,950 用求得的实际参数代换过程的形式参数 Substituting the argument supplied for the formal parameters of the procedure, -126 +124 00:06:54,680 --> 00:06:56,220 形式参数就是 the formal parameters being the names -127 +125 00:06:56,220 --> 00:06:58,220 过程定义中声明的变量名 defined by the declaration of the procedure. -128 +126 00:06:58,760 --> 00:07:00,850 然后我们对产生的新过程体求值 Then we evaluate the resulting new body, -129 +127 00:07:01,020 --> 00:07:04,770 这个新过程体是由将代换模型应用到之前的过程体而产生的 the body resulting from copying the old body with the substitutions made. -130 +128 00:07:06,990 --> 00:07:08,270 这个规则非常简单 It's a very simple rule, -131 +129 00:07:09,370 --> 00:07:11,650 并且我们会非常形式化地应用这个规则一段时间 and we're going to do it very formally for a little while. -132 +130 00:07:11,980 --> 00:07:14,420 因为接下来几节课 Because for the next few lectures, -133 +131 00:07:14,730 --> 00:07:15,740 我想让各位做的是 what I want you to do is to say, -134 +132 00:07:15,740 --> 00:07:18,930 如果有什么不能理解的 if I don't understand something,if I don't understand something, -135 +133 00:07:19,280 --> 00:07:20,930 就直接机械化地应用这个规则 be very mechanical and do this. -136 +134 00:07:23,570 --> 00:07:24,160 接下来 So let's see. -137 +135 00:07:25,970 --> 00:07:27,630 考虑这么一个求值过程 Let's consider a particular evaluation, -138 +136 00:07:27,740 --> 00:07:28,990 之前讨论过的 the one we were talking about before. -139 +137 00:07:29,210 --> 00:07:32,960 3和4的平方和 The sum of the squares of 3 and 4. -140 +138 00:07:35,560 --> 00:07:36,660 这是什么意思呢 What does that mean? -141 +139 00:07:36,660 --> 00:07:37,760 这是说 It says,take -142 +140 00:07:38,260 --> 00:07:40,660 其实 我已经知道求平方的过程是怎样的 -- well,I could find out what's on the square-- -143 +141 00:07:40,660 --> 00:07:41,860 是某个过程 it's some procedure, -144 +142 00:07:42,140 --> 00:07:43,710 但我并不关心这个过程的具体表述 and I'm not going to worry about the representation, -145 +143 00:07:43,710 --> 00:07:45,710 并且我也不会在黑板上写出来 and I'm not going to write it on the blackboard for you. -146 +144 00:07:46,810 --> 00:07:48,530 还有3 代表某个数字 And I have that 3,represents some number, -147 +145 00:07:49,250 --> 00:07:51,360 但如果要我重复这个数字 but if I have to repeat that number, -148 +146 00:07:51,360 --> 00:07:52,280 我并不知道这个数字是什么 I can't tell you the number. -149 +147 00:07:52,410 --> 00:07:54,190 这个数字本身是抽象的 The number itself is some abstract thing. -150 +148 00:07:54,490 --> 00:07:55,860 只知道有个数词能代表它 There's a numeral which represents it, -151 +149 00:07:55,890 --> 00:07:56,690 我把这个它叫作3 which I'll call 3, -152 +150 00:07:56,960 --> 00:07:59,000 并且会在代换中用到它 and I'll use that in my substitution. -153 +151 00:07:59,370 --> 00:08:01,400 4也是个数字 And 4 is also a number. -154 +152 00:08:01,570 --> 00:08:07,680 我在这个过程体中用3代换x 用4代换y I'm going to substitute 3 for x and 4 for y in the body of this procedure -155 +153 00:08:07,680 --> 00:08:08,570 看黑板这里 that you see over here. -156 +154 00:08:09,170 --> 00:08:10,580 这是过程体 Here's the body of the procedure. -157 +155 00:08:11,180 --> 00:08:13,780 它关联到这个组合式 It corresponds to this combination, -158 +156 00:08:13,790 --> 00:08:14,520 这个组合式是个求和过程 which is an addition. -159 +157 00:08:17,160 --> 00:08:18,480 而它就转化成这样一个过程 So what that reduces to, -160 +158 00:08:18,820 --> 00:08:19,870 我们把这个转化 as a reduction step, -161 +159 00:08:19,870 --> 00:08:20,370 称为归约 we call it, -162 +160 00:08:20,670 --> 00:08:28,740 这样就变成了求3和4的平方和 it's the sum of the square of 3 and the square of 4. -163 +161 00:08:30,000 --> 00:08:33,710 接下来我要做的是 Now,what's the next step I have to do here? -164 +162 00:08:33,710 --> 00:08:35,710 对它求值 I say,well,I have to evaluate this. -165 +163 00:08:35,710 --> 00:08:37,010 根据之前给出的求值规则 According to my rule, -166 +164 00:08:37,010 --> 00:08:40,640 就是各位刚刚在幻灯片上看到的 which you just saw on that overhead or slide, -167 +165 00:08:41,410 --> 00:08:45,780 我们要对运算对象求值 what we had was that we have to evaluate the operands -168 +166 00:08:45,930 --> 00:08:47,080 就是这些运算对象 -- and here are the operands, -169 +167 00:08:47,340 --> 00:08:49,080 这是第一个运算对象 这是第二个运算对象 here's one and here's the next operand-- -170 +168 00:08:49,400 --> 00:08:50,770 我们还得对运算符求值 and how we have to evaluate procedure. -171 +169 00:08:50,770 --> 00:08:51,850 对它们求值的顺序无关紧要 The order doesn't matter. -172 +170 00:08:52,510 --> 00:08:55,710 然后我们要应用这个过程 And then we're going to apply the procedure, -173 +171 00:08:55,710 --> 00:08:56,280 即求和过程 which is plus, -174 +172 00:08:56,280 --> 00:08:58,620 然后很神奇的 不知怎么回事就能得到答案 and magically somehow that's going to produce the answer. -175 +173 00:08:58,880 --> 00:09:01,050 我不会深究求和过程的实现 I'm not to open up plus and look inside of it. -176 +174 00:09:02,140 --> 00:09:04,500 然而 为了对运算对象求值 However,in order to evaluate the operand, -177 +175 00:09:04,510 --> 00:09:05,970 我们采用任意一种顺序来求值 let's pick some arbitrary order and do them. -178 +176 00:09:06,430 --> 00:09:07,580 我会从右到左求值 I'm going to go from right to left. -179 +177 00:09:08,190 --> 00:09:09,840 好了 为了对这个运算对象求值 Well,in order to evaluate this operand, -180 +178 00:09:09,840 --> 00:09:13,410 我得用同样的规则对这运算对象的每个部分求值 I have to evaluate the parts of it by the same rule. -181 +179 00:09:13,970 --> 00:09:15,850 首先需要弄清楚求平方是什么 And the parts are I have to find out what square is-- -182 +180 00:09:15,850 --> 00:09:16,580 是某个过程 it's some procedure, -183 +181 00:09:16,580 --> 00:09:18,120 有个形式参数x which has a formal parameter x. -184 +182 00:09:19,180 --> 00:09:22,580 然后还有个运算对象是4 And also,I have an operand which is 4, -185 +183 00:09:23,560 --> 00:09:27,550 用4代换求平方过程的形式参数x which I have to substitute for x in the body of square. -186 +184 00:09:28,410 --> 00:09:30,370 接下来基本上可以这么说 So the next step is basically to say that -187 +185 00:09:30,370 --> 00:09:38,180 这就变成了求3的平方以及4与4乘积的和 this is the sum of the square of 3 and the product of 4 and 4. -188 +186 00:09:40,600 --> 00:09:43,710 当然了 还可以接着深入分析乘法 Of course,I could open up asterisk if I liked -189 +187 00:09:43,720 --> 00:09:45,970 这个求乘积的操作 -- the multiplication operation-- -190 +188 00:09:46,680 --> 00:09:47,480 但现在这个不重要 but I'm not going to do that. -191 +189 00:09:47,500 --> 00:09:48,610 先把乘法过程看作是基本过程 I'm going to consider that primitive. -192 +190 00:09:49,790 --> 00:09:53,250 其实 如果你深究计算机的抽象层次 So,and,of course,at any level of detail,if you look inside this machine, -193 +191 00:09:53,250 --> 00:09:54,320 你会发现无论在具体哪一个层次 you're going to find that -194 +192 00:09:54,320 --> 00:09:56,820 在其下都还有若干个你不清楚的抽象层次 there's multiple levels below that that you don't know about. -195 +193 00:09:57,980 --> 00:10:00,650 但我们必须明白要学会忽略细节 But one of the things we have to learn how to do is ignore details. -196 +194 00:10:01,980 --> 00:10:03,960 理解复杂事物的关键是 The key to understanding complicated things -197 +195 00:10:04,280 --> 00:10:08,290 避免不必要的观察 计算和思考 is to know what not to look at and what not compute and what not to think. -198 +196 00:10:09,260 --> 00:10:10,840 所以对这个乘法过程的细节我们不深入研究 So we're going to stop this one here and say, -199 +197 00:10:10,840 --> 00:10:13,560 就直接看作两个数的乘积 oh,yes,this is the product of two things. -200 +198 00:10:13,970 --> 00:10:15,000 现在把两个4乘起来 We're going to do it now. -201 +199 00:10:15,660 --> 00:10:22,320 于是就有3的平方与16的和 So this is nothing more than the sum of the square of 3 and 16. -202 +200 00:10:22,960 --> 00:10:25,640 还有一个求平方的表达式要求值 And now I have another thing I have to evaluate, -203 +201 00:10:27,010 --> 00:10:28,120 注意这是3的平方 but that square of 3, -204 +202 00:10:28,120 --> 00:10:29,080 这与之前求4的平方是一个过程 well,it's the same thing. -205 +203 00:10:29,080 --> 00:10:35,200 于是整个表达式就变成了求3和3乘积与16的和 That's the sum of the product of 3 and 3 and 16, -206 +204 00:10:35,960 --> 00:10:41,810 也就是求9和16的和 结果是25 which is the sum of 9 and 16,which is 25. -207 +205 00:10:44,500 --> 00:10:48,570 大家现在看到了应用代换模型的基本方法 So now you see the basic method of doing substitutions. -208 +206 00:10:49,800 --> 00:10:51,730 但我提醒大家 And I warn you that -209 +207 00:10:52,110 --> 00:10:55,410 代换模型并不能准确描述计算机实际的运行方式 this is not a perfect description of what the computer does. -210 +208 00:10:56,800 --> 00:10:58,200 但这个模型对于接下来几节课我们要讨论的问题 But it's a good enough description -211 +209 00:10:58,200 --> 00:11:02,000 已经可以描述得足够清楚 for the problems that we're going to have in the next few lectures -212 +210 00:11:02,510 --> 00:11:04,120 你可以先认为这是正确的 that you should think about this religiously. -213 +211 00:11:04,820 --> 00:11:07,230 认为计算机就是这么运行的 And this is how the machine works for now. -214 +212 00:11:07,520 --> 00:11:08,920 之后我们的讨论会变得更加严谨 Later we'll get more detailed. -215 +213 00:11:11,700 --> 00:11:12,280 好了 现在可以看到 Now,of course, -216 +214 00:11:12,280 --> 00:11:15,020 我在这个例子中采用了从右到左的求值顺序 I made a specific choice of the order of evaluation here. -217 +215 00:11:15,530 --> 00:11:16,530 当然还有别的选择 There are other possibilities. -218 +216 00:11:17,020 --> 00:11:20,580 如果我们回过头来再看幻灯片 If we go back to this,ah,to the telestrator here -219 +217 00:11:20,580 --> 00:11:21,870 上所写的代换模型应用规则 and look at the substitution rule, -220 +218 00:11:22,800 --> 00:11:25,650 可以发现刚才我对运算符求值得到了一个过程 we see that I evaluated the operator to get the procedures, -221 +219 00:11:25,900 --> 00:11:28,690 对运算对象求值得到了过程的实际参数 and I evaluated the operands to get the arguments first, -222 +220 00:11:28,690 --> 00:11:29,950 然后才将参数应用到过程 before I do the application. -223 +221 00:11:30,730 --> 00:11:32,030 这样做是完全可行的 It's entirely possible, -224 +222 00:11:32,030 --> 00:11:34,650 还有别的求值顺序比如正则序求值 and there are alternate rules called normal order evaluation -225 +223 00:11:35,020 --> 00:11:37,470 应用正则序你可以先用表达式去代换 whereby you can do the substitution of the expressions -226 +224 00:11:37,470 --> 00:11:44,640 将其作为过程的形式参数的运算对象 然后再求值 which are the operands for the formal parameters inside the body first. -227 +225 00:11:45,970 --> 00:11:47,650 这么做会得到同样的答案 And you'll get also the same answer. -228 +226 00:11:48,560 --> 00:11:50,110 但是现在 考虑到实际情况 But right now,for concreteness, -229 +227 00:11:50,110 --> 00:11:52,240 因为计算机实际是这么运行的 and because this is the way our machine really does it, -230 +228 00:11:52,900 --> 00:11:54,000 我还是会给出在黑板上演示的方式 I'm going to give you this rule, -231 +229 00:11:54,000 --> 00:11:55,130 这种方式给出了明确的顺序 which has a particular order. -232 +230 00:11:56,120 --> 00:11:58,140 当然了这些顺序一定程度上也是灵活可变的 But that order is to some extent arbitrary,too. -233 +231 00:11:59,620 --> 00:12:01,340 归根到底 In the long run,in the long run, -234 +232 00:12:01,340 --> 00:12:03,760 采用哪种求值顺序来理解 都有各自的依据 there are some reasons why you might pick one order or another, -235 +233 00:12:03,760 --> 00:12:05,760 以后我们会继续讨论这个问题 and we'll get to that later in the subject. -236 +234 00:12:11,930 --> 00:12:15,130 好了 为了让大家理解这都是怎么回事 OK,well now the only other thing I have to tell you about -237 +235 00:12:15,150 --> 00:12:16,420 还需要给大家介绍一个东西 ejust to understand what's going on -238 +236 00:12:16,440 --> 00:12:18,670 来看幻灯片上对条件表达式的解释 is let's look at the rule for conditionals. -239 +237 00:12:19,470 --> 00:12:21,620 条件表达式很简单 Conditionals are very simple, -240 +238 00:12:22,490 --> 00:12:25,440 我给大家详细解释一下 and I'd like to examine this. -241 +239 00:12:26,890 --> 00:12:31,380 条件表达式就是if表达式 A conditional is something that is if -242 +240 00:12:31,980 --> 00:12:33,380 当然还有cond表达式 -- there's also cond of course-- -243 +241 00:12:33,570 --> 00:12:35,700 我会把条件表达式各部分的名称都写下来 but I'm going to give names to the parts of the expression. -244 +242 00:12:35,700 --> 00:12:36,490 首先有一个判断表达式 There's a predicate, -245 +243 00:12:38,080 --> 00:12:40,050 这是个求值结果为真或假的表达式 which is a thing that is either true or false. -246 +244 00:12:41,010 --> 00:12:42,010 然后有一个结果子句 And there's a consequent, -247 +245 00:12:45,600 --> 00:12:47,490 当判断表达式的结果为真就对这个表达式求值 which is the thing you do if the predicate is true. -248 +246 00:12:48,010 --> 00:12:49,020 还有个选择表达式 And there's an alternative, -249 +247 00:12:52,520 --> 00:12:54,610 当判断表达式结果为假 就对这个表达式求值 which is the thing you do if the predicate is false. -250 +248 00:12:55,090 --> 00:12:56,350 顺便提一句 It's important,by the way, -251 +249 00:12:56,760 --> 00:12:59,960 知道事物各部分的名称 to get names for,to get names for, -252 +250 00:13:00,260 --> 00:13:01,390 或者表达式各部分的名称 the parts of things, -253 +251 00:13:01,390 --> 00:13:02,730 很重要 or the parts of expressions. -254 +252 00:13:03,420 --> 00:13:05,660 每个“魔法师”都会告诉你 One of the things that every sorcerer will tell you -255 +253 00:13:05,690 --> 00:13:07,140 如果你能叫出一个“精灵”的名字 is if you have the name of a spirit, -256 +254 00:13:07,140 --> 00:13:08,110 你就有控制它的能力 you have power over it. -257 +255 00:13:10,050 --> 00:13:11,080 所以你得掌握这些名称 So you have to learn these names -258 +256 00:13:11,080 --> 00:13:12,560 以便之后的讨论 so that we can discuss these things. -259 +257 00:13:13,370 --> 00:13:15,410 现在我们有了一个判断表达式 So here we have a predicate, -260 +258 00:13:15,410 --> 00:13:16,100 一个结果表达式 a consequent, -261 +259 00:13:16,100 --> 00:13:16,810 还有个选择表达式 and an alternative. -262 +260 00:13:17,420 --> 00:13:21,060 用这些名称 我们发现对于一个if表达式 And,using such words,we see that an if expression, -263 +261 00:13:21,360 --> 00:13:23,500 关键的问题是对判断表达式的求值 the problems you evaluate to the predicate expression, -264 -00:13:26,810 --> 00:13:28,890 +262 +00:13:24,360 --> 00:13:25,580 如果结果为真 if that yields true, -265 -00:13:29,220 --> 00:13:29,850 +263 +00:13:26,840 --> 00:13:28,860 就对结果表达式求值 then you then go on to evaluate the consequent. -266 -00:13:30,190 --> 00:13:30,200 -就对选择表达式求值 -you evaluate the alternative expression. - -267 -00:13:30,200 --> 00:13:31,920 -否则 -Otherwise, -就对选择表达式求值 -you evaluate the alternative expression. - -268 -00:13:31,920 --> 00:13:31,930 -就对选择表达式求值 -you evaluate the alternative expression. +264 +00:13:29,280 --> 00:13:31,920 +否则 就对选择表达式求值 +Otherwise,you evaluate the alternative expression. -269 +265 00:13:34,530 --> 00:13:35,840 我想通过一个具体的程序 So I'd like to illustrate that -270 +266 00:13:36,170 --> 00:13:42,250 来进一步说明这个问题 now in the context of a particular,of a particular,little program. -271 +267 00:13:43,200 --> 00:13:45,280 就写一个我们会经常遇到的程序吧 Going to write down a program which we're going to see many times. -272 +268 00:13:51,470 --> 00:13:58,430 这是由皮亚诺算术定义的求x和y之和的过程 This is the sum of x and y done by what's called Peano arithmetic, -273 +269 00:13:58,430 --> 00:14:00,430 其实就是加1和减1 which is all we're doing is incrementing and decrementing. -274 +270 00:14:01,500 --> 00:14:02,660 来看看这个 And we're going to see this for a little bit. -275 +271 00:14:02,660 --> 00:14:03,980 这是个非常重要的程序 It's a very important program. -276 +272 00:14:06,020 --> 00:14:07,120 如果x等于0 If x equals 0, -277 +273 00:14:09,680 --> 00:14:11,260 那么结果就是y then the result is y. -278 +274 00:14:11,580 --> 00:14:12,270 否则 Otherwise, -279 +275 00:14:12,270 --> 00:14:20,350 结果就是x减1与y加1的和 this is the sum of the decrement of x and the increment of y. -280 +276 00:14:23,220 --> 00:14:26,110 以后我们会继续研究这个加法过程 We're going to look at this a lot more in the future. -281 +277 00:14:27,580 --> 00:14:28,740 现在看前面 Let's look at the overhead. -282 +278 00:14:28,980 --> 00:14:31,090 现在有了这么个过程 So here we have this procedure, -283 +279 00:14:31,090 --> 00:14:33,180 我们看看怎么应用代换模型 and we're going to look at how we do the substitutions, -284 +280 00:14:33,180 --> 00:14:34,270 以及代换的顺序 the sequence of substitutions. -285 +281 00:14:35,760 --> 00:14:37,700 我现在想要求3和4的和 Well,I'm going to try and add together 3 and 4. -286 +282 00:14:38,280 --> 00:14:40,200 根据我演示给大家步骤 Well,using the first rule that I showed you, -287 +283 00:14:40,420 --> 00:14:43,900 第一步 在过程体中 we substitute 3 for x and 4 four y -288 +284 00:14:44,180 --> 00:14:45,620 用3代换x 用4代换y in the body of this procedure. -289 +285 00:14:45,800 --> 00:14:47,360 过程体 The body of the procedure is the thing -290 +286 00:14:47,360 --> 00:14:49,690 由if表达式开始 在这儿结束 that begins with if and finishes over here. -291 +287 00:14:51,020 --> 00:14:51,940 于是我们就有了 So what we get is, -292 +288 00:14:51,940 --> 00:14:53,390 如果3是0 of course,if 3 is 0, -293 +289 00:14:53,390 --> 00:14:54,600 那么结果就是4 then the result is 4. -294 +290 00:14:55,570 --> 00:14:56,350 否则 Otherwise, -295 +291 00:14:56,450 --> 00:14:58,360 结果就是3减1 it's the sum of the decrement of 3 -296 +292 00:14:58,360 --> 00:14:59,320 与4加1的和 and the increment of 4. -297 +293 00:15:01,080 --> 00:15:02,490 但这部份我不会去管它 But I'm not going to worry about these yet -298 +294 00:15:03,020 --> 00:15:04,820 因为3不等于0 because 3 is not 0. -299 +295 00:15:05,290 --> 00:15:06,540 于是答案不是4 So the answer is not 4. -300 +296 00:15:07,970 --> 00:15:08,430 因此 Therefore, -301 +297 00:15:08,450 --> 00:15:10,960 于是整个if表达式就可以归约成 this if reduces to -302 +298 00:15:10,960 --> 00:15:12,610 对这个表达式的求值 an evaluation of the expression, -303 +299 00:15:12,850 --> 00:15:14,010 求3减1 the sum to the decrement of 3 -304 +300 00:15:14,010 --> 00:15:14,920 与4加1的和 and the increment of 4. -305 +301 00:15:16,580 --> 00:15:17,800 继续求值 Continuing with my evaluation, -306 +302 00:15:17,800 --> 00:15:19,800 先认为加1的过程是基本过程 the increment I presume to be primitive, -307 +303 00:15:20,380 --> 00:15:21,800 然后就有了5 and so I get a 5 there. -308 +304 00:15:22,680 --> 00:15:24,530 同样把减1也认为是基本过程 OK,and then the decrement is also primitive, -309 +305 00:15:24,530 --> 00:15:25,330 然后就有了2 and I get a 2. -310 +306 00:15:25,680 --> 00:15:28,080 于是我把问题化简了 And so I change the problem into a simpler problem. -311 +307 00:15:28,080 --> 00:15:30,080 不是3加4的问题了 Instead of adding 3 to 4, -312 +308 00:15:30,840 --> 00:15:32,080 而是2加5的问题了 I'm adding 2 to 5. -313 +309 00:15:33,130 --> 00:15:34,600 为什么这样就变简单了呢 The reason why this is a simpler problem -314 +310 00:15:34,600 --> 00:15:37,310 因为我在减小x is because I'm counting down on x, -315 +311 00:15:37,780 --> 00:15:40,160 最终x会变为0 and eventually,then,x will be 0. -316 +312 00:15:43,070 --> 00:15:44,990 这就是代换模型了 So,so much for the substitution rule. -317 +313 00:15:45,690 --> 00:15:46,420 通常来讲 In general, -318 +314 00:15:46,420 --> 00:15:48,900 在对if表达式使用代换模型的时候 I'm not going to write down intermediate steps -319 +315 00:15:48,900 --> 00:15:50,900 我不会写下所有中间步骤 when using substitutions having to do with ifs, -320 +316 00:15:51,500 --> 00:15:54,660 因为这么做只能把事情复杂化 because they just expand things to become complicated. -321 +317 00:15:55,120 --> 00:15:56,400 我们要做的是 What we will be doing is saying, -322 +318 00:15:56,400 --> 00:15:58,190 把求3和4的和 oh,yes,the sum of 3 and 4 -323 +319 00:15:58,190 --> 00:16:00,780 归约到求2和5的和 results in the sum of 2 and 5 -324 +320 00:16:01,390 --> 00:16:02,920 在那之后 and reduces to the sum of 2 and 5. -325 +321 00:16:03,160 --> 00:16:06,040 归约到求1和6的和 which,in fact,reduces to the sum of 1 and 6, -326 +322 00:16:06,740 --> 00:16:09,700 最后归约到求0和7的和 which reduces to the sum of 0 and 7 -327 +323 00:16:09,700 --> 00:16:10,360 看这儿 over here, -328 +324 00:16:10,860 --> 00:16:11,960 最后归约到7 which reduces to a 7. -329 +325 00:16:13,730 --> 00:16:14,680 这就是我们想要的结果 That's what we're going to be seeing. -330 +326 00:16:16,180 --> 00:16:17,850 现在 各位对第一部分的内容有疑问么 Are there any questions for the first segment yet? -331 +327 00:16:20,180 --> 00:16:20,680 有吗 Yes? -332 +328 00:16:21,010 --> 00:16:23,530 学生 你用了1+过程和-1+过程 STUDENT: You're using 1 plus and minus 1 plus -333 +329 00:16:23,530 --> 00:16:25,460 它们都是基本过程么 Are those primitive operations? -334 +330 00:16:25,460 --> 00:16:26,100 教授 是的 PROFESSOR: Yes -335 +331 00:16:26,410 --> 00:16:28,450 你们在这门课中要明白的是 One of the things you're going to be seeing in this subject -336 +332 00:16:28,970 --> 00:16:32,110 我绝对不会考虑 is I'm going to,without thinking about it, -337 +333 00:16:32,260 --> 00:16:35,680 给大家介绍越来越多的基本过程 introduce more and more primitive operations -338 +334 00:16:35,920 --> 00:16:37,710 可以假定 There's presumably some large library -339 +335 00:16:37,710 --> 00:16:39,260 有一些庞大的基本过程的程序库 of primitive operations somewhere. -340 +336 00:16:39,450 --> 00:16:41,100 不用过分追究这些基本过程 But it doesn't matter that they're primitive-- -341 +337 00:16:41,100 --> 00:16:43,100 -可能有一些手册有这些基本过程的索引便于我们查找 +可能有一些手册有这些基本过程的索引 便于我们查找 there may be some manual that lists them all. -342 +338 00:16:43,100 --> 00:16:44,860 如果我向你解释基本过程的作用 你应该说 If I tell you what they do,you should say, -343 +339 00:16:44,860 --> 00:16:46,300 啊 是这样 我明白它们的作用了 oh,yes,I know what they do. -344 +340 00:16:46,660 --> 00:16:48,300 比如 这个减1的过程 So one of them is the decrementor -345 +341 00:16:48,670 --> 00:16:49,490 -1+ -- minus 1 plus-- -346 +342 00:16:49,790 --> 00:16:50,890 另一个加1的过程 and the other operation is increment, -347 +343 00:16:50,890 --> 00:16:51,780 1+ which is 1 plus. -348 +344 00:16:53,140 --> 00:16:53,480 谢谢大家 Thank you. -349 +345 00:16:53,490 --> 00:16:54,900 第一部分结束 That's the end of the first segment. -350 +346 00:17:05,540 --> 00:17:19,170 [音乐] [MUSIC PLAYING BY J.S. BACH] -351 +347 00:17:19,310 --> 00:17:21,540 教授 现在我们有了一个合理的 机械的方式 PROFESSOR: Now that we have a reasonably mechanical way -352 +348 00:17:21,540 --> 00:17:27,300 去理解一个由过程 of understanding how a program made out of procedures -353 +349 00:17:27,300 --> 00:17:30,700 和表达式组成的程序是如何演化成计算过程的 and expressions evolves a process, -354 +350 00:17:31,160 --> 00:17:32,700 我想要大家培养出一些直觉 I'd like to develop some intuition -355 +351 00:17:33,460 --> 00:17:37,150 有关程序如何演化成计算过程的直觉 about how particular programs evolve particular processes, -356 +352 00:17:37,380 --> 00:17:39,150 有关程序到底是什么“形状”的直觉 what the shapes of programs have to be -357 +353 00:17:39,150 --> 00:17:41,150 这样才能产生特定“形状”的计算过程 in order to get particular shaped processes. -358 +354 00:17:42,430 --> 00:17:46,000 这让我想到了 画面预览 This is a question about,really,pre-visualizing. -359 +355 00:17:47,140 --> 00:17:48,670 这是个摄影专业术语 That's a word from photography. -360 +356 00:17:48,670 --> 00:17:51,020 我曾经对摄影十分着迷 I used to be interested in photography a lot, -361 +357 00:17:51,890 --> 00:17:53,700 你会发现 and one of the things you discover -362 +358 00:17:53,700 --> 00:17:55,250 初学摄影的时候 when you start trying to learn about photography -363 +359 00:17:55,250 --> 00:17:57,860 你很想成为一名有创意的摄影师 is that you say,gee,I'd like to be a creative photographer. -364 +360 00:17:58,520 --> 00:18:00,100 我知道怎么做 按快门 Now,I know the rules,I push buttons, -365 +361 00:18:00,100 --> 00:18:02,800 调整光圈什么的 and I adjust the aperture and things like that. -366 +362 00:18:03,140 --> 00:18:06,320 但要想有创意 But the key to being a creative person,partly, -367 +363 00:18:06,320 --> 00:18:09,400 一定程度上靠的是能在一定层次上进行分析 is to be able to do analysis at some level. -368 +364 00:18:09,400 --> 00:18:12,540 比如 在按下快门前 To say,how do I know what it is -369 +365 00:18:12,540 --> 00:18:15,320 我怎么知道交卷上最终是怎样的画面呢 that I'm going to get on the film before I push the button. -370 +366 00:18:16,920 --> 00:18:18,770 我能在心中 Can I imagine in my mind -371 +367 00:18:20,290 --> 00:18:23,340 非常准确清晰地刻画出拍好的照片么 the resulting image very precisely and clearly -372 +368 00:18:24,970 --> 00:18:27,700 那精心地取景 as a consequence of the particular framing, -373 +369 00:18:27,700 --> 00:18:29,120 仔细地调整光圈 of the aperture I choose, -374 +370 00:18:29,120 --> 00:18:30,140 仔细地对焦 of the focus, -375 +371 00:18:30,140 --> 00:18:31,210 而拍出来的照片 and things like that? -376 +372 00:18:32,480 --> 00:18:35,070 一定程度来说 这就是摄影的艺术 That's part of the art of doing this sort of thing. -377 +373 00:18:35,600 --> 00:18:38,480 而要掌握摄影的艺术牵涉到许多方面 And a lot of that involves,learning a lot of that -378 +374 00:18:38,480 --> 00:18:40,050 比如曝光试验片 involves things like test strips. -379 +375 00:18:40,580 --> 00:18:42,400 比如拍摄非常几张简单的画面 You take very simple images -380 +376 00:18:42,970 --> 00:18:46,050 每张都有不同的影像密度 that have varying degrees of density in them,for example, -381 +377 00:18:46,340 --> 00:18:49,330 然后把它们打印在一张纸上 and examine what those look like on a piece of paper -382 +378 00:18:49,330 --> 00:18:50,190 观察它们都有什么不同 when you print them out. -383 +379 00:18:51,180 --> 00:18:53,520 你能发现实际可见的对比度 You find out what is the range of contrasts -384 +380 00:18:53,520 --> 00:18:54,610 都是怎样的范围内 that you can actually see. -385 +381 00:18:55,470 --> 00:18:56,880 还能发现 从实际角度来讲 And what,in a real scene, -386 +382 00:18:56,880 --> 00:18:59,160 都是哪些因素与同一张照片中不同层次 would correspond to the various levels -387 +383 00:18:59,160 --> 00:19:03,640 不同区块的影像密度的产生有关 and zones that you have of density in an image. -388 +384 00:19:05,020 --> 00:19:08,540 -今天这节课我就想让大家看一些特别的“曝光试验片” +今天这节课我就想让大家看一些特别的 “曝光试验片” Well,today I want to look at some very particular test strips, -389 +385 00:19:09,290 --> 00:19:12,280 下面通过幻灯片来看第一张 and I suppose one of them I see here is up on the telestrator, -390 +386 00:19:12,280 --> 00:19:14,040 请看 so we should switch to that. -391 +387 00:19:14,810 --> 00:19:18,840 有两个非常非常重要的程序 There's a very important,very important pair of programs -392 +388 00:19:18,840 --> 00:19:24,050 用来理解在程序执行时 for understanding what's going on in the evolution of a process -393 +389 00:19:24,190 --> 00:19:25,600 计算过程中的求值都是怎样的 by the execution of a program. -394 +390 00:19:26,900 --> 00:19:30,010 大家看到的是两个几乎相同的过程 What we have here are two procedures that are almost identical. -395 +391 00:19:32,110 --> 00:19:34,580 基本上没有区别 Almost no difference between them at all. -396 +392 00:19:35,080 --> 00:19:37,260 只有几个字符不同 It's a few characters that distinguish them. -397 +393 00:19:38,610 --> 00:19:41,010 -这两个过程表示的是两种不同的求两数之和的方法 +这两个过程表示的是 两种不同的求两数之和的方法 These are two ways of adding numbers together. -398 +394 00:19:42,110 --> 00:19:43,010 第一个 The first one, -399 +395 00:19:43,610 --> 00:19:45,700 看这里 which you see here, -400 +396 00:19:46,170 --> 00:19:48,900 是求两数之和 the first one is the sum of two numbers -401 +397 00:19:48,900 --> 00:19:49,940 和我们上节课讲到的一样 -- just what we did before-- -402 +398 00:19:50,440 --> 00:19:51,650 如果第一个参数是0 is if the first one is 0, -403 +399 00:19:51,650 --> 00:19:52,960 第二个参数就是答案 it's the answer of the second one. -404 +400 00:19:53,210 --> 00:19:53,840 否则 Otherwise, -405 +401 00:19:53,840 --> 00:19:55,840 就是第一个参数减1 it's the sum of the decrement of the first -406 +402 00:19:55,840 --> 00:19:56,940 和第二个参数加1的和 and the increment of the second. -407 +403 00:19:57,610 --> 00:20:03,360 你可以这么考虑 有两堆弹珠 And you may think of that as having two piles,having two piles. -408 +404 00:20:04,090 --> 00:20:07,180 把这两堆加在一块儿形成新的一堆的办法就是 And the way I'm adding these numbers together to make a third pile is by -409 +405 00:20:07,180 --> 00:20:09,180 把弹珠从一堆挪到另一堆去 moving marbles from one to the other. -410 +406 00:20:10,130 --> 00:20:11,070 仅此而已 Nothing more than that. -411 +407 00:20:11,290 --> 00:20:12,850 最终 有一堆会被搬空 And eventually,when I run out of one -412 +408 00:20:12,850 --> 00:20:13,900 另一堆就是加和的结果了 then the other is the sum. -413 +409 00:20:15,330 --> 00:20:19,240 然而 第二个过程不是这样做的 However,the second procedure here doesn't do it that way. -414 +410 00:20:20,340 --> 00:20:22,380 它是这样做的 如果第一个参数是0 It says if the first number is 0, -415 +411 00:20:22,380 --> 00:20:23,550 答案就是第二个 then the answer is the second. -416 +412 00:20:24,020 --> 00:20:24,530 否则 Otherwise, -417 +413 00:20:24,530 --> 00:20:26,990 答案就是 第一个参数减1 it's the increment of the sum -418 +414 00:20:26,990 --> 00:20:28,900 和第二个参数的和 of the decrement of the first number -419 +415 00:20:28,900 --> 00:20:29,560 然后再加1 and the second. -420 +416 00:20:31,020 --> 00:20:32,010 这种方式就是 So what this says is -421 +417 00:20:32,020 --> 00:20:36,490 把第一个参数减1和第二个数加起来 add together,add together the decrement of the first number and the second -422 +418 00:20:36,490 --> 00:20:38,320 毫无疑问 简化了问题 -- a simpler problem,no doubt-- -423 +419 00:20:38,320 --> 00:20:41,540 然后把这个和再加1 and then change that result to increment it. -424 +420 00:20:42,780 --> 00:20:46,060 这就意味着如果你用两堆弹珠的方式来思考 And so this means that if you think about this in terms of piles, -425 +421 00:20:46,480 --> 00:20:50,100 就是说 我左手右手各有一堆弹珠 it means I'm holding in my hand the things to be added later. -426 +422 00:20:51,450 --> 00:20:53,020 然后我要把它们加在一起 And then I'm going to add them in. -427 +423 00:20:53,610 --> 00:20:56,060 然后我慢慢地把其中一堆搬空 As I slowly decrease one pile to 0, -428 +424 00:20:56,430 --> 00:20:57,710 然后把这些拿出来的弹珠 I've got what's left here, -429 +425 00:20:57,710 --> 00:20:58,890 再放回另一堆弹珠中 and then I'm going to add them back. -430 +426 00:20:59,960 --> 00:21:01,170 两种不同的加法 Two different ways of adding. -431 +427 00:21:02,160 --> 00:21:04,200 关于这两个程序 有趣的是 The nice thing about these two programs -432 +428 00:21:04,460 --> 00:21:05,850 它们几乎一样 is that they're almost identical. -433 +429 00:21:06,450 --> 00:21:08,260 唯一的区别是把加1这个过程放哪儿 The only thing is where I put the increment. -434 +430 00:21:09,140 --> 00:21:10,570 把几个字符挪挪地方 A couple of characters moved around. -435 +431 00:21:11,490 --> 00:21:14,280 现在我想弄明白 Now I want to understand,I want to understand the kind of behavior -436 +432 00:21:14,760 --> 00:21:16,820 我们会从这两个程序中看到怎样的行为 we're going to get from each of these programs. -437 +433 00:21:17,330 --> 00:21:19,040 加深大家对这个问题的印象 Just to get them firmly in your mind -438 +434 00:21:19,280 --> 00:21:21,680 我通常不想这么谨慎 -- I usually don't want to be this careful-- -439 +435 00:21:21,680 --> 00:21:23,610 但是还是要加深大家的印象 but just to get them firmly in your mind, -440 +436 00:21:23,610 --> 00:21:25,330 我在黑板上把程序再写一遍 I'm going to write the programs again on the blackboard, -441 +437 00:21:25,330 --> 00:21:27,100 然后我会演化出一个计算过程 and then I'm going to evolve a process. -442 +438 00:21:27,770 --> 00:21:29,100 大家都注意观察 看看是怎么回事 And you're going to see what happens. -443 +439 00:21:29,460 --> 00:21:31,340 看看过程演化是怎样的“形状” We're going to look at the shape of the process -444 +440 00:21:31,340 --> 00:21:32,740 这是程序导致的一个结果 as a consequence of the program. -445 +441 00:21:34,040 --> 00:21:35,730 我们从这个程序开始 So the program we started with is this: -446 +442 00:21:40,510 --> 00:21:44,160 求x与y的和 the sum of x and y says -447 +443 00:21:44,160 --> 00:21:49,410 如果x等于0 if x is 0, -448 +444 00:21:49,410 --> 00:21:50,670 那么答案就是y then the result is y. -449 +445 00:21:50,670 --> 00:21:51,520 否则 Otherwise, -450 +446 00:21:51,520 --> 00:21:58,220 答案就是x减1和y加1的和 it's the sum of the decrement of x and the increment of y. -451 +447 00:22:01,260 --> 00:22:05,490 现在 假设我们要求3与4的和 Now,supposing we wish to do this addition of 3 and 4, -452 +448 00:22:06,610 --> 00:22:09,120 3与4的和 the sum of 3 and 4, -453 +449 00:22:09,530 --> 00:22:10,300 接下来呢 well,what is that? -454 +450 00:22:10,530 --> 00:22:13,690 我要代换这两个参数 It says that I have to substitute the arguments -455 +451 00:22:14,030 --> 00:22:16,040 代换过程定义中的形式参数 for the formal parameters in the body. -456 +452 00:22:17,380 --> 00:22:18,920 我在脑海中想像这个过程 I'm doing that in my mind. -457 +453 00:22:19,560 --> 00:22:20,660 然后说 啊 这样 And I say,oh,yes, -458 +454 00:22:20,660 --> 00:22:21,920 3代换了x 3 is substituted for x, -459 +455 00:22:21,920 --> 00:22:23,280 但3不是0 but 3 is not 0, -460 +456 00:22:24,650 --> 00:22:27,000 于是直接进入这一部分 so I'm going to go directly to this part -461 +457 00:22:27,000 --> 00:22:30,400 然后只用关心这一部分就行 and write down the simplified consequent here. -462 +458 00:22:30,400 --> 00:22:31,980 因为我只关心加法过程的行为 Because I'm really interested in the behavior of addition. -463 +459 00:22:33,100 --> 00:22:33,840 好了 接下来 Well,what is that? -464 +460 00:22:34,080 --> 00:22:37,760 就变成了求2和5的过程 That therefore turns into the sum of 2 and 5. -465 +461 00:22:38,400 --> 00:22:38,610 换句话说 In other words, -466 +462 00:22:38,610 --> 00:22:40,270 我把这个问题归约成这个问题 I've reduced this problem to this problem. -467 +463 00:22:41,410 --> 00:22:46,280 然后我又把这个问题化归约成求1和6的和 Then I reduce this problem to the sum of 1 and 6, -468 +464 00:22:47,220 --> 00:22:49,340 再接着归约 and then,going around again once, -469 +465 00:22:49,600 --> 00:22:52,480 就有了0和7的和 I get the sum of 0 and 7. -470 +466 00:22:52,970 --> 00:22:54,930 于是x就等于0了 And that's one where x equals 0 -471 +467 00:22:54,930 --> 00:22:56,220 答案是y so the result is y, -472 +468 00:22:56,220 --> 00:22:58,220 也就是7 and so I write down here a 7. -473 +469 00:22:59,920 --> 00:23:02,780 这就是这个程序把3和4相加 So this is the behavior of the process evolved -474 +470 00:23:02,780 --> 00:23:05,360 而演化出的计算过程的行为 by trying to add together 3 and 4 with this program. -475 +471 00:23:07,390 --> 00:23:08,680 另一个程序 For the other program, -476 +472 00:23:09,630 --> 00:23:10,610 写在这边 which is over here, -477 +473 00:23:17,800 --> 00:23:22,250 如此定义x与y的和 I will define the sum of x and y. -478 +474 00:23:23,360 --> 00:23:23,980 怎么定义的呢 And what is it? -479 +475 00:23:26,480 --> 00:23:29,170 如果x是0 If x is 0, -480 +476 00:23:29,620 --> 00:23:30,810 答案就是y then the result is y -481 +477 00:23:30,810 --> 00:23:31,630 这和上一个程序一样 -- almost the same-- -482 +478 00:23:31,630 --> 00:23:34,800 否则就是x减1与y的和 otherwise the increment of the sum -483 +479 00:23:35,550 --> 00:23:40,260 再加上1 of the decrement of x and y. -484 +480 00:23:47,370 --> 00:23:54,640 写错了 黑板不能自动匹配括号 No.I don't have my balancer in front of me. -485 +481 00:23:56,080 --> 00:23:57,330 好了 接下来 OK,well,let's do it now. -486 +482 00:23:58,800 --> 00:24:00,100 求3与4的和 The sum of 3 and 4. -487 +483 00:24:01,160 --> 00:24:02,690 其实这样更有趣 来看 Well,this is actually a little more interesting. -488 +484 00:24:03,310 --> 00:24:05,630 3当然不等于0 和之前一样 Of course,3 is not 0 as before, -489 +485 00:24:06,160 --> 00:24:12,130 于是答案就是 x减1与y的和再加1 so that results in the increment of the sum of the decrement of x, -490 +486 00:24:12,130 --> 00:24:14,260 也就是2与4的和 which is 2 and 4, -491 +487 00:24:15,800 --> 00:24:22,650 接着变成 1与 which is the increment of the sum of 1 and-- -492 +488 00:24:22,850 --> 00:24:24,990 噢 弄错了 加1再加1 whoops: the increment of the increment. -493 +489 00:24:25,650 --> 00:24:29,170 现在要做的是计算这一部分 What I have to do now is compute what this means. -494 +490 00:24:29,470 --> 00:24:30,590 我要对这一部分求值 I have to evaluate this. -495 +491 00:24:30,810 --> 00:24:31,720 也就是 Or what that is, -496 +492 00:24:31,870 --> 00:24:34,720 用2和4代换x和y的结果 the result of substituting 2 and 4 for x and y here. -497 +493 00:24:35,200 --> 00:24:40,840 变成了1与4的和再加1 But that is the increment of the sum of 1 and 4, -498 +494 00:24:43,490 --> 00:24:45,920 现在我得展开这部份 which is-- well,now I have to expand this. -499 +495 00:24:47,160 --> 00:24:51,230 啊 然后就变成了0与4的和加1加1 Ah,but that's the increment of the increment -500 +496 00:24:53,520 --> 00:24:56,130 再加1 of the increment of the sum of 0 and 4. -501 +497 00:24:59,390 --> 00:25:02,380 现在来看看都能做些什么 Ah,but now I'm beginning to find things I can do. -502 +498 00:25:02,860 --> 00:25:06,850 加1加1再加1 The increment of the increment of the increment of-- -503 +499 00:25:06,850 --> 00:25:08,590 0和4的和是4 well,the sum of 0 and 4 is 4. -504 +500 00:25:12,030 --> 00:25:13,610 4加1是5 The increment of 4 is 5. -505 +501 00:25:14,560 --> 00:25:18,760 这就变成了5加1再加1 So this is the increment of the increment of 5, -506 +502 00:25:20,180 --> 00:25:23,200 也就是6再加1 which is the increment of 6, -507 +503 00:25:23,900 --> 00:25:24,970 最后答案就是7 which is 7. -508 +504 00:25:26,100 --> 00:25:28,060 两种不同的求和方法 Two different ways of computing sums. -509 +505 00:25:29,540 --> 00:25:30,330 现在来看看 Now,let's see. -510 +506 00:25:31,260 --> 00:25:33,260 这两种计算过程有不同的“形状” These processes have very different shapes. -511 +507 00:25:33,920 --> 00:25:35,400 我想让各位去感受这些“形状” I want you to feel these shapes. -512 +508 00:25:36,350 --> 00:25:38,370 这种感觉很关键 It's the feeling for the shapes that matters. -513 +509 00:25:40,340 --> 00:25:42,030 从这里我们能发现什么 What's some things we can see about this? -514 +510 00:25:42,660 --> 00:25:44,120 不知怎么回事 这个有点“直” Well somehow this is sort of straight. -515 +511 00:25:45,250 --> 00:25:46,600 就这样一路直着下来 It goes this way-- straight. -516 +512 00:25:47,340 --> 00:25:51,930 右边界并没有特别的变化 This right edge doesn't vary particularly in size. -517 +513 00:25:53,770 --> 00:25:54,700 而这边这个呢 Whereas this one, -518 +514 00:25:54,990 --> 00:25:58,320 右边界先变大又变小 I see that this thing gets bigger and then it gets smaller. -519 +515 00:26:00,910 --> 00:26:02,620 我还不知道这意味着什么 So I don't know what that means yet, -520 +516 00:26:02,620 --> 00:26:03,630 我们看到的是什么 but what are we seeing? -521 +517 00:26:03,630 --> 00:26:05,900 是不知怎么回事 We're seeing here that somehow -522 +518 00:26:07,170 --> 00:26:10,940 这些加1的过程先展开后收缩 these increments are expanding out and then contracting back. -523 +519 00:26:12,760 --> 00:26:14,570 我构造出一个推迟进行的操作的链条 I'm building up a bunch of them to do later. -524 +520 00:26:16,090 --> 00:26:17,070 我不能马上求值 I can't do them now. -525 +521 00:26:18,720 --> 00:26:19,860 有的过程需要推迟 There's things to be deferred. -526 +522 00:26:21,400 --> 00:26:22,160 好了 我们继续 Well,let's see, -527 +523 00:26:22,670 --> 00:26:24,290 我可以想象一个抽象的计算机 I can imagine an abstract machine. -528 +524 00:26:24,290 --> 00:26:25,480 也许能制造一些实际的计算机 There's some physical machine, -529 +525 00:26:25,480 --> 00:26:26,840 能像我这样求值 perhaps,that could be built to do it, -530 +526 00:26:26,840 --> 00:26:30,010 能像我演示给大家的一样执行程序 which,in fact,executes these programs exactly as I tell you, -531 +527 00:26:30,010 --> 00:26:32,850 像这样代换字符串 substituting character strings in like this. -532 +528 00:26:34,210 --> 00:26:35,080 这样的计算机 Such a machine, -533 +529 00:26:35,780 --> 00:26:38,750 计算中的步骤与计算所花费的时间近似相等 the number of such steps is an approximation of the amount of time it takes. -534 +530 00:26:39,730 --> 00:26:40,830 用纵轴表示时间 So this way is time. -535 +531 00:26:43,630 --> 00:26:46,420 横轴的宽度表示 And the width of the thing is -536 +532 00:26:46,970 --> 00:26:49,700 进行计算需要保存的信息量 how much I have to remember in order to continue the process. -537 +533 00:26:50,110 --> 00:26:51,150 也就是表示进行计算所需要的空间 And this much is space. -538 +534 00:26:53,540 --> 00:26:56,180 我们看到的是一个计算过程 And what we see here is a process -539 +535 00:26:57,090 --> 00:27:00,910 其时间消耗正比于参数x that takes a time which is proportional to the argument x. -540 +536 00:27:02,420 --> 00:27:04,350 因为如果我让x增大一倍 Because if I made x larger by 1, -541 +537 00:27:04,350 --> 00:27:05,520 就得把这个表示时间的纵轴再加长一倍 then I'd had an extra line. -542 +538 00:27:08,480 --> 00:27:12,260 于是这就是个计算过程 它的空间 抱歉 时间 So this is a process which is space-- sorry-- time. -543 +539 00:27:14,520 --> 00:27:19,160 这个计算过程消耗的时间为O(x) The time of this process is what we say order of x. -544 +540 00:27:20,540 --> 00:27:24,890 也就是正比于x That means it is proportional to x by some constant of proportionality -545 +541 00:27:24,890 --> 00:27:27,150 不要刻意追究这个常比例系数是多少 and I'm not particularly interested in what the constant is. -546 +542 00:27:28,360 --> 00:27:29,530 我们还会发现 The other thing we see here -547 +543 00:27:29,540 --> 00:27:32,280 这个计算过程消耗的空间是固定不变的 is that the amount of space this takes up is constant. -548 +544 00:27:33,370 --> 00:27:34,530 即与1成正比 it's proportional to 1. -549 +545 00:27:35,200 --> 00:27:40,420 于是这个计算过程的空间复杂度就是O(1)的 So the space complexity of this is order of 1. -550 +546 00:27:41,710 --> 00:27:43,360 对这样的计算过程我们有个名字 We have a name for such a process. -551 +547 00:27:43,870 --> 00:27:45,810 叫迭代计算过程 Such a process is called an iteration. -552 +548 00:27:50,800 --> 00:27:51,900 关键的地方 And what matters here -553 +549 00:27:51,900 --> 00:27:55,820 不是什么我设计的 is not that some particular machine I designed here -554 +550 00:27:56,450 --> 00:27:57,960 之前提到的 and talked to you about -555 +551 00:27:57,960 --> 00:27:59,320 代换字符计算机 and called a substitution machine -556 +552 00:27:59,320 --> 00:28:00,720 或者 代换模型 or whatever-- substitution model-- -557 +553 00:28:01,060 --> 00:28:03,880 怎么设法在常数空间复杂度内完成计算过程 managed to do this in constant space. -558 +554 00:28:04,220 --> 00:28:06,090 关键的地方是这给出了一个界限 What really matters is this tells us a bound. -559 +555 00:28:06,750 --> 00:28:09,200 -任何计算机都应该在常数空间复杂度内完成计算过程 +任何计算机 都应该在常数空间复杂度内完成计算过程 Any machine could do this in constant space. -560 +556 00:28:09,360 --> 00:28:12,030 这个过程代表的算法 This algorithm represented by this procedure -561 +557 00:28:12,440 --> 00:28:14,030 在常数空间复杂度内是可执行的 is executable in constant space. -562 +558 00:28:14,840 --> 00:28:17,330 当然了这种说法忽略了一些细节 Now,of course,the model is ignoring some things, -563 +559 00:28:17,630 --> 00:28:18,640 却量化了一些东西 standard sorts of things. -564 +560 00:28:18,640 --> 00:28:21,980 忽略了比如数值越大越消耗空间等等 Like numbers that are bigger take up more space and so on. -565 +561 00:28:21,980 --> 00:28:24,010 但那确实是抽象过程中我要忽略的一个层次 But that's a level of abstraction at which I'm cutting off. -566 +562 00:28:24,010 --> 00:28:24,920 如何表示数值 How do you represent numbers? -567 +563 00:28:24,920 --> 00:28:26,640 -我假定所有的数值都占用同样大小的存储空间 +我假定所有的数值都具有同样的字长 I'm considering every number to be the same size. -568 +564 00:28:27,710 --> 00:28:31,420 -事实上数值的增大和它们占用的存储空间的增加都非常缓慢 +事实上数值的字长的增长 和它们占用的存储空间的增长都非常缓慢 And numbers grow slowly for the amount of space they take up and their size. -569 +565 00:28:33,860 --> 00:28:36,680 而这个算法的复杂度就不同了 Now,this algorithm is different in its complexity. -570 +566 00:28:37,390 --> 00:28:38,680 我们看到 As we can see here, -571 +567 00:28:40,370 --> 00:28:43,210 这个算法时间复杂度 this algorithm has a time complexity -572 +568 00:28:44,680 --> 00:28:48,560 也是正比于输入参数x的 which is also proportional to the input argument x. -573 +569 00:28:49,130 --> 00:28:51,610 因为如果我把这个3再加1 That's because if I were to add 1 to 3, -574 +570 00:28:51,610 --> 00:28:53,320 如果我把问题规模扩大 if I made a larger problem -575 +571 00:28:53,320 --> 00:28:54,980 扩大了1 which is larger by 1 here, -576 +572 00:28:54,980 --> 00:28:56,170 那我就要在上面再加一行 then I'd add a line at the top -577 +573 00:28:56,170 --> 00:28:57,290 在底下再加一行 and I'd add a line at the bottom. -578 +574 00:29:00,280 --> 00:29:01,980 而且事实上这是个常量 And the fact that it's a constant amount, -579 +575 00:29:01,980 --> 00:29:04,050 就像这个计算过程的步骤数是那个的两倍 like this is twice as many lines as that -580 +576 00:29:04,290 --> 00:29:06,600 我不会关心这个层次的细节 也不会关心那个常量 is not interesting at the level of detail I'm talking about right now. -581 +577 00:29:07,730 --> 00:29:12,020 所以这个计算过程的时间复杂度也是O(x) So this is a time complexity order of the input argument x. -582 +578 00:29:12,640 --> 00:29:13,950 而空间复杂度呢 And space complexity, -583 +579 00:29:15,880 --> 00:29:17,060 这个比较有趣 well,this is more interesting. -584 +580 00:29:17,390 --> 00:29:19,970 有一些东西是固定不变的 I happen to have some overhead, -585 +581 00:29:20,180 --> 00:29:21,000 比如这些 which you see over here, -586 +582 00:29:21,000 --> 00:29:22,720 是近似保持不变的 which is constant approximately. -587 +583 00:29:23,290 --> 00:29:24,090 是常量的 Constant overhead. -588 +584 00:29:24,340 --> 00:29:27,210 但也有一些增加和减少 But then I have something which increases and decreases -589 +585 00:29:27,210 --> 00:29:29,460 是正比于参数x的 and is proportional to the input argument x. -590 +586 00:29:29,710 --> 00:29:31,060 输入参数x是3 The input argument x is 3. -591 +587 00:29:31,060 --> 00:29:34,690 也就是这里为何有3个推迟操作的加1过程 That's why there are three deferred increments sitting around here. -592 +588 00:29:36,360 --> 00:29:36,650 发现了么 See? -593 +589 00:29:37,400 --> 00:29:40,060 于是这个计算过程的空间复杂度也是O(x)的 So the space complexity here is also order x. -594 +590 00:29:41,400 --> 00:29:43,150 这种计算过程 And this kind of process, -595 +591 00:29:43,410 --> 00:29:44,640 给它取个名字 named for the kind of process, -596 +592 00:29:44,640 --> 00:29:45,850 是递归计算过程 this is a recursion. -597 +593 00:29:50,410 --> 00:29:51,280 更准确地来说 A linear recursion -598 +594 00:29:51,280 --> 00:29:51,870 是线性递归计算过程 I will call it. -599 +595 00:29:54,600 --> 00:29:56,680 因为它的时间和空间复杂度 because of the fact that it's proportional -600 +596 00:29:56,680 --> 00:29:58,520 都是正比于输入参数的 to the input argument in both time and space. -601 +597 00:30:01,360 --> 00:30:03,070 那么这个就可以称为一个线性迭代过程 This could have been a linear iteration. -602 +598 00:30:13,630 --> 00:30:15,070 那么 这两种计算过程的本质是什么 So then what's the essence of this matter? -603 +599 00:30:16,190 --> 00:30:17,780 不太容易看出来 This matter isn't so obvious. -604 +600 00:30:18,750 --> 00:30:19,690 可能还有别的模型 Maybe there are other models -605 +601 00:30:19,690 --> 00:30:21,630 可以用来描述 by which we can describe the differences -606 +602 00:30:21,630 --> 00:30:23,390 迭代和递归计算过程的不同之处 between iterative and recursive processes. -607 +603 00:30:23,390 --> 00:30:24,410 但这就有些困难了 Because this is hard now. -608 +604 00:30:25,150 --> 00:30:25,390 记住 Remember, -609 +605 00:30:25,390 --> 00:30:27,140 这两种计算过程都是递归定义的 we have-- those are both recursive definitions. -610 +606 00:30:28,010 --> 00:30:28,940 我们看到的是 What we're seeing there -611 +607 00:30:29,980 --> 00:30:31,380 两种过程在定义上都使用了递归 are both recursive definitions, -612 +608 00:30:31,680 --> 00:30:34,370 也就是两个过程的定义中都引用了该过程本身 definitions that refer to the thing being defined in the definition. -613 +609 00:30:35,050 --> 00:30:36,620 但是它们却产生了不同“形状”的计算过程 But they lead to different shape processes. -614 +610 00:30:37,440 --> 00:30:41,250 过程定义是递归的 There's nothing special about the fact -615 +611 00:30:41,250 --> 00:30:42,540 而产生的计算过程也是递归的 that the definition is recursive -616 +612 00:30:42,540 --> 00:30:44,170 这没什么特别的 that leads to a recursive process. -617 +613 00:30:45,770 --> 00:30:49,490 好了 来看看另一种有趣的解释 OK.Let's think of another model. -618 +614 00:30:49,850 --> 00:30:51,490 我会和大家谈谈 官僚主义 I'm going to talk to you about bureaucracy. -619 +615 00:30:52,620 --> 00:30:54,290 官僚主义 挺有趣儿的 Bureaucracy is sort of interesting. -620 +616 00:30:54,650 --> 00:30:58,890 在幻灯片上我们看到有关迭代计算过程的解释 Here we see on a slide an iteration. -621 +617 00:31:00,120 --> 00:31:02,220 迭代是一种挺好玩儿的计算过程 An iteration is sort of a fun kind of process. -622 +618 00:31:03,820 --> 00:31:06,210 想象一下有个叫GJS的家伙 Imagine that there's a fellow called GJS -623 +619 00:31:06,330 --> 00:31:07,160 也就是我 -- that stands for me-- -624 +620 00:31:08,400 --> 00:31:09,260 遇到了一个问题 and he's got a problem: -625 +621 00:31:09,260 --> 00:31:10,770 他想求3与4的和 he wants to add together 3 and 4. -626 +622 00:31:13,010 --> 00:31:14,820 这个家伙想把3和4加一块儿 This fella here wants to add together 3 and 4. -627 +623 00:31:16,260 --> 00:31:17,170 他打算这么办 Well,the way he's going to do it -628 +624 00:31:17,170 --> 00:31:17,900 他挺懒的 -- he's lazy-- -629 +625 00:31:18,260 --> 00:31:20,180 他想找别人帮忙 is he's going to find somebody else to help him do it. -630 +626 00:31:21,060 --> 00:31:22,210 他是这样找人帮忙的 The way he finds someone else to-- -631 +627 00:31:22,210 --> 00:31:24,210 他找了个人帮忙 对他说 he finds someone else to help him do it and says, -632 +628 00:31:24,210 --> 00:31:25,970 算出3和4的和 well,give me the answer to 3 and 4 -633 +629 00:31:26,010 --> 00:31:27,260 然后把答案返回给我 and return the result to me. -634 +630 00:31:28,010 --> 00:31:30,180 他写了一张小纸条递给这个人然后说 He makes a little piece of paper and says, -635 +631 00:31:30,460 --> 00:31:31,770 给 这是一张纸条 here,here's a piece of paper-- -636 +632 00:31:31,770 --> 00:31:32,620 你去解决这个问题 you go ahead and solve this problem -637 +633 00:31:32,620 --> 00:31:33,800 然后把答案返回给我 and give the result back to me. -638 +634 00:31:35,480 --> 00:31:36,000 而这个找来帮忙的家伙呢 And this guy, -639 +635 00:31:36,000 --> 00:31:36,930 也很懒 of course,is lazy,too. -640 +636 00:31:38,010 --> 00:31:39,630 他可不想再看见这张小纸条了 He doesn't want to see this piece of paper again. -641 +637 00:31:40,960 --> 00:31:42,210 他说 啊 好吧 He says,oh,yes, -642 +638 00:31:44,450 --> 00:31:45,480 然后出了个新问题 produce a new problem -643 +639 00:31:45,480 --> 00:31:46,940 是求2和5的和 which is the sum of 2 ad 5 -644 +640 00:31:46,940 --> 00:31:49,120 并把答案返回给GJS and return the result back to GJS. -645 +641 00:31:50,080 --> 00:31:51,210 我不想再看见这张纸条了 I don't want to see it again. -646 +642 00:31:52,000 --> 00:31:53,700 这个家伙不想再看见这张纸条了 This guy does not want to see this piece of paper. -647 +643 00:31:55,860 --> 00:31:59,940 于是就出了一个新问题 And then this fellow makes a new problem, -648 +644 00:32:00,260 --> 00:32:02,030 也就是求1与6的和的问题 which is the addition of the sum of 1 and 6, -649 +645 00:32:02,030 --> 00:32:04,260 他又把纸条传给了这个家伙 并说到 and he give it to this fella and says, -650 +646 00:32:04,260 --> 00:32:06,820 算出答案然后把答案告诉GJS produce that answer and returned it to GJS. -651 +647 00:32:08,100 --> 00:32:09,740 那个家伙又接着出了个新问题 And that produces a problem, -652 +648 00:32:09,740 --> 00:32:11,310 求0与7的和 which is to add together 0 and 7 -653 +649 00:32:11,310 --> 00:32:13,090 然后把答案返回给GJS and give the result to GJS. -654 +650 00:32:13,810 --> 00:32:15,010 最后这个家伙直接说 This fella finally just says, -655 +651 00:32:15,010 --> 00:32:16,120 啊 好吧 答案是7 oh,yeah,the answer is 7, -656 +652 00:32:16,120 --> 00:32:17,340 然后把答案返回给了GJS and sends it back to GJS. -657 +653 00:32:18,140 --> 00:32:19,170 迭代计算过程就是这样的 That's what an iteration is. -658 +654 00:32:19,870 --> 00:32:20,800 相对而言 By contrast, -659 +655 00:32:21,260 --> 00:32:23,570 递归计算过程就有些不同了 a recursion is a slightly different kind of process. -660 +656 00:32:26,060 --> 00:32:27,610 它更加“官僚主义” This one involves more bureaucracy. -661 +657 00:32:28,140 --> 00:32:29,380 它使得更多的人变得忙碌 It keeps more people busy. -662 +658 00:32:30,190 --> 00:32:31,900 更多的人被雇佣 It keeps more people employed. -663 +659 00:32:32,400 --> 00:32:34,860 当然了能增加就业也许更好 Perhaps it's better for that reason. -664 +660 00:32:35,600 --> 00:32:36,160 请看幻灯片 But here it is: -665 +661 00:32:36,160 --> 00:32:38,160 我想要知道3与4的和 I want the answer to the problem 3 and 4. -666 +662 00:32:38,520 --> 00:32:39,900 于是写了张纸条 说到 So I make a piece of paper that says, -667 +663 00:32:39,900 --> 00:32:40,990 把答案返回给我 give the result back to me. -668 +664 00:32:42,840 --> 00:32:43,610 然后我把纸条给了这个家伙 Give it to this fella. -669 +665 00:32:44,330 --> 00:32:45,080 这个家伙说 This fellow says, -670 +666 00:32:45,220 --> 00:32:45,860 好吧我会记得 oh,yes,I will remember -671 +667 00:32:46,440 --> 00:32:48,170 一会儿要加1 that I have to add later, -672 +668 00:32:49,040 --> 00:32:51,560 而我又得知道2加4的和是多少 and I want to get the answer the problem 2 plus 4, -673 +669 00:32:52,020 --> 00:32:54,740 于是把这个问题给了Harry give that one to Harry, -674 +670 00:32:54,740 --> 00:32:57,040 然后把结果返回给我 Joe and have the results sent back to me-- I'm Joe. -675 +671 00:32:58,520 --> 00:33:00,260 一旦Harry返回了答案 When the answer comes back from Harry, -676 +672 00:33:00,260 --> 00:33:01,050 也就是6 which is a 6, -677 +673 00:33:01,300 --> 00:33:02,500 我就会把答案加1 I will then do the increment -678 +674 00:33:04,120 --> 00:33:05,650 然后把结果返回给GJS and give that 7 back to GJS. -679 +675 00:33:07,290 --> 00:33:09,050 大家可以发现 在递归计算过程中 需要保留的纸条 So there are more pieces of paper outstanding -680 +676 00:33:09,660 --> 00:33:12,290 比迭代计算过程的多 in the recursive process than the iteration. -681 +677 00:33:16,540 --> 00:33:18,660 还有一种方法可以用来理解迭代计算过程 There's another way to think about what an iteration is -682 +678 00:33:19,300 --> 00:33:20,990 以及迭代计算过程和递归计算过程的不同之处 and the difference between an iteration and a recursion. -683 +679 00:33:21,420 --> 00:33:22,760 问题的关键在于 You see,the question is, -684 +680 00:33:22,760 --> 00:33:24,780 有多少东西是不可见的 how much stuff is under the table? -685 +681 00:33:26,000 --> 00:33:27,550 如果我要停止 something,if I were to stop-- -686 +682 00:33:28,360 --> 00:33:32,130 假设我现在要关闭计算机 supposing I were to kill this computer right now,OK? -687 +683 00:33:32,130 --> 00:33:35,180 此时我丢失了事务的状态 And at this point I lose the state of affairs, -688 +684 00:33:36,680 --> 00:33:38,690 但我能接着进行运算 well,I could continue the computation from this point -689 +685 00:33:39,390 --> 00:33:44,290 因为继续运算所需要的所有信息都在参数变量中 cause everything I need to continue the computation is in the variables -690 +686 00:33:44,290 --> 00:33:48,540 即程序员编写过程时定义的变量 that were defined in the procedure that the programmer wrote for me. -691 +687 00:33:49,440 --> 00:33:53,810 -迭代计算过程能用明确的变量保存计算过程中的状态 +迭代计算过程能用明确的变量 保存计算过程中的状态 An iteration is a system that has all of its state in explicit variables. -692 +688 00:33:56,670 --> 00:34:00,020 这一点递归计算过程就不同 Whereas the recursion is not quite the same. -693 +689 00:34:00,940 --> 00:34:04,040 如果我弄丢了这堆垃圾 If I were to lose this pile of junk over here -694 +690 00:34:04,880 --> 00:34:06,850 那就只剩下求1与4的和了 and all I was left with was the sum of 1 and 4, -695 +691 00:34:07,000 --> 00:34:09,530 这些信息不足以继续 that's not enough information to continue the process -696 +692 00:34:09,530 --> 00:34:11,870 从初始的求3与4的和的问题 of computing out the 7 from the original problem -697 +693 00:34:11,870 --> 00:34:13,130 求出7的计算过程 of adding together 3 of 4. -698 +694 00:34:14,580 --> 00:34:18,350 除了保存在过程中的 Besides the information that's in the variables -699 +695 00:34:19,290 --> 00:34:21,420 形式参数变量中的信息 of the formal parameters of the program, -700 +696 00:34:22,830 --> 00:34:25,920 计算机还保存了一些不可见的信息 there is also information under the table belonging to the computer, -701 +697 00:34:26,500 --> 00:34:28,280 也就是都有哪些过程被推迟计算了 which is what things have been deferred for later. -702 +698 00:34:30,100 --> 00:34:30,680 当然了 And,of course, -703 +699 00:34:30,930 --> 00:34:32,680 有个实际的比喻 there's a physical analogy to this, -704 +700 00:34:33,220 --> 00:34:37,080 比如微分方程 which is in differential equations,for example, -705 +701 00:34:37,740 --> 00:34:40,560 当我们说到画一个圆的时候 when we talk about something like drawing a circle. -706 +702 00:34:41,760 --> 00:34:42,990 试图画一个圆 Try to draw a circle, -707 +703 00:34:42,990 --> 00:34:46,030 你从一个微分方程中解得它 you make that out of a differential equation -708 +704 00:34:46,030 --> 00:34:52,440 即状态的改变是我当前状态的一个函数 which says the change in my state as a function of my current state. -709 +705 00:34:52,890 --> 00:34:56,920 于是如果我当前的状态由y和x的特定值表征 So if my current state corresponds to particular values of y and x -710 +706 00:34:57,630 --> 00:35:01,100 那么我就能求出一个导数 表征状态如何改变 then I can compute from them a derivative which says how the state must change. -711 +707 00:35:03,150 --> 00:35:03,570 并且 实际上 And,in fact -712 +708 00:35:03,580 --> 00:35:05,630 大家可以看出这是一个圆 this,you can see this was a circle -713 +709 00:35:06,010 --> 00:35:09,760 因为如果碰巧 because if I happen to be, -714 +710 00:35:09,760 --> 00:35:11,100 在这个点 say,at this place over here, -715 +711 00:35:11,770 --> 00:35:15,500 比如在(1,0)这个点 at (1,0),for example -716 +712 00:35:15,530 --> 00:35:16,210 在这个图像上 on this graph, -717 +713 00:35:16,730 --> 00:35:22,180 那就是说y的导数是x then it means that the derivative of y is x, -718 +714 00:35:22,240 --> 00:35:23,160 我们也看见了 which we see over here. -719 +715 00:35:23,160 --> 00:35:24,040 是1 That's 1, -720 +716 00:35:24,040 --> 00:35:25,100 接着向上走 so I'm going up. -721 +717 00:35:25,780 --> 00:35:28,640 x的导数是-y And the derivative of x is minus y, -722 +718 00:35:28,640 --> 00:35:29,700 意味着要走回来 which means I'm going backwards. -723 +719 00:35:30,860 --> 00:35:32,520 实际在这点什么也没做 I'm actually doing nothing at this point, -724 +720 00:35:32,680 --> 00:35:36,220 然后开始走回来使y增加 then I start going backwards as y increases. -725 +721 00:35:37,720 --> 00:35:39,440 这样就画出了一个圆 So that's how you make a circle. -726 +722 00:35:39,720 --> 00:35:41,940 有趣的是 And the interesting thing to see is -727 +723 00:35:41,940 --> 00:35:44,900 这儿有一个程序能用这种方法画出一个圆 a little program that will draw a circle by this method. -728 +724 00:35:44,940 --> 00:35:46,180 实际上 不会画一出一个圆 Actually,this won't draw a circle -729 +725 00:35:46,200 --> 00:35:48,320 因为这个程序实际是一个积分器 because it's a forward oil or integrator -730 +726 00:35:48,320 --> 00:35:50,480 并且最终会陷入死循环 and will eventually spiral out and all that. -731 +727 00:35:50,480 --> 00:35:52,900 但陷入死循环前还是能画出一个圆的 But it'll draw a circle for a while before it starts spiraling. -732 +728 00:35:54,080 --> 00:35:55,920 然而 我们在这儿看到的是两个表征状态的变量 However,what we see here is two state variables, -733 +729 00:35:55,920 --> 00:35:56,660 x和y x and y. -734 +730 00:35:57,740 --> 00:35:59,220 还有个迭代计算过程表明 And there's an iteration that says, -735 +731 00:35:59,600 --> 00:36:00,520 要画一个圆 in order to circle, -736 +732 00:36:00,760 --> 00:36:01,740 已经有了x和y given an x and y, -737 +733 00:36:02,100 --> 00:36:04,520 我想要的是x和y的下一组取值以用来画圆 what I want is to circle with the next values of x and y -738 +734 00:36:04,520 --> 00:36:08,580 也就是x原有的值减去y和dt的乘积 being the old value of x decrement by y times dt -739 +735 00:36:08,580 --> 00:36:09,760 dt就是时间变化 where dt is the time step -740 +736 00:36:10,320 --> 00:36:15,780 还有y原有的值加上x的dt的乘积 and the old value of y being implemented by x times dt, -741 +737 00:36:16,080 --> 00:36:17,780 然后就得到了x和y的新的取值 giving me the new values of x and y. -742 +738 00:36:21,160 --> 00:36:22,940 好了 现在大家对于两种不同的计算过程的感觉 So now you have a feeling -743 +739 00:36:23,620 --> 00:36:25,740 应该都有了一个直观的感觉 for at least two different kinds of processes -744 +740 00:36:26,920 --> 00:36:29,720 而这两种计算过程却可能是由几乎相同的程序产生的 that can be evolved by almost the same program. -745 +741 00:36:32,300 --> 00:36:34,560 像这样做了一点扰动分析之后 And with a little bit of perturbation analysis like this, -746 +742 00:36:35,340 --> 00:36:37,880 如何小小地修改一下程序观察计算过程是怎样变化的 how you change a program a little bit and see how the process changes, -747 +743 00:36:38,680 --> 00:36:39,800 这样才能有直观的感受 that's how we get some intuition. -748 -00:36:43,540 --> 00:36:45,160 -来构建大型的 恼人的 复杂的系统 -to build big,hairy,complicated systems. - -749 -00:36:45,160 --> 00:36:45,180 +744 +00:36:41,620 --> 00:36:43,080 之后我们就会用到这样的直观感受 Pretty soon we're going to use that intuition -750 +745 +00:36:43,520 --> 00:36:45,180 +来构建大型的 恼人的 复杂的系统 +to build big,hairy,complicated systems. + +746 00:36:46,360 --> 00:36:46,800 谢谢 Thank you. -751 +747 00:36:56,880 --> 00:37:06,660 [音乐] [MUSIC PLAYING BY J.S.BACH] -752 +748 00:37:06,660 --> 00:37:06,920 教授 好了 PROFESSOR: Well, -753 +749 00:37:06,920 --> 00:37:11,100 -各位之前已经见过一个对若干程序的简单的扰动分析 +各位之前已经见过一次 对若干程序的简单的扰动分析 you've just seen a simple perturbational analysis of some programs. -754 +750 00:37:11,600 --> 00:37:14,300 我拿一个程序与另一个很相似的程序进行比较 I took a program that was very similar to another program -755 +751 00:37:14,300 --> 00:37:17,340 观察他们如何演化出计算过程 and looked at them both and saw how they evolved processes. -756 +752 00:37:18,220 --> 00:37:19,520 我想再给大家介绍点不同的东西 I want to show you some variety -757 +753 00:37:19,520 --> 00:37:22,720 -向大家展示程序可能具有的别的一些计算过程和“形状” +向大家展示程序可能具有的 别的一些计算过程和“形状” by showing you some other processes and shapes they may have. -758 +754 00:37:24,280 --> 00:37:25,860 同样地 我们将要举例非常简单的程序 Again,we're going to take very simple things, -759 +755 00:37:26,140 --> 00:37:27,860 简单到你都不想写的程序 programs that you wouldn't want to ever write. -760 +756 00:37:29,040 --> 00:37:30,460 那可能是最糟糕的方式 They would be probably the worst way -761 +757 00:37:31,420 --> 00:37:33,060 去进行一些计算过程 of computing some of the things we're going to compute. -762 +758 00:37:33,740 --> 00:37:35,060 但我还是要展示给大家 But I'm just going to show you these things -763 +759 00:37:35,400 --> 00:37:36,520 是想要展现 for the purpose of feeling out -764 +760 00:37:37,420 --> 00:37:41,320 一个程序如何 how a program represents itself -765 +761 00:37:41,600 --> 00:37:44,520 在计算过程的求值中表现出自己的相应规则 as the rule for the evolution of a process. -766 +762 00:37:46,760 --> 00:37:48,800 考虑一个有趣的问题 So let's consider a fun thing, -767 +763 00:37:48,800 --> 00:37:49,920 斐波那契数列 the Fibonacci numbers. -768 +764 00:37:50,400 --> 00:37:52,020 大家可能已经知道斐波那契数列了 You probably know about the Fibonacci numbers. -769 +765 00:37:53,000 --> 00:37:54,280 我记不清是谁了 Somebody,I can't remember who, -770 +766 00:37:54,660 --> 00:37:59,060 有一个人对兔群的繁殖问题很感兴趣 was interested in the growth of piles of rabbits. -771 +767 00:37:59,920 --> 00:38:01,140 不管怎样 And for some reason or other, -772 +768 00:38:01,380 --> 00:38:04,820 我们都知道兔群趋向于指数式增长 the piles of rabbits tend to grow exponentially,as we know. -773 +769 00:38:05,520 --> 00:38:08,620 对于这个过程我们有个不错的模型来解释 And we have a nice model for this process, -774 +770 00:38:08,620 --> 00:38:11,520 由两个数开始算起 is that we start with two numbers, -775 +771 00:38:11,520 --> 00:38:12,360 0和1 0 and 1. -776 +772 00:38:13,420 --> 00:38:16,560 之后的每个数都是前两个数之和 And then every number after this is the sum of the two previous. -777 +773 00:38:17,840 --> 00:38:19,600 于是接下来就是1 So we have here a 1. -778 +774 00:38:20,100 --> 00:38:21,900 然后这两数之和就是2 Then the sum of these two is 2. -779 +775 00:38:22,380 --> 00:38:23,900 那两数之和是3 The sum of those two is 3. -780 +776 00:38:24,300 --> 00:38:25,900 那两数之和是5 The sum of those two is 5. -781 +777 00:38:26,400 --> 00:38:27,900 那两数之和是8 The sum of those two is 8. -782 +778 00:38:28,280 --> 00:38:30,540 那两数之和是13 The sum of those two is 13. -783 +779 00:38:31,360 --> 00:38:33,260 这是21 This is 21. -784 +780 00:38:34,360 --> 00:38:37,740 34 55 34.55. -785 +781 00:38:38,000 --> 00:38:38,520 等等 Et cetera. -786 +782 00:38:40,300 --> 00:38:41,900 如果给这些数字标上序号 If we start numbering these numbers, -787 +783 00:38:42,360 --> 00:38:43,720 第0个 say this is the zeroth one, -788 +784 00:38:43,720 --> 00:38:44,380 第1个 the first one, -789 +785 00:38:44,380 --> 00:38:45,060 第2个 the second one, -790 +786 00:38:45,060 --> 00:38:45,620 第3个 the third one, -791 +787 00:38:45,620 --> 00:38:46,580 第4个 the fourth one, -792 +788 00:38:46,580 --> 00:38:47,040 等等 et cetera. -793 +789 00:38:47,420 --> 00:38:48,420 这是第10个 This is the 10th one, -794 +790 00:38:49,000 --> 00:38:50,040 第10个斐波那契数 the 10th Fibonacci number. -795 +791 00:38:51,620 --> 00:38:53,000 这些数字增长很快 These numbers grow very fast -796 +792 00:38:54,620 --> 00:38:55,360 就像兔子们一样 Just like rabbits. -797 +793 00:38:55,640 --> 00:38:56,760 为什么兔群是这样增长的呢 Why rabbits grow this way -798 +794 00:38:56,760 --> 00:38:58,140 我不会瞎猜 I'm not going to hazard a guess. -799 +795 00:38:59,320 --> 00:39:00,960 接下来我要为大家展示 Now,I'm going to try to write for you -800 +796 00:39:01,280 --> 00:39:05,260 写一个极简单的程序来计算斐波那契数 the very simplest program that computes Fibonacci numbers. -801 +797 00:39:07,040 --> 00:39:09,620 我想要的是一个程序 It's,ah,what I want is a program that, -802 +798 00:39:09,620 --> 00:39:11,160 输入一个n given an n, -803 +799 00:39:12,320 --> 00:39:13,900 就能输出第n个斐波那契数 will produce for me Fibonacci event. -804 +800 00:39:17,860 --> 00:39:22,320 我就写在这儿了 OK?I'll write it right here. -805 +801 00:39:27,860 --> 00:39:30,740 想要得到第n个斐波那契数 I want the Fibonacci of n, -806 +802 00:39:30,740 --> 00:39:32,500 也就是说 这是那个n which means the-- this is the n, -807 +803 00:39:33,860 --> 00:39:34,880 这是第n个斐波那契数 and this is Fibonacci of n. -808 +804 00:39:36,080 --> 00:39:37,080 我这么写 And here's the story. -809 +805 00:39:37,400 --> 00:39:42,280 如果n小于2 If n is less than 2, -810 +806 00:39:42,740 --> 00:39:43,940 那么答案就是n then the result is n. -811 +807 00:39:45,060 --> 00:39:46,040 因为此时斐波那契数和序号相等 Because that's what these are. -812 +808 00:39:46,920 --> 00:39:47,660 这是最开始的规定 That's how you start it up. -813 +809 00:39:48,700 --> 00:39:49,420 否则 Otherwise, -814 +810 00:39:49,760 --> 00:39:55,180 答案就是第n-1个斐波那契数 the result is the sum of Fib of n minus 1 -815 +811 00:39:58,220 --> 00:40:00,420 和第n-2个斐波那契数的和 and the Fibonacci number,n minus 2. -816 +812 00:40:10,220 --> 00:40:11,260 这是个很简单的 So this is a very simple, -817 +813 00:40:11,440 --> 00:40:12,680 很直接的 direct specification -818 +814 00:40:12,960 --> 00:40:14,900 求斐波那契数的实现 of the description of Fibonacci numbers -819 +815 00:40:15,540 --> 00:40:17,140 就是根据斐波那契数的定义直接写出这个过程 that I gave you when I introduced those numbers. -820 +816 00:40:19,260 --> 00:40:21,940 -用最简单可行的方式直接描述了定义中的递归关系 +用最简单可行的方式 直接描述了定义中的递归关系 It represents the recurrence relation in the simplest possible way. -821 +817 00:40:23,260 --> 00:40:24,720 现在 我们又如何解释这个过程呢 Now,how do we use such a thing? -822 +818 00:40:25,080 --> 00:40:25,980 我们来分析一下这个过程所演化出的计算过程 Let's draw this process. -823 +819 00:40:26,840 --> 00:40:27,980 让我们来看看这个过程都做了些什么 Let's figure out what this does. -824 +820 00:40:29,240 --> 00:40:32,080 举一个简单的例子 求第4个斐波那契数 Let's consider something very simple by computing Fibonacci of 4. -825 +821 00:40:35,840 --> 00:40:37,020 要求第4个斐波那契数 To compute Fibonacci of 4, -826 +822 00:40:37,020 --> 00:40:37,620 我应该怎么做呢 what do I do? -827 +823 00:40:38,720 --> 00:40:41,960 根据过程的定义 4并不比2小 Well,it says I have-- it's not less than 2. -828 +824 00:40:42,740 --> 00:40:44,320 因此答案就是两个数之和 Therefore it's the sum of two things -829 +825 00:40:45,160 --> 00:40:47,000 然后 我又要求出 Well,in order to compute that I have to compute,then, -830 +826 00:40:47,460 --> 00:40:52,520 第3个和第2个斐波那契数 Fibonacci of 3 and Fibonacci of 2 -831 +827 00:40:56,880 --> 00:40:58,620 要求出第3个斐波那契数 In order to compute Fibonacci of 3, -832 +828 00:40:58,960 --> 00:41:04,020 我就得求出第2个斐波那契数和第1个斐波那契数 I have to compute Fibonacci of 2 and Fibonacci of 1. -833 +829 00:41:07,640 --> 00:41:09,280 而为了求出第2个斐波那契数 In order to compute Fibonacci of 2, -834 +830 00:41:09,800 --> 00:41:12,140 就得求出第1个和第0个斐波那契数 I have to compute Fibonacci of 1 and Fibonacci of 0. -835 +831 00:41:16,500 --> 00:41:17,900 要求第1个斐波那契数 In order to compute Fibonacci of 1, -836 +832 00:41:17,900 --> 00:41:18,960 答案就是1 well,the answer is 1. -837 +833 00:41:19,880 --> 00:41:24,520 这正是递归的边界条件 That's from the base case of this recursion. -838 +834 00:41:25,740 --> 00:41:28,140 而第0个斐波那契数 And in order to compute Fibonacci of 0, -839 +835 00:41:28,140 --> 00:41:29,080 是0 well,that answer is 0, -840 +836 00:41:29,080 --> 00:41:29,860 也是递归的边界条件 from the same base. -841 +837 00:41:30,100 --> 00:41:31,660 然后这儿有个1 And here is a 1. -842 +838 00:41:32,800 --> 00:41:36,140 第2个斐波那契数就是第1个斐波那契数 And Fibonacci of 2 is really the sum of Fibonacci of 1. -843 +839 00:41:38,300 --> 00:41:39,220 和第0个斐波那契数的和 And Fib of 0, -844 +840 00:41:40,580 --> 00:41:41,440 分别求出它们 in order to compute that, -845 +841 00:41:41,480 --> 00:41:42,180 就有了一个1 I get a 1, -846 +842 00:41:42,840 --> 00:41:43,860 和一个0 and here I've got a 0. -847 +843 00:41:46,680 --> 00:41:47,500 我画出了一棵树 I've built a tree. -848 +844 00:41:49,880 --> 00:41:52,440 通过这棵树我们能看出一些结果 Now,we can observe some things about this tree. -849 +845 00:41:53,300 --> 00:41:53,920 我们能发现为何 We can see why -850 +846 00:41:53,920 --> 00:41:55,540 这是个非常糟糕的 this is an extremely bad way -851 +847 00:41:55,540 --> 00:41:56,720 求斐波那契数的办法 to compute Fibonacci numbers. -852 +848 00:41:58,120 --> 00:42:00,020 因为 为了求第4个斐波那契数 Because in order to compute Fibonacci of 4, -853 +849 00:42:00,180 --> 00:42:02,740 就得求两次第2个斐波那契数 I had to compute Fibonacci of 2's sub-tree twice. -854 +850 00:42:07,300 --> 00:42:09,000 实际上 如果再多求一个斐波那契数 In fact,in order to add one more, -855 +851 00:42:09,400 --> 00:42:11,000 假设我想要求第5个斐波那契数 supposing I want to do Fibonacci of 5, -856 +852 00:42:12,480 --> 00:42:13,540 那我要做的就是 what I really have to do then -857 +853 00:42:13,540 --> 00:42:16,180 求第4个斐波那契数和第3个斐波那契数的和 is compute Fibonacci of 4 plus Fibonacci of 3. -858 +854 00:42:17,760 --> 00:42:20,900 但第3个斐波那契数已经求过一遍了 But Fibonacci of 3's sub-tree has already been built. -859 +855 00:42:24,500 --> 00:42:29,200 -整个递归树描述了一个指数时间复杂度的递归计算过程 +整个递归树描述了一个 指数时间复杂度的递归计算过程 This is a prescription for a process that's exponential in time. -860 +856 00:42:30,220 --> 00:42:31,100 只要多求一个斐波那契数 To add 1, -861 +857 00:42:31,360 --> 00:42:32,800 时间复杂度就得乘上一个数 I have to multiply by something -862 +858 00:42:32,800 --> 00:42:33,920 因为多求一个斐波那契数 需要把已经计算过的过程 because I take a proportion -863 +859 00:42:33,920 --> 00:42:34,660 再计算一次 of the existing thing -864 +860 00:42:34,660 --> 00:42:38,660 时间复杂度就成比例地急剧增长 and add it to itself to add one more step. -865 +861 00:42:40,200 --> 00:42:47,760 于是这个过程的时间复杂度就是 So this is a thing whose time complexity is order of -866 +862 00:42:47,940 --> 00:42:50,600 实际上就是O(Fib(n)) （Fib(n)指第n个斐波那契数） -- actually,it turns out to be Fibonacci-- of n. -867 +863 00:42:55,940 --> 00:42:58,240 这个过程的时间复杂度就是按斐波那契数列增长 There's a thing that grows exactly at Fibonacci numbers. -868 +864 00:43:00,760 --> 00:43:01,560 这很恐怖 It's a horrible thing. -869 +865 00:43:02,240 --> 00:43:03,040 你都不想这么求斐波那契数 You wouldn't want to do it. -870 +866 00:43:03,440 --> 00:43:05,180 时间复杂度这么增长 The reason why the time has to grow that way -871 +867 00:43:05,180 --> 00:43:06,680 是因为在这个模型中 is because we're presuming in the model -872 +868 00:43:06,680 --> 00:43:08,140 之前给出的代换模型 --the substitution model that I gave you, -873 +869 00:43:08,140 --> 00:43:09,580 我没在这儿形式化地应用代换模型 which I'm not doing formally here, -874 +870 00:43:10,280 --> 00:43:13,200 只是简单地表示了一下 I sort of now spit it out in a simple way-- -875 +871 00:43:14,000 --> 00:43:16,200 我们假设所有事情都是按顺序完成的 but presuming that everything is done sequentially. -876 +872 00:43:17,560 --> 00:43:21,180 即这个递归树中每个节点都会被检查 That every one of these nodes in this tree has to be examined. -877 +873 00:43:24,420 --> 00:43:25,580 于是 由于树的节点数 And so since the number of nodes -878 +874 00:43:25,580 --> 00:43:27,660 是按指数增长的 in this tree grows exponentially, -879 +875 00:43:28,080 --> 00:43:30,020 因为要多求一个斐波那契数 because I add a proportion of the existing nodes -880 +876 00:43:30,760 --> 00:43:32,440 就要把已经遍历过的节点再遍历一遍 to the nodes I already have to add 1, -881 +877 00:43:33,880 --> 00:43:36,800 然后就发现时间复杂度呈爆炸式的指数增长 then I know I've got an exponential explosion here. -882 +878 00:43:38,460 --> 00:43:39,840 现在 再来考虑 Now,let's see if we can think of -883 +879 00:43:39,840 --> 00:43:41,200 这个过程的空间复杂度 how much space this takes up. -884 +880 00:43:44,160 --> 00:43:45,180 空间复杂度不算太高 Well,it's not so bad. -885 +881 00:43:45,800 --> 00:43:47,360 它主要取决于我们要记录多少信息 It depends on how much we have to remember -886 +882 00:43:47,360 --> 00:43:48,620 才能使整个计算过程得以持续进行 in order to continue this thing running. -887 +883 00:43:50,280 --> 00:43:51,220 这个并不难 Well,that's not so hard. -888 +884 00:43:51,500 --> 00:43:52,220 从这里看出 It says,gee, -889 +885 00:43:52,400 --> 00:43:54,220 想要知道我现在处于树中的什么位置 in order to know where I am in this tree, -890 +886 00:43:54,340 --> 00:43:55,880 就要记录一条回到根节点的路径 I have to have a path back to the root. -891 +887 00:43:56,720 --> 00:43:57,160 也就是说 In other words, -892 +888 00:43:57,340 --> 00:43:57,800 为了 in order to-- -893 +889 00:43:57,800 --> 00:43:58,720 考虑一下路径 let's consider the path. -894 +890 00:43:58,720 --> 00:43:59,680 我要把这个过程执行一遍 I would have to execute this. -895 +891 00:44:00,400 --> 00:44:01,180 我说 啊 好吧 I'd say,oh,yes -896 +892 00:44:01,180 --> 00:44:02,220 我向下走到这儿 I'm going to go down here. -897 +893 00:44:02,580 --> 00:44:03,940 不用管方向 I don't care which direction I go. -898 +894 00:44:04,620 --> 00:44:05,660 我这么走 I have to do this. -899 +895 00:44:06,100 --> 00:44:06,880 这么走 I have to then do this. -900 +896 00:44:06,880 --> 00:44:08,880 用一种有趣的方式遍历这棵树 I have to traverse this tree in a sort of funny way. -901 +897 00:44:11,660 --> 00:44:13,180 又接着走这条有趣的小路 I'm going to walk this nice little path. -902 +898 00:44:13,580 --> 00:44:14,680 回到了这里 I come back to here. -903 +899 00:44:15,400 --> 00:44:17,420 好了我已经记住接下来要去哪儿了 Well,I've got to remember where I'm going to be next. -904 +900 00:44:18,180 --> 00:44:19,200 我记在心里了 I've got to keep that in mind. -905 +901 00:44:19,800 --> 00:44:20,860 于是我得知道已经遍历过的节点 So I have to know what I've done. -906 +902 00:44:20,860 --> 00:44:22,020 还得知道未遍历的节点 I have to know what's left. -907 +903 00:44:22,400 --> 00:44:25,600 为了计算出第4个斐波那契数 In order to compute Fibonacci of 4, -908 +904 00:44:25,600 --> 00:44:27,600 某一时刻我会向下走到这儿 at some point I'm going to have to be down here. -909 +905 00:44:28,260 --> 00:44:29,740 我还要记得 And I have to remember -910 +906 00:44:30,080 --> 00:44:31,360 要回到这儿 that I have to go back to here -911 +907 00:44:31,540 --> 00:44:32,960 又回到这儿做一次加法 and then go back to here to do an addition. -912 +908 00:44:33,380 --> 00:44:34,420 然后又走到这儿做一次加法 And then go back to here to do an addition -913 +909 00:44:34,420 --> 00:44:35,440 之前还需要检查这个尚未检查的节点 to something I haven't touched yet. -914 +910 00:44:37,880 --> 00:44:40,340 消耗的空间就是路径的长度 The amount of space that takes up is the path, -915 +911 00:44:40,340 --> 00:44:41,160 最长路径的长度 the longest path. -916 +912 00:44:42,400 --> 00:44:43,360 它有多长 How long it is. -917 +913 00:44:45,620 --> 00:44:46,860 长度就是n And that grows as n. -918 +914 00:44:48,080 --> 00:44:48,860 于是消耗的空间 So the space -919 +915 00:44:50,640 --> 00:44:53,620 因为那是树的最长下降深度 -- because that's the length of the deepest line through the tree-- -920 +916 00:44:54,280 --> 00:44:56,220 空间复杂度就是O(n) the space is order of n. -921 +917 00:44:58,880 --> 00:44:59,920 可以看出这个过程时空效率都很低 It's a pretty bad process. -922 +918 00:45:08,660 --> 00:45:10,260 我想让各位获得的是 Now,one thing I want to see from this -923 +919 00:45:11,860 --> 00:45:13,520 一种直观的感觉 is a feeling -924 +920 00:45:13,540 --> 00:45:14,440 关于整个计算过程的直观感觉 of what's going on here. -925 +921 00:45:15,380 --> 00:45:16,560 为何 Why are there-- -926 +922 00:45:17,140 --> 00:45:19,420 这个程序是如何与整个计算过程联系起来的 how is this program related to this process? -927 +923 00:45:20,800 --> 00:45:21,620 在这儿 我们看到了什么 Well,what are we seeing here? -928 +924 00:45:21,800 --> 00:45:22,820 这个程序实际上 There really are -929 +925 00:45:23,740 --> 00:45:25,640 只做了两件事 only two sorts of things this program does. -930 +926 00:45:27,080 --> 00:45:28,620 这个程序由两条规则组成 This program consists of two rules, -931 +927 00:45:28,620 --> 00:45:29,100 假如你也这么认为的话 if you will. -932 +928 00:45:29,560 --> 00:45:31,700 第一条规则是 第n个斐波那契数 One rule that says Fibonacci of n -933 +929 00:45:32,200 --> 00:45:36,200 就是这里求和过程的结果 is this sum that you see over here, -934 +930 00:45:37,240 --> 00:45:39,400 对应递归树中这样的一个节点 which is a node that's shaped like this. -935 +931 00:45:41,900 --> 00:45:44,880 它指出要把整个过程分为两部分 It says that I break up something into two parts. -936 +932 00:45:46,340 --> 00:45:49,340 在有的情况下 看这里 Under some condition,under some condition over here -937 +933 00:45:49,860 --> 00:45:51,040 n比2大 that n is greater than 2, -938 +934 00:45:56,500 --> 00:45:56,520 比2小 Less than 2. -939 +935 00:45:56,520 --> 00:45:57,040 递归树中对应节点就分为两部分 then the node breaks up into two parts. 比2小 Less than 2. -940 +936 00:45:57,580 --> 00:45:58,560 不 比2大 是的 No.Greater than 2.Yes. -941 +937 00:46:01,600 --> 00:46:03,600 还有一种可能 The other possibility is that -942 +938 00:46:03,600 --> 00:46:05,140 有的归约导致节点没有分成两部分 I have a reduction that looks like this. -943 +939 00:46:08,920 --> 00:46:09,780 也就是这样的情况 And that's this case. -944 +940 00:46:10,960 --> 00:46:11,740 如果比2小 If it's less than 2, -945 +941 00:46:11,740 --> 00:46:12,780 答案就是n本身 the answer is n itself. -946 +942 00:46:14,200 --> 00:46:15,200 我们在这儿看到的是 So what we're seeing here is that -947 +943 00:46:15,360 --> 00:46:16,860 构建起的计算过程 the process that got built -948 +944 00:46:17,440 --> 00:46:18,740 局部来看 每个节点 locally at every place -949 +945 00:46:19,260 --> 00:46:20,440 都是这条规则的实际体现 is an instance of this rule. -950 +946 00:46:21,920 --> 00:46:23,220 这里是规则的一次实际体现 Here's one instance of the rule. -951 +947 00:46:23,800 --> 00:46:25,220 这里是规则的又一次实际体现 Here is another instance of the rule. -952 +948 00:46:26,080 --> 00:46:27,380 人们之所以认为 And the reason why people think of -953 +949 00:46:27,380 --> 00:46:29,260 编程很难 确实挺难的 programming as being hard,of course, -954 +950 00:46:29,900 --> 00:46:32,160 是因为编程实际是在编写一种通用规则 is because you're writing down a general rule, -955 +951 00:46:34,720 --> 00:46:36,340 这种通用规则会被应用于很多实际情况 which is going to be used for lots of instances, -956 +952 00:46:37,000 --> 00:46:38,200 而某一个特定的实际情况 that a particular instance-- -957 +953 00:46:38,840 --> 00:46:41,540 而编写好的程序会为你处理每种特定的实际情况 it's going to control each particular instance for you. -958 +954 00:46:43,560 --> 00:46:44,640 你必须写出这样的程序 You've got to write down something -959 +955 00:46:44,640 --> 00:46:47,180 它是通用的 并且考虑到变量 that's a general and in terms of variables, -960 +956 00:46:47,180 --> 00:46:48,160 你要考虑那些变量 and you have to think of all the things -961 +957 00:46:48,160 --> 00:46:49,540 所有可能的取值 that could possibly fit in those variables, -962 +958 00:46:49,800 --> 00:46:50,760 所有这些最终都必须通向 and all those have to lead to -963 +959 00:46:50,760 --> 00:46:52,180 你想要实现的计算过程 the process you want to work. -964 +960 00:46:53,200 --> 00:46:55,400 局部看来 你又得把计算过程分成 Locally,you have to break up your process -965 +961 00:46:56,100 --> 00:46:58,240 可以表示出来的若干部分 into things that can be represented -966 +962 00:46:58,240 --> 00:47:00,340 考虑到这些非常特定的局部规则 in terms of these very specific local rules. -967 +963 00:47:03,220 --> 00:47:03,800 好了 来看一下 Well,let's see. -968 +964 00:47:04,660 --> 00:47:06,800 斐波那契数 当然了 不太有趣 Fibonaccis are,of course,not much fun. -969 +965 00:47:07,820 --> 00:47:08,440 不 其实还是挺有趣的 Yes,they are. -970 +966 00:47:09,160 --> 00:47:10,920 大家还会学习黄金分割等概念 You get something called the golden ratio, -971 +967 00:47:12,100 --> 00:47:14,060 -并且说不定什么时候我们还有可能接触更多的类似的东西 +并且说不定什么时候我们还有可能接触 更多的类似的东西 and we may even see a lot of that some time. -972 +968 00:47:15,080 --> 00:47:16,140 好了 下面来讨论另一件事 Well,let's talk about another thing. -973 +969 00:47:16,600 --> 00:47:19,640 有个很有名的游戏叫汉诺塔 There's a famous game called the Towers of Hanoi, -974 +970 00:47:19,640 --> 00:47:20,280 举这个例子是因为我教会大家 because I want to teach you -975 +971 00:47:20,280 --> 00:47:21,980 如何递归地思考问题 how to think about things recursively. -976 +972 00:47:23,860 --> 00:47:25,940 游戏是这样的 The problem is this one: -977 +973 00:47:26,860 --> 00:47:28,040 我有一堆盘子 I have a bunch of disks, -978 +974 00:47:29,040 --> 00:47:30,060 又有几根柱子 I have a bunch of spikes, -979 +975 00:47:31,500 --> 00:47:34,300 传说在遥远的东方某地 and it's rumored that somewhere in the Orient -980 +976 00:47:35,100 --> 00:47:36,760 有一个有64个盘子的汉诺塔 there is a 64-high tower, -981 +977 00:47:37,080 --> 00:47:39,100 僧侣们每天的工作就是 and the job of various monks or something -982 +978 00:47:39,200 --> 00:47:40,600 移动盘子从一个柱子到另一个 is to move these spikes -983 +979 00:47:40,600 --> 00:47:41,860 移动的规则很复杂 in some complicated pattern -984 +980 00:47:42,040 --> 00:47:44,540 然后最终这些盘子 so eventually-- these disks-- -985 +981 00:47:44,940 --> 00:47:48,520 最终我把所有的盘子 so eventually I moved all of the disks -986 +982 00:47:48,660 --> 00:47:49,840 从一个柱子移到了另一个柱子 from one spike to the other. -987 +983 00:47:50,380 --> 00:47:51,540 如果有64个盘子 And if it's 64 high, -988 +984 00:47:52,000 --> 00:47:54,180 就要移动2的64次方（减1 译者注）次 and it's going to take 2 to the 64th moves, -989 +985 00:47:54,980 --> 00:47:56,180 要花费很长时间 then it's a long time. -990 +986 00:47:57,820 --> 00:48:01,500 僧侣们声称盘子移完之时即是宇宙终结之时 They claim that the universe ends when this is done. -991 +987 00:48:03,420 --> 00:48:04,060 好了 Well,let's see. -992 +988 00:48:05,580 --> 00:48:08,120 构造一个递归过程的方法其实很简单 The way in which you would construct a recursive process -993 +989 00:48:08,720 --> 00:48:09,660 就是“想当然” is by wishful thinking. -994 +990 00:48:11,580 --> 00:48:12,440 你们要相信 You have to believe. -995 +991 00:48:14,340 --> 00:48:15,240 这个主意 So,the idea. -996 +992 00:48:15,320 --> 00:48:18,360 假如我想把这一堆盘子从这儿移到这儿 Supposing I want to move this pile from here to here, -997 +993 00:48:19,500 --> 00:48:21,360 从1号柱移动到2号柱 from spike one to spike two, -998 +994 00:48:23,460 --> 00:48:24,540 好吧 其实不怎么难 well,that's not so hard. -999 +995 00:48:25,100 --> 00:48:26,540 假如 不知怎么回事 See,supposing somehow, -1000 +996 00:48:27,000 --> 00:48:27,780 好像有魔法在作用 by some magic-- -1001 +997 00:48:27,860 --> 00:48:28,860 因为有个更简单的问题 because I've got a simpler problem -1002 +998 00:48:29,020 --> 00:48:30,600 我把3个盘子移到这里 -- I move a three-high pile to here-- -1003 +999 00:48:30,860 --> 00:48:32,000 一次只能移动一个盘子 I can only move one disk at a time, -1004 +1000 00:48:32,000 --> 00:48:32,960 不管我是怎么做到的 so I don't even think how I did it. -1005 +1001 00:48:33,640 --> 00:48:34,800 但假设我做到了 But supposing I could do that, -1006 +1002 00:48:36,300 --> 00:48:38,080 那我只用拿起这个盘子 well,then I could just pick up this disk -1007 +1003 00:48:38,080 --> 00:48:38,680 把它移到这里 and move it here. -1008 +1004 00:48:41,140 --> 00:48:42,520 现在问题就简单了 And now I have a simple problem, -1009 +1005 00:48:42,620 --> 00:48:43,940 把这3个盘子移到这里 I have to move a three-high tower to here, -1010 +1006 00:48:44,680 --> 00:48:45,420 之前已经做过了 which is no problem. -1011 +1007 00:48:45,780 --> 00:48:47,780 于是通过两次对3个盘子的移动 So by two moves of a three high tower -1012 +1008 00:48:47,780 --> 00:48:49,180 再加上1次对一个盘子的移动 plus one move of a single object. -1013 +1009 00:48:50,700 --> 00:48:52,760 我就能把整堆盘子从这儿移动到这儿 I can move the tower from here to here. -1014 +1010 00:48:55,260 --> 00:48:56,440 不管怎样 Now,whether or not-- -1015 +1011 00:48:57,180 --> 00:48:58,440 不管探究到什么程度 this is not obvious -1016 +1012 00:48:59,240 --> 00:49:00,980 都不能明显看出整个过程是正确的 in any deep way that this works. -1017 +1013 00:49:02,360 --> 00:49:02,980 还有 为什么呢 And why? -1018 +1014 00:49:03,900 --> 00:49:06,520 为什么我就能假设 Now why is it the case that I can presume,maybe, -1019 +1015 00:49:06,640 --> 00:49:07,920 假设我能把成功移动这三个盘子 that I can move the three-high tower. -1020 +1016 00:49:11,080 --> 00:49:12,260 好吧 这是因为 Well the answer is because -1021 +1017 00:49:12,540 --> 00:49:13,660 我总能减小问题的规模 I'm always counting down, -1022 +1018 00:49:14,080 --> 00:49:15,960 然后最终是0个盘子的移动问题 and eventually I get down to zero-high tower, -1023 +1019 00:49:16,200 --> 00:49:17,800 0个盘子不需要移动 and a zero-high tower requires no moves. -1024 +1020 00:49:19,800 --> 00:49:21,680 现在可以写出整个过程的算法了 So let's write the algorithm for that. -1025 +1021 00:49:23,820 --> 00:49:24,400 很简单 Very easy. -1026 +1022 00:49:26,120 --> 00:49:28,720 我会给这些柱子编号 I'm going to label these towers with numbers, -1027 +1023 00:49:28,720 --> 00:49:30,120 但它们各自编号成什么无关紧要 but it doesn't matter what they're labelled with. -1028 +1024 00:49:30,780 --> 00:49:32,800 问题就是把n个盘子 And the problem is to move an n-high tower -1029 +1025 00:49:33,800 --> 00:49:36,440 从一个起始柱移到终点柱 from a spike called From to a spike called To -1030 +1026 00:49:36,620 --> 00:49:38,240 另一个柱子作为中转 称作中转柱 with a particular spike called Spare. -1031 +1027 00:49:39,520 --> 00:49:40,200 这就是主要的过程 That's what we're going to do. -1032 +1028 00:49:48,700 --> 00:49:52,120 用这个我非形式化描述的算法 Using the algorithm I informally described to you, -1033 +1029 00:49:52,720 --> 00:50:01,700 把n个盘子从起始柱移到终点柱 另一个柱子作为中转 move of a n-high tower from From to To with a Spare. -1034 +1030 00:50:05,940 --> 00:50:08,480 接着就有两种情况了 Well,I've got two cases, -1035 +1031 00:50:09,220 --> 00:50:10,480 就要分析都是怎样的情况 and this is a case analysis, -1036 +1032 00:50:10,940 --> 00:50:14,360 就像之前所做的分析一样 just like it is in all the other things we've done. -1037 +1033 00:50:20,920 --> 00:50:21,840 如果n等于0 那么 If n is 0,then -1038 +1034 00:50:22,780 --> 00:50:24,940 我会输出一些东西 输出"Done" -- I'm going to put out some answers-- Done,we'll say. -1039 +1035 00:50:26,560 --> 00:50:27,560 我不知道那代表什么意思 I don't know what that means -1040 +1036 00:50:29,360 --> 00:50:31,160 因为我们也不会遇到这样的情况 Because we'll never use that answer for anything. -1041 +1037 00:50:31,640 --> 00:50:32,620 我们还是会移动盘子的 We're going to do these moves. -1042 +1038 00:50:34,000 --> 00:50:34,380 否则 Else. -1043 +1039 00:50:36,280 --> 00:50:37,320 就移动一次 I'm going to do a move. -1044 +1040 00:50:39,900 --> 00:50:42,360 移动少于n个盘子 Move a tower of height less than n, -1045 +1041 00:50:44,160 --> 00:50:46,300 也就是n-1个盘子 the decrement of n height. -1046 +1042 00:50:47,740 --> 00:50:50,080 把它们移到中转柱上 Now,I'm going to move it to the Spare tower. -1047 +1043 00:50:50,720 --> 00:50:51,720 整体的思想就是 The whole idea now -1048 +1044 00:50:52,020 --> 00:50:53,880 把它们从这里移到这里 is to move this from here to here, -1049 +1045 00:50:54,940 --> 00:50:55,600 移到中转柱 to the Spare tower -1050 +1046 00:50:55,860 --> 00:50:57,160 从起始柱到中转柱 -- so from From to Spare-- -1051 +1047 00:51:02,340 --> 00:51:04,380 用终点柱作为中转 using To as a spare tower. -1052 +1048 00:51:08,560 --> 00:51:10,080 之后某个时候 Later,somewhere later, -1053 +1049 00:51:10,080 --> 00:51:14,800 我就会移动之前的那n个盘子 I'm going to move that same n-high tower, -1054 +1050 00:51:15,240 --> 00:51:16,160 在那之前已经完成了这个盘子的移动 after I've done this. -1055 +1051 00:51:17,340 --> 00:51:20,200 要移动n个盘子同样的道理就要上面的n-1个盘子 Going to move that same n minus one-high tower -1056 +1052 00:51:20,620 --> 00:51:23,540 从中转柱移到终点柱 from the Spare tower to the To tower -1057 +1053 00:51:23,940 --> 00:51:25,860 用起始柱作为中转 using the From tower as my spare. -1058 +1054 00:51:29,060 --> 00:51:39,040 编写成代码就是从中转柱移动到终点柱 So the Spare tower to the To tower -1059 +1055 00:51:40,040 --> 00:51:44,140 用起始柱作为中转 using the From as the spare. -1060 +1056 00:51:48,440 --> 00:51:49,580 当进行到这种情况 All I have to do now -1061 +1057 00:51:50,020 --> 00:51:51,840 剩下要做的事情就是 is when I've gotten it in this condition, -1062 +1058 00:51:52,100 --> 00:51:55,080 现在就是这种情况 between these two moves of a whole tower -1063 +1059 00:51:56,260 --> 00:51:57,420 在两次移动同一堆盘子之间 -- I've got it into that condition-- -1064 +1060 00:51:57,580 --> 00:52:00,480 只需移动一个盘子 now I just have to move one disk. -1065 +1061 00:52:02,820 --> 00:52:03,380 所以我想要说的就是 So I'm going to say that -1066 +1062 00:52:03,380 --> 00:52:04,300 -有个什么过程输出一条信息表示做了一次盘子的移动 +有个什么过程输出一条信息 表示做了一次盘子的移动 some things are printing a move -1067 +1063 00:52:04,300 --> 00:52:05,200 不管这个过程具体怎么实现的 and I don't care how it works. -1068 +1064 00:52:11,340 --> 00:52:13,900 把盘子从起始柱移动到终点柱 From to To. -1069 +1065 00:52:17,420 --> 00:52:19,780 现在大家就能明白为何我在这个时候举这样一个例子 Now,you see the reason why I'm bringing this up -1070 +1066 00:52:19,780 --> 00:52:23,060 是因为在某种程度上 at this moment is this is an almost identical program -1071 +1067 00:52:23,060 --> 00:52:25,060 这个程序和之前这个程序几乎一样 to this one in some sense. -1072 +1068 00:52:26,600 --> 00:52:28,940 尽管它们所求解的问题不一样 It's not computing the same mathematical quantity, -1073 +1069 00:52:29,240 --> 00:52:30,940 演化出的递归树也不尽相同 it's not exactly the same tree, -1074 +1070 00:52:31,220 --> 00:52:32,320 但不管怎样都会在计算过程中演化出一棵递归树 but it's going to produce a tree. -1075 +1071 00:52:34,300 --> 00:52:36,180 移动这些盘子的过程 The general way of making these moves -1076 +1072 00:52:36,920 --> 00:52:39,580 会导致生成一棵指数树 is going to lead to an exponential tree. -1077 +1073 00:52:41,380 --> 00:52:43,060 好了 接下来试着来移动4个盘子 Well,let's do this four-high. -1078 +1074 00:52:44,100 --> 00:52:47,220 我还是得看着小抄来移盘子 I have my little crib sheet here -1079 +1075 00:52:49,240 --> 00:52:50,400 不然会弄错 otherwise I get confused. -1080 +1076 00:52:54,360 --> 00:52:56,480 好了 下面形式化地提出这个问题 Well,what I'm going to put in is the question of -1081 +1077 00:52:56,700 --> 00:53:05,400 把4个摞在一起的盘子 move a tower of height four -1082 +1078 00:53:06,240 --> 00:53:08,500 从1号柱子移动到2号 from one to spike two -1083 +1079 00:53:08,660 --> 00:53:10,460 用3号柱子作为中转 using spike three as a spare. -1084 +1080 00:53:11,680 --> 00:53:12,720 这就是接下来我要做的 That's all I'm really going to do. -1085 +1081 00:53:13,860 --> 00:53:15,140 好吧 开始吧 You know,let's just do it. -1086 +1082 00:53:15,140 --> 00:53:16,120 我不打算写出 I'm not going to worry about -1087 +1083 00:53:16,120 --> 00:53:17,580 过程执行的具体步骤 writing out the trace of this. -1088 +1084 00:53:17,620 --> 00:53:18,540 你自己可以做到 You can do that yourself -1089 +1085 00:53:19,560 --> 00:53:20,380 因为那很简单 because it's very simple. -1090 +1086 00:53:21,620 --> 00:53:24,580 我要把1号柱子上的一个盘子移动到3号柱子上 I'm going to move disk one to disk three. -1091 +1087 00:53:26,300 --> 00:53:28,100 这是怎么回事呢 And how do I get to move disk one to disk three? -1092 +1088 00:53:28,280 --> 00:53:28,940 我怎么知道要这么做呢 How do I know that? -1093 +1089 00:53:29,180 --> 00:53:30,820 好吧 我想还是得查看一下程序执行的具体步骤 Well,I suppose I have to look at the trace a little bit. -1094 +1090 00:53:32,420 --> 00:53:33,220 才能明白我这是在做什么 What am I doing here? -1095 +1091 00:53:33,520 --> 00:53:34,440 好吧 这个不等于 Well,and this is not-- -1096 +1092 00:53:34,440 --> 00:53:35,340 n不等于0 n is not zero. -1097 +1093 00:53:36,400 --> 00:53:37,860 那我看一下这里过程的定义 So I'm going to look down here. -1098 +1094 00:53:38,340 --> 00:53:40,220 这得需要两次移动 This is going to require doing two moves. -1099 +1095 00:53:40,640 --> 00:53:41,680 只观察第一次吧 I'm only going to look at the first one. -1100 +1096 00:53:41,940 --> 00:53:43,160 第一次是说把 It's going to require moving-- -1101 +1097 00:53:47,500 --> 00:53:48,880 我怎么写成了移动一座塔 why do I have move tower? -1102 +1098 00:53:48,880 --> 00:53:50,340 塔可不好移 It makes it harder for me to move. -1103 +1099 00:53:52,580 --> 00:53:56,280 我要移动摞成一摞的3个盘子 I'm going to move a three-high tower -1104 +1100 00:53:57,560 --> 00:53:59,460 从起始柱 from the From place, -1105 +1101 00:53:59,580 --> 00:54:00,340 也就是4号柱（教授不小心犯错 译者注） which is four, -1106 +1102 00:54:01,400 --> 00:54:02,160 移动到中转柱 to the Spare, -1107 +1103 00:54:02,320 --> 00:54:03,140 即2号柱 which is two, -1108 +1104 00:54:03,900 --> 00:54:06,400 用3号柱 using three as my -1109 +1105 00:54:07,960 --> 00:54:09,800 不 用 用 -- no,using,using-- -1110 +1106 00:54:09,980 --> 00:54:15,740 学生 [听不清] STUDENT: [INAUDIBLE PHRASE] -1111 +1107 00:54:15,820 --> 00:54:16,920 教授 啊 是这样 抱歉 PROFESSOR: Yes,I'm sorry. -1112 +1108 00:54:17,540 --> 00:54:18,880 从2号柱 From two-- -1113 +1109 00:54:19,340 --> 00:54:25,680 从1号柱到3号柱用2号柱作为中转 from one to three using two as my spare. -1114 +1110 00:54:25,900 --> 00:54:26,200 这就对了 That's right. -1115 +1111 00:54:27,140 --> 00:54:31,200 之后这儿还有一次移动 And then there's another move over here afterwards. -1116 +1112 00:54:32,800 --> 00:54:33,580 现在可以说 So now I say, -1117 +1113 00:54:33,880 --> 00:54:34,320 啊 是的 oh,yes, -1118 +1114 00:54:34,420 --> 00:54:37,880 接下来还得把之上两个盘子 that requires me moving a two-high tower -1119 +1115 00:54:38,680 --> 00:54:42,140 从1号柱移动到2号柱 用3号柱作为中转 from one to two using three as a spare. -1120 +1116 00:54:43,000 --> 00:54:43,960 同样的还得再移一次 And so,are the same, -1121 +1117 00:54:44,580 --> 00:54:45,980 接下来 又要移动 and that's going to require me moving -1122 +1118 00:54:47,540 --> 00:54:52,000 一个盘子 从1号柱到2号柱 a one-high tower from one to three -1123 +1119 00:54:52,180 --> 00:54:53,220 用2号柱作为中转 using two as a spare. -1124 +1120 00:54:57,500 --> 00:54:59,300 好了 当然之后还得再移一次 Well,and then there's lots of other things to be done. -1125 +1121 00:55:03,260 --> 00:55:05,080 于是我把1个盘子 So I move my one-high tower -1126 +1122 00:55:06,160 --> 00:55:09,220 从1号柱移动到3号柱 用2号柱作为中转 from one to three using two as a spare, -1127 +1123 00:55:09,220 --> 00:55:10,300 实际上没用到2号柱 which I didn't do anything with. -1128 +1124 00:55:11,140 --> 00:55:12,800 好了 整个过程还是非常简单的 Well,this thing just proceeds very simply. -1129 +1125 00:55:15,300 --> 00:55:16,660 接下来我把这个盘子从1号柱移到2号柱 I move this from one to two. -1130 +1126 00:55:17,360 --> 00:55:19,260 然后把这个盘子从3号柱移动到2号柱 And I move this disk from three to two. -1131 +1127 00:55:20,980 --> 00:55:22,120 其实我不想移了 And I don't really want to do it, -1132 +1128 00:55:22,520 --> 00:55:23,800 但还得继续 从1到3 but I move from one to three. -1133 +1129 00:55:24,680 --> 00:55:26,480 然后从2到1 Then I move two to one. -1134 +1130 00:55:29,040 --> 00:55:30,400 从2到3 Then I move two to three. -1135 +1131 00:55:31,940 --> 00:55:34,900 从1到3 Then one to three. -1136 +1132 00:55:35,940 --> 00:55:36,820 1到2 One to two. -1137 +1133 00:55:39,360 --> 00:55:41,040 3到2 Three to two. -1138 +1134 00:55:41,800 --> 00:55:42,560 3到1 Three to one. -1139 +1135 00:55:44,060 --> 00:55:45,540 当然了这是因为我之前已经准备好了 This all got worked out beforehand,of course. -1140 +1136 00:55:46,040 --> 00:55:46,700 2到1 Two to one. -1141 +1137 00:55:47,480 --> 00:55:48,980 3到2 Three to two. -1142 +1138 00:55:49,780 --> 00:55:51,420 1到3 One to three. -1143 +1139 00:55:52,920 --> 00:55:54,040 学生 [听不清] STUDENT: [INAUDIBLE PHRASE]. -1144 +1140 00:55:54,060 --> 00:55:54,840 教授 噢 从1到3 PROFESSOR: Oh,one to three. -1145 +1141 00:55:54,840 --> 00:55:55,780 抱歉 谢谢提醒 Excuse me.Thank you. -1146 +1142 00:55:56,080 --> 00:55:56,840 1到2 One to two. -1147 +1143 00:55:58,720 --> 00:56:01,520 然后3到2 呼 And then three to two.Whew. -1148 +1144 00:56:03,880 --> 00:56:05,920 现在请大家思考 Now what I'd like you to think about, -1149 +1145 00:56:07,120 --> 00:56:08,820 你们刚才看到的是解决这个问题的递归算法 you just saw a recursive algorithm for doing this, -1150 +1146 00:56:09,100 --> 00:56:10,340 显然要耗费指数级的时间 and it takes exponential time,of course. -1151 +1147 00:56:11,120 --> 00:56:12,220 不知道是否有别的算法 Now,I don't know if there's any algorithm -1152 +1148 00:56:12,220 --> 00:56:14,060 不用耗费指数级的时间 它一定会耗费指数级的时间 that doesn't take exponential time-- it has to. -1153 +1149 00:56:14,440 --> 00:56:15,920 因为我每次只能移动一个盘子 As I'm doing one operation -1154 +1150 00:56:15,920 --> 00:56:17,240 每次只能移动一个盘子 -- I can only move one thing at a time-- -1155 +1151 00:56:17,540 --> 00:56:19,240 没有什么算法不用耗费指数级的时间 there's no algorithm that's not going to take exponential time. -1156 +1152 00:56:21,420 --> 00:56:22,640 但是 你能写出迭代的算法 But can you write an iterative algorithm -1157 +1153 00:56:23,640 --> 00:56:25,160 而不是递归的算法么 rather than a recursive algorithm for doing this? -1158 +1154 00:56:28,360 --> 00:56:30,260 我一直爱琢磨 One of the sort of little things I like to think about -1159 +1155 00:56:32,320 --> 00:56:34,840 能否给出一个算法 Can you write one that, -1160 +1156 00:56:35,040 --> 00:56:39,060 不是像我描述的这样 in fact,doesn't break this problem -1161 +1157 00:56:39,080 --> 00:56:40,560 把一个问题分解成两个子问题 into two sub-problems the way I described, -1162 +1158 00:56:41,020 --> 00:56:43,200 而是用一个更局部的规则 but rather proceeds a step at a time -1163 +1159 00:56:43,260 --> 00:56:44,800 一次演化出整个计算过程 using a more local rule. -1164 +1160 00:56:47,760 --> 00:56:48,480 那可能会很有趣 That might be fun. -1165 +1161 00:56:50,340 --> 00:56:51,720 谢谢大家 第三部分结束 Thank you so much for the third segment. -1166 +1162 00:56:55,880 --> 00:56:56,380 有什么问题要问么 Are there questions? -1167 +1163 00:56:57,400 --> 00:56:58,280 学生 我想知道有没有什么办法 STUDENT: I wonder if there's a way -1168 +1164 00:56:58,280 --> 00:57:01,660 能减小递归过程的代价 to reduce a tree or recursion problem, -1169 +1165 00:57:02,060 --> 00:57:05,660 能否把中间过程 how do you save the intermediate work -1170 +1166 00:57:05,760 --> 00:57:08,180 计算出的斐波那契数保存下来 you have done in computing the Fibonacci number? -1171 +1167 00:57:09,000 --> 00:57:10,100 教授 呃 好吧 实际上 PROFESSOR: Oh,well,in fact,one way,that -1172 +1168 00:57:11,200 --> 00:57:13,120 你刚才说的就是一种办法 one of the ways to do is what you just said. -1173 +1169 00:57:13,580 --> 00:57:16,500 你说 把中间过程的结果保存下来 是吧 You said,I save the intermediate work.OK? -1174 +1170 00:57:16,620 --> 00:57:18,360 好的 我告诉你 Well,let me tell you -1175 +1171 00:57:18,940 --> 00:57:20,100 再次说一下 我们后面会见到 -- this,again,we'll see later-- -1176 +1172 00:57:20,720 --> 00:57:21,820 但现在假设就是这种情形 but suppose it's the case -1177 +1173 00:57:22,360 --> 00:57:24,060 不论何时做怎样的计算 that anytime I compute anything, -1178 +1174 00:57:24,340 --> 00:57:25,720 所有的这些斐波那契数 any one of these Fibonacci numbers, -1179 +1175 00:57:26,400 --> 00:57:27,620 我记录一个表 I remember the table -1180 +1176 00:57:27,900 --> 00:57:28,920 在表中查询 that takes only linear time -1181 +1177 00:57:28,920 --> 00:57:31,580 只用线性的时间 to look up the answer. -1182 +1178 00:57:32,720 --> 00:57:33,660 如果我已经计算过了 Then if I ever see it again, -1183 +1179 00:57:33,660 --> 00:57:35,340 就不再递归地求解了 instead of doing the expansional tree, -1184 +1180 00:57:35,620 --> 00:57:36,100 是直接在表中查询 I look it up. -1185 +1181 00:57:36,700 --> 00:57:37,940 我把问题 I've just transformed my problem -1186 +1182 00:57:38,780 --> 00:57:39,840 变得简单多了 into a problem that's much simpler. -1187 +1183 00:57:40,980 --> 00:57:41,620 当然 Now,of course, -1188 +1184 00:57:42,260 --> 00:57:43,560 这种方法已经被使用了 there are the way to do this,as well. -1189 +1185 00:57:44,240 --> 00:57:45,340 这种方法叫作记忆化 That one's called memoization, -1190 +1186 00:57:45,340 --> 00:57:47,620 这学期你们就会遇到 and you'll see it sometime later in this term. -1191 +1187 00:57:48,240 --> 00:57:53,280 但我认为还有个方法是线性时间复杂度的 But I suppose there's a very simple linear time and, -1192 +1188 00:57:53,360 --> 00:57:55,920 实际上 是用迭代模型计算斐波那契数 in fact,iterative model for computing Fibonaccis, -1193 +1189 00:57:56,660 --> 00:57:58,200 这也是需要各位坐下来仔细思考的 and that's another thing you should sit down and work out. -1194 +1190 00:58:00,080 --> 00:58:00,720 这很重要 That's important. -1195 +1191 00:58:01,260 --> 00:58:02,620 明白怎么运用迭代模型很重要 It's important to see how to do this. -1196 +1192 00:58:05,200 --> 00:58:06,060 希望大家多练习一下 I want you to practice. -1197 +1193 00:58:06,500 --> 00:58:08,100 MIT OpenCourseWare http://ocw.mit.edu -1198 +1194 00:58:08,100 --> 00:58:10,240 本项目主页 https://github.com/FoOTOo/Learning-SICP