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Anyone want to ask anything about that before we start adding? All right. So finally, weeks ago, I said, well, what if you had some hardware device to do addition? And in the meantime, since that first discussion, we've actually gone and filled in. I know the truth tables for adding. I'll walk through it again in a minute. So I think you know mostly how to do this. And probably, you could do it without my showing you. | [
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So I think you know mostly how to do this. And probably, you could do it without my showing you. But let's walk through and do it. So we're going to do it based on the human approach. So remember, we write down numbers as binary numbers. And we add them just like we do in base 10, except it's binary. So 1 plus 1 is 10, and you have to carry. But in general, this approach of, well, let's start with a human design will often give you a pretty good design. | [
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So 1 plus 1 is 10, and you have to carry. But in general, this approach of, well, let's start with a human design will often give you a pretty good design. So you can start that way. It's usually pretty easy, because if you know what you're doing as a human, turning that into logic should be pretty straightforward compared with making up something abstract and trying to figure out the details. | [
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So you can start that way. It's usually pretty easy, because if you know what you're doing as a human, turning that into logic should be pretty straightforward compared with making up something abstract and trying to figure out the details. And it often does lead to a good design, because humans are pretty smart. The way we do addition was the result of thousands of years of thinking about, well, what's the best way to teach kids how to do this efficiently kind of thing? So usually, the way we do things is not a bad way to do it. | [
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And it often does lead to a good design, because humans are pretty smart. The way we do addition was the result of thousands of years of thinking about, well, what's the best way to teach kids how to do this efficiently kind of thing? So usually, the way we do things is not a bad way to do it. So you may remember this slide. This is just an example slide from the first time we talked about binary addition. | [
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So you may remember this slide. This is just an example slide from the first time we talked about binary addition. So we did it before. 0 plus 0 is 0. 1 plus 0 is 1. 1 plus 1 is 0. Carry the 1. 1 plus 1 plus 0. 0. Carry the 1, and so forth. So we got that. We got the right answer. We were happy. And now, we need to put some labels. So in order to build a system that'll do, say, 5-bit addition, we're going to need labels. | [
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And now, we need to put some labels. So in order to build a system that'll do, say, 5-bit addition, we're going to need labels. So I've already called this number A, called this number B. This will be our sum S. And then up here are going to be our carry bits. Now, this is a digital system. So when I was doing base 2 addition by hand, I was kind of lazy. When the carry was 0, I didn't go right to 0. | [
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So when I was doing base 2 addition by hand, I was kind of lazy. When the carry was 0, I didn't go right to 0. But there's no blank bit. So those things, there's no blank bit. Those carry bits are going to be 0s, not blanks. And so just fill that in. Make sure we know that. There's also this carry bit here. So if we're going to design one piece of logic that adds one column, well, that piece of logic also needs a carry. | [
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There's also this carry bit here. So if we're going to design one piece of logic that adds one column, well, that piece of logic also needs a carry. And it can't be a blank. So we're just going to assume that we're going to put a 0 into the lowest, least significant bit carry | [
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So those things, there's no blank bit. Those carry bits are going to be 0s, not blanks. And so just fill that in. Make sure we know that. There's also this carry bit here. So if we're going to design one piece of logic that adds one column, well, that piece of logic also needs a carry. And it can't be a blank. So we're just going to assume that we're going to put a 0 into the lowest, least significant bit carry. | [
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And it can't be a blank. So we're just going to assume that we're going to put a 0 into the lowest, least significant bit carry. So just flesh out our human approach with the details that we usually just don't bother to write down. I mean, if I'd asked you, oh, what's the carry here? You'd say, well, it's 0, right? Obviously, I left it blank. But there's nothing obvious to a computer. | [
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I mean, if I'd asked you, oh, what's the carry here? You'd say, well, it's 0, right? Obviously, I left it blank. But there's nothing obvious to a computer. All right, so two extra assumptions. For the least significant bits, we're going to set c to 0. And for the other bits, the carry input is going to come from the next least significant bit. So this is just adapting our human approach to digital systems. | [
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And for the other bits, the carry input is going to come from the next least significant bit. So this is just adapting our human approach to digital systems. So let's spell that out. So we've got inputs and outputs for this full adder. So a full adder is going to add one bit. The name is historical. There was a half adder that added two bits. And then if you put two of those together, you could add three bits. | [
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There was a half adder that added two bits. And then if you put two of those together, you could add three bits. But a full adder will have three inputs. So there's going to be one bit of the number A, which we'll also call A, one bit of the number B, which we'll also call B, a carry input from the next least significant bit, or 0 for bit 0, which we'll call C in. And a full adder will then produce two outputs. | [
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So there's going to be one bit of the number A, which we'll also call A, one bit of the number B, which we'll also call B, a carry input from the next least significant bit, or 0 for bit 0, which we'll call C in. And a full adder will then produce two outputs. So one is the carry out, which will go to the next most significant bit, or if this is the most significant bit, that'll be the carry out that tells us overflow for unsigned or just carry out for just complement. | [
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And a full adder will then produce two outputs. So one is the carry out, which will go to the next most significant bit, or if this is the most significant bit, that'll be the carry out that tells us overflow for unsigned or just carry out for just complement. And then one bit of the sum S. So those are the inputs and outputs for our full adder. So here's our full adder the way we might draw it. | [
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] |
So those are the inputs and outputs for our full adder. So here's our full adder the way we might draw it. So you can think of it as a bit slice. You can think of it as a full adder. But here's a picture of it. So it's got two inputs coming in. Those will be the m-th bit of A and B. So this will be one bit slice for the m-th column of our addition. We'll get the carry. I put it as a superscript here to differentiate it from the in and the out. | [
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We'll get the carry. I put it as a superscript here to differentiate it from the in and the out. But otherwise, putting it a superscript doesn't really mean anything. It's just the m-th bit of the carry. The m plus 1-th bit of the carry is an output. And some bit m is also an output. Yeah. So the question is, do you have to pass the carry? It gets passed from the left. So the one coming in is from the one less significant digit. | [
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So the question is, do you have to pass the carry? It gets passed from the left. So the one coming in is from the one less significant digit. And the one going out is to the next most significant digit. But if you're all the way at the end, then I can go back to the yeah, here. So if you're this last bit here, so in a five-bit addition, we would have five of these full adders. So the last one will produce the carry out of the whole adder. | [
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So if you're this last bit here, so in a five-bit addition, we would have five of these full adders. So the last one will produce the carry out of the whole adder. That make sense? OK. Anything else? Yeah | [
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So the one coming in is from the one less significant digit. And the one going out is to the next most significant digit. But if you're all the way at the end, then I can go back to the yeah, here. So if you're this last bit here, so in a five-bit addition, we would have five of these full adders. So the last one will produce the carry out of the whole adder. That make sense? OK. Anything else? Yeah. | [
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That make sense? OK. Anything else? Yeah. Yeah, so for unsigned, the carry out is the overflow, as you might remember. In order to calculate the overflow for 2's complement, we would need to add some additional logic to our adder. And in practice, most adders in real processors would have that extra logic, and it will give you an overflow bit as well. In the LC3 design in the textbook, they didn't bother to add it. | [
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And in practice, most adders in real processors would have that extra logic, and it will give you an overflow bit as well. In the LC3 design in the textbook, they didn't bother to add it. So you can calculate it other ways. It's just a little more onerous. We won't add it to our design. It's one extra XOR gate. All right, so here's our design. So now in order to implement this bit, we're going to use a KMAP, of course. | [
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All right, so here's our design. So now in order to implement this bit, we're going to use a KMAP, of course. And first, we'll fill in our truth table. But we need to add A, B, and C in, and then produce S and C out. So let's go ahead and do that. So let's calculate our outputs. Again, the inputs are A, B, and C in. The outputs are C out and S. I've written these so that we can just add the two numbers and then get a 2-bit sum. | [
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The outputs are C out and S. I've written these so that we can just add the two numbers and then get a 2-bit sum. And then the high bit will be the carry, and the low bit will be the sum output bit. So we solved this a few weeks ago. Let's go ahead and do it again. So if I do 0 plus 0 plus 0, what do I get? 0, 0. Yeah, 0, 0, 1. 1. 0, 1, 0. Good. 0, 1, 1. Yeah, 0 carry the 1, right? 1, 0, 0. | [
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0, 1, 1. Yeah, 0 carry the 1, right? 1, 0, 0. 1 and 0. 1, 0, 1. Yeah, 1, 0. So carry is 1. 1, 1, 0. Sorry, I jumped ahead. And then 1, 1, 1. 1, 1. Good. So everyone remembers how to do this, hopefully. So then we can copy over to the KMAP. I don't remember if we've done this before in class, so let me just remind you. In our truth table, we're generally going to write binary order. In our KMAPs, we're going to write grade code order. | [
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In our truth table, we're generally going to write binary order. In our KMAPs, we're going to write grade code order. And so the order of filling things in, if you're copying, here I put the two ladder variables there on the top. So B and C go this way. So reading downwards, we're first going to go across the KMAP. If you write the variables in different order, you would go down first. | [
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So reading downwards, we're first going to go across the KMAP. If you write the variables in different order, you would go down first. But in this KMAP ordering, we'll go B and C first, and then we'll do the second row as the second half of the truth table. And then the binary to grade code, since only this direction has two variables, we're going to go here, here, and then jump over to this one, which is the 1, 0 case of BC, and then fill this one last. | [
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But in this KMAP ordering, we'll go B and C first, and then we'll do the second row as the second half of the truth table. And then the binary to grade code, since only this direction has two variables, we're going to go here, here, and then jump over to this one, which is the 1, 0 case of BC, and then fill this one last. So that said, we can just read them off. So 0, 0, 0, 1, and then fill them in. | [
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So that said, we can just read them off. So 0, 0, 0, 1, and then fill them in | [
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And then the binary to grade code, since only this direction has two variables, we're going to go here, here, and then jump over to this one, which is the 1, 0 case of BC, and then fill this one last. So that said, we can just read them off. So 0, 0, 0, 1, and then fill them in. So 0 goes there, 0, and then we'll skip over, put the third 0 there, and then the 1 goes in the 1, 1 position. | [
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So 0, 0, 0, 1, and then fill them in. So 0 goes there, 0, and then we'll skip over, put the third 0 there, and then the 1 goes in the 1, 1 position. So make sure you get this, because every time you create something, if you make a mistake copying from your truth table to your KMAP, you're only going to catch it after you've solved the whole KMAP. You go back and you say, is my logic right? Did I get the right expression? Does it work? And the answer is no. | [
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So make sure you get this, because every time you create something, if you make a mistake copying from your truth table to your KMAP, you're only going to catch it after you've solved the whole KMAP. You go back and you say, is my logic right? Did I get the right expression? Does it work? And the answer is no. And so then you'll go back and realize, oh, shoot, I forgot to flip these. | [
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You go back and you say, is my logic right? Did I get the right expression? Does it work? And the answer is no. And so then you'll go back and realize, oh, shoot, I forgot to flip these. I'll have to start over, basically. So just be careful when you're copying. So C out on the bottom is 0, 1, 1, 1. So we'll fill it in the same order. Even though the 1s, you could swap them. It doesn't make any difference. | [
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Even though the 1s, you could swap them. It doesn't make any difference. But just do things in the right order. Make sense? Question? Yeah. So remember, in a truth table, we're just listing all possible input combinations. C in is an input. Yeah. Yeah, yeah, we have to consider all possible combinations of inputs. And the A, B, and C in are inputs. And so we've got, in binary order, 0, 0, 0, 0, 0, 1, all the way through 1, 1, 1. | [
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And the A, B, and C in are inputs. And so we've got, in binary order, 0, 0, 0, 0, 0, 1, all the way through 1, 1, 1. Yeah, remember, in the truth table, to the left of the line, typically, is our inputs. And to the right of the line are the outputs. All right, so there came up a question. All right, so there came out, so we can go ahead and solve this one. So let's find loops. So where do we have loops? 1, 1. | [
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So let's find loops. So where do we have loops? 1, 1. Good answer. All right, there's a 1, 1. So which one is that? B, C, N. Yeah, OK, here's another loop. Which one is that? A, C, N. And which one is this? A, B. So you can write that down. That's called a majority function, by the way, because we've got three inputs. And whenever two of them are 1, then the output is 1. Check that. | [
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And whenever two of them are 1, then the output is 1. Check that. It's not terribly important you know that. But if you're interested, it's called a majority function. So that's our carry out. For the sum, we're not going to do a KMAP, because it's not so easy. I'll show you in a second. But it's not easy to just immediately know you've got an XOR coming out. So I wanted to remind you that when you do the sum, the output bit, the low bit, is the odd function of the number of inputs. | [
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0.006705561187118292,
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0.03810737654566765,
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-0.0012694101314991713,
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-0.0002550937933847308,
0.01188014168292284,
0.017930522561073303,
-0.015791160985827446,
0.00041533695184625685,
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0.024736367166042328,
-0.013036733493208885,
-0.0073473695665597916,
0.03291942551732063,
0.0076281605288386345,
0.03163580596446991,
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0.011218276806175709,
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0.014587771147489548,
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0.0008758010808378458,
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0.0009301208192482591,
0.005782961379736662,
0.00274105672724545,
0.017609618604183197,
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0.02468288317322731,
0.024749737232923508,
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0.0059902118518948555,
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0.0014716467121616006,
0.030539384111762047,
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0.025538627058267593,
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0.026902470737695694,
0.005806360859423876,
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0.0013061804929748178,
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0.0020407503470778465,
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0.031047482043504715,
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0.013150387443602085,
0.011231647804379463,
0.010141910053789616,
0.0015652436995878816,
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0.0032508266158401966,
0.002463608281686902,
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0.015496999025344849,
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0.020417530089616776,
0.010777032934129238,
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0.002440209034830332,
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0.0062308902852237225,
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0.029148800298571587,
0.029683640226721764,
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0.01619229093194008,
0.004933902528136969,
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0.037465568631887436,
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0.008503961376845837,
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0.0011031082831323147,
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0.004078157711774111,
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0.007888895459473133,
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0.009032116271555424,
0.02214239165186882,
0.013591630384325981,
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0.010242193005979061,
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0.006143978796899319,
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0.018732784315943718,
0.022289471700787544,
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0.02143372781574726,
0.0034146213438361883,
0.01730208471417427,
0.006993037648499012,
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0.003834136761724949,
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0.03027196414768696,
0.011652834713459015,
0.003190656891092658,
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0.012441723607480526,
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0.012669030576944351,
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0.014360463246703148,
0.008938519284129143,
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0.03904334455728531,
0.028293054550886154,
0.013511404395103455,
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0.021112823858857155,
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0.038668956607580185,
0.02281094156205654,
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-0.0281326025724411,
0.013838994316756725,
-0.0065785362385213375,
0.014266866259276867,
0.03802714869379997,
0.007180231623351574,
0.07573339343070984,
0.0164062287658453,
-0.0027226717211306095,
0.008791438303887844,
-0.013598316349089146,
-0.005428629461675882,
0.016045210883021355,
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0.005709420889616013,
-0.02611358091235161,
0.01798400655388832,
0.0010078398045152426,
-0.014200011268258095,
-0.01641959883272648,
-0.002502050017938018,
0.004138327203691006,
0.01328409742563963,
0.00006471359665738419,
0.00002736878377618268,
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0.0037271687760949135,
-0.01339775137603283,
0.0017633017851039767,
-0.0014532615896314383,
-0.0027443994767963886,
-0.015202837064862251,
0.022690601646900177,
-0.0015869715716689825,
-0.01498890109360218,
-0.017917152494192123,
-0.015069127082824707,
-0.003324367105960846,
-0.030378932133316994,
-0.01960189826786518,
-0.014801707118749619,
-0.0011565922759473324,
0.015309805050492287,
-0.018050862476229668,
0.04915183037519455,
0.010161967016756535,
-0.00818305741995573,
-0.014975530095398426,
-0.01166620571166277,
-0.01573767699301243,
-0.00809614546597004,
-0.005167894996702671,
0.004990729037672281,
-0.00005113366205478087,
-0.034470461308956146
] |
But it's not easy to just immediately know you've got an XOR coming out. So I wanted to remind you that when you do the sum, the output bit, the low bit, is the odd function of the number of inputs. So if you're adding 0's and 1's, it doesn't matter how many you add. The answer you get for the low digit depends on whether they're an odd number of 1's in your sum or not. Here we've got three. | [
0.018732290714979172,
-0.010238809511065483,
-0.001951835467480123,
-0.014162464998662472,
-0.02508741430938244,
0.017053579911589622,
-0.02391498163342476,
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-0.04447253793478012,
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0.01991804875433445,
0.031549122184515,
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0.005142719950526953,
0.018625706434249878,
0.009286207146942616,
0.021943161264061928,
0.005019481293857098,
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0.003723809029906988,
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0.005655659828335047,
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0.002949403366073966,
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0.0038936787750571966,
0.014455573633313179,
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0.025593692436814308,
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0.03192216902971268,
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0.042420778423547745,
0.001893546897917986,
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0.04250071942806244,
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0.0008118769619613886,
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0.005225989501923323,
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0.005499113351106644,
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0.019704878330230713,
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0.011384597048163414,
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0.01002564001828432,
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0.024461228400468826,
0.007580849342048168,
0.002125035971403122,
0.022822486236691475,
0.010831687599420547,
0.0005096089444123209,
0.008500143885612488,
0.015521422028541565,
0.020837342366576195,
0.006018714513629675,
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0.006481692660599947,
0.005595705937594175,
0.014722035266458988,
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0.019531678408384323,
0.020650818943977356,
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0.019691554829478264,
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0.010765071958303452,
0.0364786721765995,
0.0141224954277277,
0.04196779429912567,
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0.017852965742349625,
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0.004576487932354212,
0.005918791051954031,
0.03935646638274193,
0.011517828330397606,
0.02773871272802353,
0.01228390634059906,
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0.02571360021829605,
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0.006641570013016462,
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0.0001076257394743152,
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0.03122936747968197,
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0.0037671092431992292,
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0.012124028988182545,
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0.00617859186604619,
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] |
The answer you get for the low digit depends on whether they're an odd number of 1's in your sum or not. Here we've got three. So whenever there's an odd number in those three, we'll get a 1. So you can check that assertion. So here it's even. You get a 0 | [
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So if you're adding 0's and 1's, it doesn't matter how many you add. The answer you get for the low digit depends on whether they're an odd number of 1's in your sum or not. Here we've got three. So whenever there's an odd number in those three, we'll get a 1. So you can check that assertion. So here it's even. You get a 0. Here it's odd. You get a 1. Odd, 1, even, 0, so forth. But we write that as A XOR B XOR C. | [
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] |
Odd, 1, even, 0, so forth. But we write that as A XOR B XOR C. That doesn't come so easily out of a KMAP. So here's the KMAP. Just for your own benefit, I mean, we're not going to really expect you to do this. But if you ever notice it, you can write it, which is if you see this kind of checkerboard pattern, and they'll vary depending on what particular XOR combination. But the full checkerboard means XOR of all the variables. | [
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But if you ever notice it, you can write it, which is if you see this kind of checkerboard pattern, and they'll vary depending on what particular XOR combination. But the full checkerboard means XOR of all the variables. So here you see a checkerboard. And the answer is A XOR B XOR C. Checkerboard of 0's and 1's. OK, so we can design our full adder. So what I've done is just done the majority function up top with the C out coming out of that. | [
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OK, so we can design our full adder. So what I've done is just done the majority function up top with the C out coming out of that. So you've got the A ended with B on top, A ended with C in here, and B ended with C in from this one, or those three AND gates together. We create C out. And then S is just one XOR gate with A, B, and C coming into it. Yeah. Yeah, so for example, if you flip the 0's and 1's, that would be XNOR. | [
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Yeah. Yeah, so for example, if you flip the 0's and 1's, that would be XNOR. So it would be an XOR gate followed by an inverter, for example. You could also have a partial checkerboard where you had 1 1 0 0 0 0 1 1. And that would be an XOR of not all of the variables. And again, you don't have to recognize those. That's why I put the stars up here. If you're interested or you want to be able to pull the XORs out. | [
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That's why I put the stars up here. If you're interested or you want to be able to pull the XORs out. OK, so I wanted to also, so this was our circuit. I wanted to show you in CMOS what this looks like. So typically in CMOS, we will build the XOR gate. And make that kind of a primitive, just because it takes a handful fewer transistors. So typically, those would be available as gates built out of transistors, kind of like we did the strange gate yesterday. | [
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And make that kind of a primitive, just because it takes a handful fewer transistors. So typically, those would be available as gates built out of transistors, kind of like we did the strange gate yesterday. If you want to see how that works, you can go to the tool and build it yourself. I would suggest doing the two input, because the three input is kind of a pain. But the two input's not so bad. | [
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I would suggest doing the two input, because the three input is kind of a pain. But the two input's not so bad. And then you can check that you got the right answer. But again, it's not something you need to do. But if you're interested in seeing why you get fewer transistors, you can do that. So all we need to do for the top part, remember that any AND OR circuit, any SOP circuit, we can just replace the ANDs and the OR with NAND gates. | [
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But if you're interested in seeing why you get fewer transistors, you can do that. So all we need to do for the top part, remember that any AND OR circuit, any SOP circuit, we can just replace the ANDs and the OR with NAND gates. And we get the same circuit. So that's all I've done here. So this is the CMOS implementation of the full ladder. Yeah. Yes. What? So OR AND becomes NOR NOR. | [
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Yes. What? So OR AND becomes NOR NOR. So remember, yeah, I could pull up this other slide deck | [
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So all we need to do for the top part, remember that any AND OR circuit, any SOP circuit, we can just replace the ANDs and the OR with NAND gates. And we get the same circuit. So that's all I've done here. So this is the CMOS implementation of the full ladder. Yeah. Yes. What? So OR AND becomes NOR NOR. So remember, yeah, I could pull up this other slide deck. But remember that the first step is to replace this OR gate with DeMorgan equivalent, which is complemented inputs and outputs of an AND gate. | [
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So remember, yeah, I could pull up this other slide deck. But remember that the first step is to replace this OR gate with DeMorgan equivalent, which is complemented inputs and outputs of an AND gate. So you have an AND gate here with a complement on the output. That's an AND gate. And all the inputs are complemented. You then slide those inverters down to the other AND gates. And those become NAND also. | [
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You then slide those inverters down to the other AND gates. And those become NAND also. So it's because DeMorgan's law is this OR gate is complemented inputs AND followed by inverter, if that makes sense. And if you look back, I think, to last Friday's lecture that Professor Verdaen gave, those slides will illustrate that for you. Any other questions? Yeah. So again, it just takes a few fewer transistors in CMOS to do an XOR out of transistors. | [
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Any other questions? Yeah. So again, it just takes a few fewer transistors in CMOS to do an XOR out of transistors. And so typically, because of that savings, people will make the XOR gate available built out of transistors as opposed to built out of NAND NOR. So that's the answer. I didn't want to illustrate that for you. I mean, it's going a little too far into the details of how gates are built. | [
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I didn't want to illustrate that for you. I mean, it's going a little too far into the details of how gates are built. But you can do it in the online tool if you're interested. And again, I would do the two-input version. I myself did not bother with the three-input version. I did do the two-input. It's fine. All right, so we have this one-bit adder that we just designed using what we learned in the last week or two. | [
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It's fine. All right, so we have this one-bit adder that we just designed using what we learned in the last week or two. So then how do we actually build an adder for n bits? So when we add stuff, we add one column at a time. And one of these adders, these full adders, is going to add one column for us. So all we need to do is then hook them together. So we'll feed 0 into the carry input for the least significant bit. | [
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So all we need to do is then hook them together. So we'll feed 0 into the carry input for the least significant bit. The carry out of the most significant bit is the adder's carry out. And then for the other signals, we'll connect C out to C in for adjacent bits. Then we'll take A and B and divide them up. And then feed one bit of each into each of the full adders and collect the bits of S from the full adders. | [
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Then we'll take A and B and divide them up. And then feed one bit of each into each of the full adders and collect the bits of S from the full adders. So let me show you that. So here is a chain of n full adders with a dot, dot, dot in the middle, since we don't know what n is. But you can see we're feeding a 0. Is this big enough to see in the back? Can you see this? OK. You're feeding a 0 into the carry in of the first one, the 0-th bit. | [
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Is this big enough to see in the back? Can you see this? OK. You're feeding a 0 into the carry in of the first one, the 0-th bit. And then between them, we're taking carry out, feeding it to carry in, all the way down the chain. And then the output here is the carry out. You can see A of n minus 1 and B of n minus 1 going to the bit slice and so forth, all the way down to A sub 0, B sub 0. | [
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] |
And then the output here is the carry out. You can see A of n minus 1 and B of n minus 1 going to the bit slice and so forth, all the way down to A sub 0, B sub 0. Those are the least significant bits of A and B | [
-0.02243545651435852,
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0.002725754166021943,
0.014994270168244839,
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0.008336367085576057,
0.03432457149028778,
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You're feeding a 0 into the carry in of the first one, the 0-th bit. And then between them, we're taking carry out, feeding it to carry in, all the way down the chain. And then the output here is the carry out. You can see A of n minus 1 and B of n minus 1 going to the bit slice and so forth, all the way down to A sub 0, B sub 0. Those are the least significant bits of A and B. And then down here, S sub 0 comes out, S sub 1, all the way up to S sub n minus 1. | [
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Those are the least significant bits of A and B. And then down here, S sub 0 comes out, S sub 1, all the way up to S sub n minus 1. So this is the most significant bit on the left, least significant bit on the right. So all we have to do is take n copies of our full adder, wire them together, and now we have an ended adder. So this is a ripple carry adder. Why is it called that? The word ripple is referring to something like a ripple on a pond. | [
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So this is a ripple carry adder. Why is it called that? The word ripple is referring to something like a ripple on a pond. So you throw a rock into a pond, and you see these ripples spreading out from where the rock hit the water. And they move kind of slowly. So the carry information is moving kind of slowly between bit slice to bit slice, until finally it gets over here. So the speed of this ripple carry adder is not great. | [
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So the carry information is moving kind of slowly between bit slice to bit slice, until finally it gets over here. So the speed of this ripple carry adder is not great. It's a simple design. It's an easy to build design. In practice, we use actually tree adders that are significantly faster than a carry adder for when you get to 32 bits or 64 bits. So we may get to look at that at the end of the class. | [
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In practice, we use actually tree adders that are significantly faster than a carry adder for when you get to 32 bits or 64 bits. So we may get to look at that at the end of the class. But for now, it's a perfectly good way to build an adder. You can also think of it as a bit sliced adder, because for each of the bits, we have just the same piece of logic. And we just copy it. You want a 10-bit adder, just make 10 copies. | [
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And we just copy it. You want a 10-bit adder, just make 10 copies. You want a 20-bit adder, 32-bit adder, 100-bit adder. Make the appropriate number of copies, wire them together, and you're done. So fairly simple design. Loop, you mean to speed up the carry? Yeah, so let me move forward a couple of slides. So let me move forward a couple of slides. Let me show you first what an n-bit adder would look like in a circuit diagram. | [
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So let me move forward a couple of slides. Let me show you first what an n-bit adder would look like in a circuit diagram. So once you build the n-bit adder, doesn't actually matter how you build it. You can represent it this way. So you've got this sort of funny-looking B thing. Typically, you've got to label it as an adder. The n bits, you see the input with a crosshatch n. That means n bits wide. | [
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The n bits, you see the input with a crosshatch n. That means n bits wide. So n bits of A, n bits of B, they're added together. Sometimes people just put a plus. The sum is n bits wide. The carry in is one bit, no crosshatch. Carry out is one bit also. So there's a shape. You can also then, sorry, illustration of the crosshatching. You can also then hook them together. If you have an n-bit adder and you want a two-n-bit adder, you can simply put them side by side. | [
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You can also then hook them together. If you have an n-bit adder and you want a two-n-bit adder, you can simply put them side by side. Again, the implementation doesn't matter. Once you've got the adder implemented, you can put two of them together pretty easily. You can also, as Sasha mentioned, do this virtually in software | [
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Carry out is one bit also. So there's a shape. You can also then, sorry, illustration of the crosshatching. You can also then hook them together. If you have an n-bit adder and you want a two-n-bit adder, you can simply put them side by side. Again, the implementation doesn't matter. Once you've got the adder implemented, you can put two of them together pretty easily. You can also, as Sasha mentioned, do this virtually in software. | [
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] |
Once you've got the adder implemented, you can put two of them together pretty easily. You can also, as Sasha mentioned, do this virtually in software. So if you take the carry out bit of one physical adder and then somehow manage to put it back as the carry in, so instead of putting the carry in as a zero for the second part, you can add the next higher sets of bits and then continue that as often as you want. | [
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You can also, as Sasha mentioned, do this virtually in software. So if you take the carry out bit of one physical adder and then somehow manage to put it back as the carry in, so instead of putting the carry in as a zero for the second part, you can add the next higher sets of bits and then continue that as often as you want. So in a typical processor, you might have a 32- or 64-bit adder. | [
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So if you take the carry out bit of one physical adder and then somehow manage to put it back as the carry in, so instead of putting the carry in as a zero for the second part, you can add the next higher sets of bits and then continue that as often as you want. So in a typical processor, you might have a 32- or 64-bit adder. But you can use that to add arbitrarily large numbers. So you can write software libraries that will do arbitrarily large or even quasi-infinite arithmetic by simply dynamically using the adder to keep adding the bits until you finish for as much as you need. | [
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But you can use that to add arbitrarily large numbers. So you can write software libraries that will do arbitrarily large or even quasi-infinite arithmetic by simply dynamically using the adder to keep adding the bits until you finish for as much as you need. So usually, in practice, you would use that to check whether there was overflow. And that would go into a carry register. So I mean, at this point, we haven't seen how to store bits or anything. | [
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And that would go into a carry register. So I mean, at this point, we haven't seen how to store bits or anything. That's another week, week and a half out. So at the end, right now, all we know how to do is build combinational logic. So there's a signal coming out, and we can look at it as humans. But in a real design, you would end up storing that somewhere. And then, for example, software could look at it and see if there was a carry out. | [
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But in a real design, you would end up storing that somewhere. And then, for example, software could look at it and see if there was a carry out. Did that answer your question earlier? OK, good. Yeah. OK. AUDIENCE MEMBER 2 Yeah, you're getting the right guy. You're getting the right guy. Yeah. Yeah, so the question is, well, wouldn't it be better if we had a smaller adder and reused it over and over again? It's smaller, but slower. | [
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Yeah. Yeah, so the question is, well, wouldn't it be better if we had a smaller adder and reused it over and over again? It's smaller, but slower. So actually, this trade-off is something we'll spend a fair bit of time on in about a week and a half to two weeks. So we will look at it in detail at the circuit design level. But you can then do the same thing at a broader level. Yeah. Happened if you start with 1 instead of 0. | [
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Yeah. Happened if you start with 1 instead of 0. That's a good question. What does happen? So what difference would it make if you put a 1 in? So you would still get a plus b plus 1, right? Well, the answer would be a plus b plus 1. So I mean, if what you want is a plus b, then it's 1 too high. But what if you wanted a plus b plus 1? Then it's the right answer. There's a reason I mentioned that, but I won't tell you why now. | [
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But what if you wanted a plus b plus 1? Then it's the right answer. There's a reason I mentioned that, but I won't tell you why now. The crosshatch, I'm sorry. Yeah, this thing? So this means that there are actually n bits of signal coming in from that wire. So b is actually n bits wide. a is n bits wide. The sum is n bits wide. Yeah. Yeah, so your typical processor will have either 32 or 64-bit adders in it. | [
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Yeah. Yeah, so your typical processor will have either 32 or 64-bit adders in it. But they will not be implemented as ripple carry. They'll be implemented as tree-based adders. Yeah, yeah, yeah | [
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The crosshatch, I'm sorry. Yeah, this thing? So this means that there are actually n bits of signal coming in from that wire. So b is actually n bits wide. a is n bits wide. The sum is n bits wide. Yeah. Yeah, so your typical processor will have either 32 or 64-bit adders in it. But they will not be implemented as ripple carry. They'll be implemented as tree-based adders. Yeah, yeah, yeah. | [
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They'll be implemented as tree-based adders. Yeah, yeah, yeah. So what I showed you is not really the way you build an adder. It's a simple way to build an adder. Not in practice, really, anymore. OK, so now that we have an adder, I want you to think about what we did. So how many of you can add two-digit numbers? OK, come on, raise your hands. I know you can do it. What about five-digit numbers? What about 5,000-digit numbers? Yeah. | [
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I know you can do it. What about five-digit numbers? What about 5,000-digit numbers? Yeah. Does it matter? Does it mean that? It doesn't matter, right? Does it matter if I just say some finite number of digits? You can do it, right? You think you can do it? OK. Yeah, I mean, I can make it arbitrary, right? So I can make it infinite. You wouldn't want to do it, but you can do it. Have you ever seen a proof you're correct? How do you know you can do it? What kind of proof? Proof by induction, maybe? So I think when you learned to add, probably you hadn't seen proof by induction, right? You were probably in elementary school, and probably no one said, let me show you the proof by induction so you don't get worried about digits, right? They probably just said, look, it works. | [
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You wouldn't want to do it, but you can do it. Have you ever seen a proof you're correct? How do you know you can do it? What kind of proof? Proof by induction, maybe? So I think when you learned to add, probably you hadn't seen proof by induction, right? You were probably in elementary school, and probably no one said, let me show you the proof by induction so you don't get worried about digits, right? They probably just said, look, it works. You can tell. | [
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Have you ever seen a proof you're correct? How do you know you can do it? What kind of proof? Proof by induction, maybe? So I think when you learned to add, probably you hadn't seen proof by induction, right? You were probably in elementary school, and probably no one said, let me show you the proof by induction so you don't get worried about digits, right? They probably just said, look, it works. You can tell. You can keep adding digits. | [
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You can tell. You can keep adding digits. It's OK. But if you really wanted to prove it, well, what do you need? You need to know how to some base case, right? So hey, I've memorized an addition table, and I verified it for one-digit numbers, and it works. So I'm good. That's my base case. And I know that if I can add n-digit numbers, then I can show based, for example, in place value that I can do one more column, and then I can add n plus one-digit numbers. | [
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That's my base case. And I know that if I can add n-digit numbers, then I can show based, for example, in place value that I can do one more column, and then I can add n plus one-digit numbers. And I'll get the right answers. And so you could prove both of those, and then you'd have a proof by induction that, in fact, addition works for arbitrary finite number of digits. Turns out it doesn't work so well for infinite digits, but that's a different story. | [
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] |
And so you could prove both of those, and then you'd have a proof by induction that, in fact, addition works for arbitrary finite number of digits. Turns out it doesn't work so well for infinite digits, but that's a different story. So when we designed a ripple caryata, we kind of also assumed proof by induction. I didn't prove it to you. I just said, well, we'll just base it on the human approach, and you think the human approach works, so should this thing that we're doing, right? Well, you could prove it by induction. | [
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I didn't prove it to you. I just said, well, we'll just base it on the human approach, and you think the human approach works, so should this thing that we're doing, right? Well, you could prove it by induction. We know how to add a bit. We made a truth table, binary addition table. It's very simple, just four cases or eight cases if you want to do three bits. And we went through that a couple of times. | [
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] |
It's very simple, just four cases or eight cases if you want to do three bits. And we went through that a couple of times | [
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I didn't prove it to you. I just said, well, we'll just base it on the human approach, and you think the human approach works, so should this thing that we're doing, right? Well, you could prove it by induction. We know how to add a bit. We made a truth table, binary addition table. It's very simple, just four cases or eight cases if you want to do three bits. And we went through that a couple of times. | [
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It's very simple, just four cases or eight cases if you want to do three bits. And we went through that a couple of times. Given that we can build an n-bit adder, we then have to show we can build an n plus one-bit adder by attaching one more full adder, a one-bit adder, to our n-bit adder. So those two steps are also something that I think if I put it on your homework, you would come back having done it and say, OK, great, I've proven that I can do arbitrarily large adder design using ripple carry approach. | [
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Given that we can build an n-bit adder, we then have to show we can build an n plus one-bit adder by attaching one more full adder, a one-bit adder, to our n-bit adder. So those two steps are also something that I think if I put it on your homework, you would come back having done it and say, OK, great, I've proven that I can do arbitrarily large adder design using ripple carry approach. So in 220, so one reason I want to mention this, one is to get you understanding that for these bit slice design, we're basically just doing proof by induction. | [
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So those two steps are also something that I think if I put it on your homework, you would come back having done it and say, OK, great, I've proven that I can do arbitrarily large adder design using ripple carry approach. So in 220, so one reason I want to mention this, one is to get you understanding that for these bit slice design, we're basically just doing proof by induction. So any time you have a problem where you can prove it works by induction, you can do a bit slice design and design a small piece of logic for one or two bits or four bits or whatever, and then put a bunch of copies down, and that'll work. | [
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So in 220, so one reason I want to mention this, one is to get you understanding that for these bit slice design, we're basically just doing proof by induction. So any time you have a problem where you can prove it works by induction, you can do a bit slice design and design a small piece of logic for one or two bits or four bits or whatever, and then put a bunch of copies down, and that'll work. You need to make sure that that's an applicable approach or a useful approach for your problem. | [
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So any time you have a problem where you can prove it works by induction, you can do a bit slice design and design a small piece of logic for one or two bits or four bits or whatever, and then put a bunch of copies down, and that'll work. You need to make sure that that's an applicable approach or a useful approach for your problem. But when you can do that, it's a simple approach. The other reason I mention it is in 220, you're going to write software that does what's called recursion. | [
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But when you can do that, it's a simple approach. The other reason I mention it is in 220, you're going to write software that does what's called recursion. So the recursive functions are going to call themselves. A lot of people end up finding this confusing. So it's the same thing. So you say, well, there's some base case. We call them stopping conditions in software, for which you know the answer. | [
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So you say, well, there's some base case. We call them stopping conditions in software, for which you know the answer. And so you need to have that base case. And then you say, well, given that you can write a function that works for input of size n, you have to prove that you can write a function that works for input of size n plus 1 by handling whatever the extra 1 is and then calling the function recursively for the n. | [
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And so you need to have that base case. And then you say, well, given that you can write a function that works for input of size n, you have to prove that you can write a function that works for input of size n plus 1 by handling whatever the extra 1 is and then calling the function recursively for the n. So it's exactly the same thing mathematically, but it tends to confuse people. | [
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And then you say, well, given that you can write a function that works for input of size n, you have to prove that you can write a function that works for input of size n plus 1 by handling whatever the extra 1 is and then calling the function recursively for the n. So it's exactly the same thing mathematically, but it tends to confuse people. And I think the reason that it tends to confuse people is there's this assumption in the inductive step. | [
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