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So when we talk about bitwise operations, we usually use the words. The reason is that it can get confusing, because when we talk about sets of bits, we can do things like plus | [
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So this is what you typically use in a computer where you've got names representing sets of bits, and then you might AND whole sets of bits together. So it's another way to generalize the idea of the Boolean expressions, or the Boolean functions that we've talked about. Be careful not to mix the algebras. So when we talk about bitwise operations, we usually use the words. The reason is that it can get confusing, because when we talk about sets of bits, we can do things like plus. | [
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So when we talk about bitwise operations, we usually use the words. The reason is that it can get confusing, because when we talk about sets of bits, we can do things like plus. We talked about how we define addition on unsigned and two-complement numbers, and if we use plus to mean addition, then having plus also mean or gets very confusing very quickly. So when we talk about bitwise operations, try just to use the words. | [
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We talked about how we define addition on unsigned and two-complement numbers, and if we use plus to mean addition, then having plus also mean or gets very confusing very quickly. So when we talk about bitwise operations, try just to use the words. When we're talking about Boolean expressions, plus will mean or, and multiplication will mean AND. So try not to mix those, otherwise people will not understand what you're trying to do, or worse, you'll get confused as you try to derive something. | [
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When we're talking about Boolean expressions, plus will mean or, and multiplication will mean AND. So try not to mix those, otherwise people will not understand what you're trying to do, or worse, you'll get confused as you try to derive something. I thought about going back and showing you another tool. There's another truth table tool. A lot of that is actually not things you need to know until after the first midterm, but the first tab in there will help you fill in truth tables. | [
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There's another truth table tool. A lot of that is actually not things you need to know until after the first midterm, but the first tab in there will help you fill in truth tables. So if you feel like, OK, I want to get some exercise looking at Boolean expressions and filling in truth tables, there is a tool that will help you do that, give you the same sort of feedback, same interface, basically. | [
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A lot of that is actually not things you need to know until after the first midterm, but the first tab in there will help you fill in truth tables. So if you feel like, OK, I want to get some exercise looking at Boolean expressions and filling in truth tables, there is a tool that will help you do that, give you the same sort of feedback, same interface, basically. And that's just the next tool down, two tools down. | [
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So if you feel like, OK, I want to get some exercise looking at Boolean expressions and filling in truth tables, there is a tool that will help you do that, give you the same sort of feedback, same interface, basically. And that's just the next tool down, two tools down. OK, so now I have a question for you. So how many different functions exist on n bits of input? So I only showed you four, right? And after class on Friday, people came up and said, well, what about this function? What about that function? I said, well, those are all nice functions. | [
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OK, so now I have a question for you. So how many different functions exist on n bits of input? So I only showed you four, right? And after class on Friday, people came up and said, well, what about this function? What about that function? I said, well, those are all nice functions. So how are you going to find that answer? So let's start with n equals one. How many functions are there for, let's say, I have an input a. | [
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So how are you going to find that answer? So let's start with n equals one. How many functions are there for, let's say, I have an input a. How many functions can I define on a? I'm hearing a bunch of different numbers. Let's name them. So what can I define on a? So I hear a zero, right? So I don't care what a is, the answer is zero. So that's one function. And one, that's two functions. | [
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So that's one function. And one, that's two functions. What else can I do with a? Zero and one don't depend on a. There must be functions that depend on a. Not a, and then just a by itself. So I think that's it. So you've got four functions. OK, I'm running out of fingers, so we're going to have to do this another way. So what about n equals two? How can I figure out how many functions there are? There are four on n equals one. | [
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] |
OK, I'm running out of fingers, so we're going to have to do this another way. So what about n equals two? How can I figure out how many functions there are? There are four on n equals one. For n equals two, try this. So here's a truth table. Got some function c as a function of f on a and b. I'm sorry, f is the function, a and b are the inputs. So instead of writing values, let me give those outputs names. | [
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I'm sorry, f is the function, a and b are the inputs. So instead of writing values, let me give those outputs names. Let's call them c sub i. C zero, one, two, and three. Now depending on how I pick c zero, one, two, and three, every combination gives me a unique function. So if I were to put four zeros, well that's the zero function on two inputs | [
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Got some function c as a function of f on a and b. I'm sorry, f is the function, a and b are the inputs. So instead of writing values, let me give those outputs names. Let's call them c sub i. C zero, one, two, and three. Now depending on how I pick c zero, one, two, and three, every combination gives me a unique function. So if I were to put four zeros, well that's the zero function on two inputs. | [
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Now depending on how I pick c zero, one, two, and three, every combination gives me a unique function. So if I were to put four zeros, well that's the zero function on two inputs. If I were to put four ones, that's the one function on two inputs. Every combination I pick gives me a different function. It's a one-to-one mapping between choices and functions. So how many choices do I have? Sixteen, right? So I've got two choices for c zero. | [
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It's a one-to-one mapping between choices and functions. So how many choices do I have? Sixteen, right? So I've got two choices for c zero. The choice of c zero doesn't affect my choice of c one, so I also pick c one. Each of these is just a bit, right? Zero, one. So I get two choices for c zero, two choices for c one, two choices for c two, two choices for c three. I multiply all those choices out. | [
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So I get two choices for c zero, two choices for c one, two choices for c two, two choices for c three. I multiply all those choices out. Overall, I have sixteen two-to-the-four different ways I could fill in this truth table. Each truth table is a different function. So there are sixteen functions on two inputs. So what about three? Good. So let's write our truth table, and we'll call the outputs d sub i, and we've got eight of them. | [
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So what about three? Good. So let's write our truth table, and we'll call the outputs d sub i, and we've got eight of them. For each of them, we have two choices. So two times two times two, I'm going to lose count, eight times. So two to the eighth. I want to point out that's two to the two to the three, right? Eight is two to the three. Four to the n. Maybe. Yes, you can do four to the n. | [
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Maybe. Yes, you can do four to the n. All right. So can we generalize that to n bits? Please don't make me write a truth table. All right. So what do we do? So without drawing the truth table, n bits means two to the n rows, right? If you were to write a truth table on n bits of input, it's got two to the n possible input combinations, right? And so for each of those rows, we have an output variable. | [
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All right. So what do we do? So without drawing the truth table, n bits means two to the n rows, right? If you were to write a truth table on n bits of input, it's got two to the n possible input combinations, right? And so for each of those rows, we have an output variable. We have two choices for that output variable. So you multiply all those twos together, you get two to the two to the n, or four to the n is another way to write that. | [
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We have two choices for that output variable. So you multiply all those twos together, you get two to the two to the n, or four to the n is another way to write that. So two to the two to the n functions on n bits. But I only showed you four, right? Yeah. You have a question? Okay. Eric? So you said that you can have two to the n possibilities of putting those bits together. Two to the n rows, yeah. | [
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Eric? So you said that you can have two to the n possibilities of putting those bits together. Two to the n rows, yeah. Two to the n possible ways to have input. But there's only eight outcomes in each of these. Yeah, so here n equals three, right? So what's the total possible outcome? When n equals three, we have eight rows. There are eight different patterns, right? And when n equals four, we would have 16 rows. | [
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Yeah, so here n equals three, right? So what's the total possible outcome? When n equals three, we have eight rows. There are eight different patterns, right? And when n equals four, we would have 16 rows. n equals five, we'd have 32. If we did n equals 10, you'd have 1,000 rows in your truth table. You'd have 1,000 variables to define your function. 1,024. That's just the rows of them, what about the others? So for each row, you have one of these variables, and for each variable, you have two choices. | [
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1,024. That's just the rows of them, what about the others? So for each row, you have one of these variables, and for each variable, you have two choices. So you multiply all those twos together | [
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n equals five, we'd have 32. If we did n equals 10, you'd have 1,000 rows in your truth table. You'd have 1,000 variables to define your function. 1,024. That's just the rows of them, what about the others? So for each row, you have one of these variables, and for each variable, you have two choices. So you multiply all those twos together. There are two to the n of them. So you get this one. | [
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There are two to the n of them. So you get this one. That's where that comes from. You're all one. Yes, each output, you choose zero or one, and all of those choices specify a unique function. Yeah? So is this a row, column expression that we did before? No, it might take us a minute to work it out in our head. Yes? So a four to the two? Yes, yeah. Yeah, so you can massage this algebraically as you like, right? So two to the two is four, and you should know that if I have this power to a power, that you can actually do the first one. | [
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Yes? So a four to the two? Yes, yeah. Yeah, so you can massage this algebraically as you like, right? So two to the two is four, and you should know that if I have this power to a power, that you can actually do the first one. And then if you plug in a specific value of n, you can also do the other one. They're associative. Do the two to the n one, which we did already on the previous slides. | [
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They're associative. Do the two to the n one, which we did already on the previous slides. OK. OK, so now the hard part. So I only gave you four functions. Got a lot of paper? I can see the headline now, Illinois professor single-handedly destroys rainforest. All right, I can't have that. So your alternate homework. Instead, you can understand logical completeness. It's your choice, really. | [
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Instead, you can understand logical completeness. It's your choice, really. But I'm not sure you really want to use that much paper. All right, so I have a claim. If you give me enough two input and, two input or, and not functions, I can compose those mathematically to make any function you want on any number of variables. Do you believe me? Really? Why? Why do you believe me? I'm a professor. | [
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If you give me enough two input and, two input or, and not functions, I can compose those mathematically to make any function you want on any number of variables. Do you believe me? Really? Why? Why do you believe me? I'm a professor. The functions, you can set them up back to back where you can keep it. Yes. That's long logic tree. Yes, yes. So the question is, how am I going to do this? We're going to have to be able to put the functions together mathematically. | [
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Yes, yes. So the question is, how am I going to do this? We're going to have to be able to put the functions together mathematically. In math, we call that composition. So we put the output of one function into another. I'm actually going to show you the proofs using gates. So we're going to draw it as circuit diagrams, but it's equivalent to mathematical composition of functions. Yeah. | [
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So we're going to draw it as circuit diagrams, but it's equivalent to mathematical composition of functions. Yeah. So you're saying that those three functions, if, like, let's say I want to do some kind of actually use any of these three functions, and I could get that. Yes. So the question is, I'm saying, regardless of what function you want to implement, no matter what the function is, I claim I can do it with just these three functions. | [
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Yes. So the question is, I'm saying, regardless of what function you want to implement, no matter what the function is, I claim I can do it with just these three functions. And yes, that's absolutely right. And so I'll come after I show you how. So the proof will be what we call by construction. So in other words, I will show you a way, regardless of what the function you want to implement is, I will show you exactly how to do it using these three kinds of gates. | [
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So the proof will be what we call by construction. So in other words, I will show you a way, regardless of what the function you want to implement is, I will show you exactly how to do it using these three kinds of gates. A way to do it. It won't be the best way to do it. This is just proof of concept proof. But it will be a way that you can always build any function. So we'll come back to why that's important after we walk through the details. | [
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But it will be a way that you can always build any function. So we'll come back to why that's important after we walk through the details | [
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So the proof will be what we call by construction. So in other words, I will show you a way, regardless of what the function you want to implement is, I will show you exactly how to do it using these three kinds of gates. A way to do it. It won't be the best way to do it. This is just proof of concept proof. But it will be a way that you can always build any function. So we'll come back to why that's important after we walk through the details. | [
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But it will be a way that you can always build any function. So we'll come back to why that's important after we walk through the details. Let's start the proof. So here's a diagram. So you remember, hopefully, this is the AND gate. So if I hook these together, what do I get out of the right side? So ABC, right? A and B and C. Good. So it's a three input AND, in other words. And if I have three inputs on the left, I put three values into that gate formation, what comes out is the AND of those three variables. | [
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So it's a three input AND, in other words. And if I have three inputs on the left, I put three values into that gate formation, what comes out is the AND of those three variables. So out of those two, two input ANDs, I produce the three input AND. So what about this one? Yeah, four input AND, right? So I've shown you how to build a three input AND. Take one of those three input ANDs, hook it together with one more two input AND, now I have a four input AND. | [
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So what about this one? Yeah, four input AND, right? So I've shown you how to build a three input AND. Take one of those three input ANDs, hook it together with one more two input AND, now I have a four input AND. So you probably remember proof by induction from high school or something like that, right? With like AB? Oh, you have to do AB then? Ah. So does it matter what order you do? No, because AND is associative and commutative. | [
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So you probably remember proof by induction from high school or something like that, right? With like AB? Oh, you have to do AB then? Ah. So does it matter what order you do? No, because AND is associative and commutative. So you could reorder these and do the diagrams a different way and you'd get the same answer. OK, so you remember proof by induction? What do I need to prove? OK, so what I'm hearing is if it's true for some N, then I have to prove it's also true for N plus 1. | [
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So you could reorder these and do the diagrams a different way and you'd get the same answer. OK, so you remember proof by induction? What do I need to prove? OK, so what I'm hearing is if it's true for some N, then I have to prove it's also true for N plus 1. Is there something else I need to prove? Yeah, I better have a base case, right? So I have an example from Yale Pat for showing why if you forget your base case, your proof doesn't work. | [
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OK, so you remember proof by induction? What do I need to prove? OK, so what I'm hearing is if it's true for some N, then I have to prove it's also true for N plus 1. Is there something else I need to prove? Yeah, I better have a base case, right? So I have an example from Yale Pat for showing why if you forget your base case, your proof doesn't work. But I'll skip that one because I think people remember, got to have a base case. | [
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Is there something else I need to prove? Yeah, I better have a base case, right? So I have an example from Yale Pat for showing why if you forget your base case, your proof doesn't work. But I'll skip that one because I think people remember, got to have a base case. So here's our proof by induction. So by induction, we've got base case of N equals 2, for example. You want a two input AND? Well, that you gave me already. | [
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So by induction, we've got base case of N equals 2, for example. You want a two input AND? Well, that you gave me already. So I'll take one off the shelf and I'll put it down. There's your two input AND. That's the base case. You want a bigger, more input AND? We'll do the construct we just did. So we'll start assuming I know how to build an input AND. I'll take one of those. I'll hook together one more two input AND down here. | [
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I'll take one of those. I'll hook together one more two input AND down here. And that will give me my N plus 1 input AND. So that's our proof by induction, that for any finite number of inputs, it can give you that AND gate. Happy? OK. Yeah, you should believe me when I prove things. I don't believe it just because I say it. I occasionally make a mistake. All right. So a couple comments. | [
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All right. So a couple comments. So first of all, the functional form is here. So if you want to see it algebraically, you can write it that way instead of drawing circuit diagrams. But this is not a practical way to build bigger gates. So don't go back and say, OK, now I know how to build 10 input AND gates. This is not the right way to do it | [
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I don't believe it just because I say it. I occasionally make a mistake. All right. So a couple comments. So first of all, the functional form is here. So if you want to see it algebraically, you can write it that way instead of drawing circuit diagrams. But this is not a practical way to build bigger gates. So don't go back and say, OK, now I know how to build 10 input AND gates. This is not the right way to do it. | [
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So don't go back and say, OK, now I know how to build 10 input AND gates. This is not the right way to do it. I just want to show you that it can be done so that later we can talk about, well, once we have this logical completeness idea, now we know what we need to build at the technology layer so that people at higher abstraction layers can make use of any function they want. So I'm kind of giving you a hint at why it's relevant. | [
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I just want to show you that it can be done so that later we can talk about, well, once we have this logical completeness idea, now we know what we need to build at the technology layer so that people at higher abstraction layers can make use of any function they want. So I'm kind of giving you a hint at why it's relevant. But it's not practically the right way to approach the problem. | [
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So I'm kind of giving you a hint at why it's relevant. But it's not practically the right way to approach the problem. It's just a proof right now. OK, so now I can simplify my claim. So before I said I had to use two input AND, two input ORs. But I just showed you how to build those arbitrary input ANDs and ORs. So now I just cross out the two input because all I need to prove is that with arbitrary number of, well, finite number of input AND, finite number of input OR, and not gates, I can build any function. | [
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But I just showed you how to build those arbitrary input ANDs and ORs. So now I just cross out the two input because all I need to prove is that with arbitrary number of, well, finite number of input AND, finite number of input OR, and not gates, I can build any function. For OR, it's the same idea. Just go through the same proof structure, replace AND with OR. So let's start by thinking about functions that produce 1, 1 in their truth table. | [
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Just go through the same proof structure, replace AND with OR. So let's start by thinking about functions that produce 1, 1 in their truth table. So a whole bunch of rows, maybe. I might have 10 inputs, 1,000 rows. One row produces output of 1. So that's a tiny number of functions, a very small number of functions. But let me start with those, and I'll show you how to build those. So each of these functions, exactly one combination of inputs will produce a 1. | [
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But let me start with those, and I'll show you how to build those. So each of these functions, exactly one combination of inputs will produce a 1. Any other combination of inputs produces 0. So let me give you an example. So here's Q. Here is one row here. All the other rows produce 0. But tell me, when is Q equal to 1? Yeah, so you just say, well, I mean, the inputs have to be here, right? So A has to be 1, B has to be 0, C has to be 1. | [
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All the other rows produce 0. But tell me, when is Q equal to 1? Yeah, so you just say, well, I mean, the inputs have to be here, right? So A has to be 1, B has to be 0, C has to be 1. So B is 0 when not B is 1. I need that expression to be true. So if I write Q equals A B prime C, then that's the function. That and is equal to 1 whenever A is 1 and B is 0 and C is 1. So any time you have this subset of functions, I can just use one and gate and maybe some not gates and build the function for you. | [
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That and is equal to 1 whenever A is 1 and B is 0 and C is 1. So any time you have this subset of functions, I can just use one and gate and maybe some not gates and build the function for you. So to build an arbitrary function, we need one more step. So the first step is, well, look at the truth table. And for every row, build the and function that produces the 1 for that row. For every row that produces a 1, I should say. | [
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And for every row, build the and function that produces the 1 for that row. For every row that produces a 1, I should say. Yeah. AUDIENCE MEMBER 2. What's the problem with the 1? So that's a good question. So if you plug it in, remember this is an and function. So all of them have to be 1. And so A has to be 1. So A can't have any other value. B prime has to be 1, which means B has to be 0. | [
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] |
So A can't have any other value. B prime has to be 1, which means B has to be 0. So B can't have any other value. And C has to be 1. So C can't have any other value | [
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Yeah. AUDIENCE MEMBER 2. What's the problem with the 1? So that's a good question. So if you plug it in, remember this is an and function. So all of them have to be 1. And so A has to be 1. So A can't have any other value. B prime has to be 1, which means B has to be 0. So B can't have any other value. And C has to be 1. So C can't have any other value. So if they have any other values, this combination produces 0. | [
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So C can't have any other value. So if they have any other values, this combination produces 0. Yeah. So again, it's an and function. So remember, and means all. So all of these have to be true when I use an and. So for each one of them, that constrains that input to one value. And each input appears exactly once. So there's a name for this that you'll learn later. If you want to learn it now, it's fine. | [
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So there's a name for this that you'll learn later. If you want to learn it now, it's fine. It's called a minterm. It's the thing that produces a 1 for exactly one row of the truth table. And it's a product of all of the inputs or their complements. All right, so we'll produce those minterms. So for every combination where we produce a 1 output, we'll produce that and. Then we'll take all those ands, and we'll put them all together into one big or gate. | [
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So for every combination where we produce a 1 output, we'll produce that and. Then we'll take all those ands, and we'll put them all together into one big or gate. That's our function. That's it. So that gives us what we call a sum of products, because the or is a sum. We use plus notation for the or. And each of the things that we're adding together is a product. It's an and. So we use multiplication notation and algebra for that. | [
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It's an and. So we use multiplication notation and algebra for that. That's a really inefficient way to build a real system. We'll talk about that in about a week and a half, how can you do a better job. But it always works. So in other words, if I give you those gates, you can build anything. And that's kind of the point of logical completeness. So we say that the set and or not is logically complete, because as we showed, you can build any function on any number of inputs using only those three. | [
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And that's kind of the point of logical completeness. So we say that the set and or not is logically complete, because as we showed, you can build any function on any number of inputs using only those three. If you want to show that something else is logically complete, you don't have to go to these lengths. If you can show me that given some set of functions, you can build and, you can build or, and you can build not, then you're done. | [
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If you want to show that something else is logically complete, you don't have to go to these lengths. If you can show me that given some set of functions, you can build and, you can build or, and you can build not, then you're done. You don't have to show anything else. So for example, when we talk about circuit technology and the CMOS devices that we use today in all digital electronics, not all, but almost all, what they actually produce is the NAND function and the NOR function. | [
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You don't have to show anything else. So for example, when we talk about circuit technology and the CMOS devices that we use today in all digital electronics, not all, but almost all, what they actually produce is the NAND function and the NOR function. So the AND gate followed by a NOT gate is kind of the natural thing to build in those technologies. Turns out that NAND all by itself is logically complete. | [
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So the AND gate followed by a NOT gate is kind of the natural thing to build in those technologies. Turns out that NAND all by itself is logically complete. So it may be something to think about. How can you take NAND, two input NAND, and build and or and not from that? Because that one function all by itself can build anything. Yeah, question? It's NAND. NOR is also logically complete. | [
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Yeah, question? It's NAND. NOR is also logically complete. So this stands for not and. And it's A and B complemented or by DeMorgan's law, that. They're the same. That one function, you can build anything. Yeah? Can't you just plug in the NAND to make it NAND? First, you have to show me how to build an inverter with NAND. Yes, I mean, it's not very hard to build | [
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Yeah, question? It's NAND. NOR is also logically complete. So this stands for not and. And it's A and B complemented or by DeMorgan's law, that. They're the same. That one function, you can build anything. Yeah? Can't you just plug in the NAND to make it NAND? First, you have to show me how to build an inverter with NAND. Yes, I mean, it's not very hard to build. But that's how you would prove that NAND by itself is logically complete, not to go through the whole constructive thing again, but just to show you can get these three. | [
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Yes, I mean, it's not very hard to build. But that's how you would prove that NAND by itself is logically complete, not to go through the whole constructive thing again, but just to show you can get these three. All right, so why do you care? So imagine you're working on a new device technology. Maybe it's based on DNA. Maybe you're working with Ogitsa Milankovitch. Maybe it's based on new semiconductors with John Dallasassie. | [
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Maybe you're working with Ogitsa Milankovitch. Maybe it's based on new semiconductors with John Dallasassie. Maybe it's based on carbon nanotubes with Joe Leiding. Maybe you're still finishing your degree. That would be really embarrassing for me if you're working on new technologies, like before you finish your Illinois degree. So no, I'm kidding. You have lots of opportunities here, all these technologies. | [
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So no, I'm kidding. You have lots of opportunities here, all these technologies. So let's say you're doing that. What do you need to do to make your technology useful? Well, what you need to do is you need to be able to build AND, OR, and NOT. If you can build AND, OR, and NOT, then people who are designing circuits and components and computer architectures, they can take what you've got and build everything on top. | [
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What do you need to do to make your technology useful? Well, what you need to do is you need to be able to build AND, OR, and NOT. If you can build AND, OR, and NOT, then people who are designing circuits and components and computer architectures, they can take what you've got and build everything on top. If you can't provide those three functions, then how can they use it? So that's the abstraction. | [
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If you can build AND, OR, and NOT, then people who are designing circuits and components and computer architectures, they can take what you've got and build everything on top. If you can't provide those three functions, then how can they use it? So that's the abstraction. If you can provide those three functions, you can build anything, any function. If you can't build those three, then it's going to be a lot harder to use. | [
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If you can provide those three functions, you can build anything, any function. If you can't build those three, then it's going to be a lot harder to use. So that's the abstraction boundary. So the technology has to be able to provide those. Again, lots of opportunities here for new technologies, lots of people working on cool stuff. Think about it in your four years here. OK. So actually, in the notes, if you look online, the slides that I posted, I did three input XOR. | [
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OK. So actually, in the notes, if you look online, the slides that I posted, I did three input XOR. But someone asked me after class about this function. So I thought, OK, let's just put this one in the lecture. Like I said, any function is fine. So someone said, well, what if I want a function where I have some number of inputs, and exactly one of those inputs has to be a 1 to produce a 1? So we'll do that on three. | [
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Like I said, any function is fine. So someone said, well, what if I want a function where I have some number of inputs, and exactly one of those inputs has to be a 1 to produce a 1? So we'll do that on three. So let's write the truth table. So actually, I just make it pop up. So get a feeling for what you think it should be, and I'll make it pop up. So again, if output's a 1, if and only if exactly one input is 1 or true. | [
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So get a feeling for what you think it should be, and I'll make it pop up. So again, if output's a 1, if and only if exactly one input is 1 or true. So here's what I thought was the answer. So if you have c equal to 1, but a and b is 0, you get a 1. If you have b equal 1, but a and c is 0, you get a 1. If you have a equal to 1, but b and c is 0, you get a 1. Anything else, you get a 0. | [
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If you have a equal to 1, but b and c is 0, you get a 1. Anything else, you get a 0. So let's go through and follow our constructive proof. So what is the function that produces that yellow row up there? Yeah, so a prime or a prime b prime c, right? Good. OK | [
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If you have b equal 1, but a and c is 0, you get a 1. If you have a equal to 1, but b and c is 0, you get a 1. Anything else, you get a 0. So let's go through and follow our constructive proof. So what is the function that produces that yellow row up there? Yeah, so a prime or a prime b prime c, right? Good. OK. What about this row? a prime b c prime? OK, good. And one more row with 1, right? What about that one? a b prime c prime. | [
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What about this row? a prime b c prime? OK, good. And one more row with 1, right? What about that one? a b prime c prime. OK, good. So those are the only ones with 1, right? So for each of those, we produce this minterm, this product of the literals. We call the individual inputs literals, or they're complement, or also called literals. So we produce these three functions, and then we OR them together. | [
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We call the individual inputs literals, or they're complement, or also called literals. So we produce these three functions, and then we OR them together. So if we OR them together, we get our function. So you can follow these steps for any function you want, and that will give you the function using only AND, OR, and NOT. So now we're ready to start thinking about, well, how would we actually build that adder we talked about in the first couple days of class? We can do any Boolean function we want. | [
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So you can follow these steps for any function you want, and that will give you the function using only AND, OR, and NOT. So now we're ready to start thinking about, well, how would we actually build that adder we talked about in the first couple days of class? We can do any Boolean function we want. Addition is just a Boolean function. You take some bits, bit pattern representing an unsigned or two's complement number, take another one, and you spit out the sum. | [
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Addition is just a Boolean function. You take some bits, bit pattern representing an unsigned or two's complement number, take another one, and you spit out the sum. It's just Boolean algebra. So now you have a way of thinking about how would you actually build that using just AND, OR, and NOT. Any questions on that before we play the floating point? Yeah. So those plus signs, won't that be the AND? OR. | [
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Any questions on that before we play the floating point? Yeah. So those plus signs, won't that be the AND? OR. Plus is OR. Yeah. So yes, these are ORs, and the multiplication notation is AND. So this is A prime. So A prime means not A. B prime means not B. C means just C by itself. Those are ANDed together, and then the output of that AND is ORed with the output of these two ANDs. And it has the same precedence as your normal algebra. | [
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Those are ANDed together, and then the output of that AND is ORed with the output of these two ANDs. And it has the same precedence as your normal algebra. So the AND happens before the OR, which is also actually important that I should have mentioned. Yes. Yes. So if I were to write this for you in your normal algebra, you would multiply before you'd add, right? So same thing. All right, so let's go forward. | [
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] |
So if I were to write this for you in your normal algebra, you would multiply before you'd add, right? So same thing. All right, so let's go forward. All right, so in binary, we have a binary point. I know internationally, people sometimes draw this as a comma. So sorry if this is not the way you'd normally write your decimal point. So if I want to write down pi, for example, I would have that period there. | [
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So sorry if this is not the way you'd normally write your decimal point. So if I want to write down pi, for example, I would have that period there. And then after the period, I would have 1 for the tenths and 4 for the hundredths and so forth. So if I write binary base 2 in human terms, well, I can have a binary point. After the binary point, I have the 2 to the minus 1, so the halves place, the quarters place, the eighths place, and so on. | [
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So if I write binary base 2 in human terms, well, I can have a binary point. After the binary point, I have the 2 to the minus 1, so the halves place, the quarters place, the eighths place, and so on. So that's pretty straightforward. So let's say that I'm not happy with integers. We've talked a lot about how we represent integers. What if I want fractions? What if I want real numbers? What do I do? Well, one thing I can do is I can use what's called fixed point. | [
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We've talked a lot about how we represent integers. What if I want fractions? What if I want real numbers? What do I do? Well, one thing I can do is I can use what's called fixed point. So I say, well, in the middle of my representation, I've got a binary point | [
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So that's pretty straightforward. So let's say that I'm not happy with integers. We've talked a lot about how we represent integers. What if I want fractions? What if I want real numbers? What do I do? Well, one thing I can do is I can use what's called fixed point. So I say, well, in the middle of my representation, I've got a binary point. You can just pick something. It can be supported by software. | [
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You can just pick something. It can be supported by software. It can be supported by hardware. It can be supported by both. It doesn't really matter that much. It's relatively easy to do it this way, because everything is still a power of 2. And so it turns out I can use the same adders and multipliers as long as I line things up properly. And sometimes maybe I'll even do that in software. | [
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And so it turns out I can use the same adders and multipliers as long as I line things up properly. And sometimes maybe I'll even do that in software. So there have been a lot of generations of processors that just did all of this stuff in software to have fixed point representations. Some signal processing and embedded processors have hardware support for fixed point. So it's a fairly simple extension, again, just based on human notions of representations for binary. | [
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Some signal processing and embedded processors have hardware support for fixed point. So it's a fairly simple extension, again, just based on human notions of representations for binary. So we could do that. So do we need anything else? Or is that good enough? Yeah? Is irrational hard? Yeah, irrational is hard. But the thing is, part of the reason it's hard is there's so many of them. | [
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So do we need anything else? Or is that good enough? Yeah? Is irrational hard? Yeah, irrational is hard. But the thing is, part of the reason it's hard is there's so many of them. And so we kind of have to make do with an approximation, because to represent irrationals well, one could argue we need infinite number of bits. So that's not quite true. There's some substantially more complicated representations. | [
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So that's not quite true. There's some substantially more complicated representations. Actually, I was talking with another professor, John Tallon, about them a few nights ago, because in his mathematical physics class, he's introducing people to them. There's something called continued fraction representations. People have tried to put those in hardware. They're fairly complicated to build hardware. | [
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People have tried to put those in hardware. They're fairly complicated to build hardware. And so what we tend to do is approximate. So what you'll see is an approximation. Good question. Yeah, that's a good point. So we do need negative numbers. So we could probably add a sine bit, kind of the way we did with two's complement. Or we could do sine magnitude fixed point. Yeah, that's a good question, though. | [
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Or we could do sine magnitude fixed point. Yeah, that's a good question, though. Yeah? What about our repeating fractions? Yeah, so repeating fractions, we could try to come up with a representation that handled them well. There's a little bit of that that's natural. If you think about translating just decimal to binary, then some numbers that are finite will become infinite in the other one, because of the primality of the factorization. | [
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There's a little bit of that that's natural. If you think about translating just decimal to binary, then some numbers that are finite will become infinite in the other one, because of the primality of the factorization. So people haven't, I think, come up with a representation that focuses on trying to represent repeating fractions. But it would be an interesting thing to do to see if there'd be any leverage. | [
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So people haven't, I think, come up with a representation that focuses on trying to represent repeating fractions. But it would be an interesting thing to do to see if there'd be any leverage. Yeah? That's just not a good idea. Yeah, yeah. So if you mean just having more range, we could throw more bits at it. Or we could think about, well, is there a way with the same number of bits we could get a bigger range of possible numbers? So let's think about all of those. | [
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So if you mean just having more range, we could throw more bits at it. Or we could think about, well, is there a way with the same number of bits we could get a bigger range of possible numbers? So let's think about all of those. And we'll roll forward and kind of see what floating point gives us. So let me just start by saying, well, what's the range of 32-bit two's complement? You guys know it, right? Everyone knew that? If I say, OK, write down on a piece of paper? OK, good, I'm glad you did. | [
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