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If I put a zero into an OR gate, I need to know what the other input is, right? Zero or whatever is whatever. So for both of those zero inputs to the NOR gates, I don't change anything in my circuit. So then we could start the way we did in class and say, okay, well, let's just say there's a zero in Q. What does that imply about P? So the bottom gate then would have two zeros going into it. | [
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So then we could start the way we did in class and say, okay, well, let's just say there's a zero in Q. What does that imply about P? So the bottom gate then would have two zeros going into it. It's a NOR gate, so out will come a one, right? So P is one. In that case, Q, I'll put a little implication arrow here, means P is one. And if P is one, what does that imply about Q? Q is zero, right? So it also goes back. | [
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In that case, Q, I'll put a little implication arrow here, means P is one. And if P is one, what does that imply about Q? Q is zero, right? So it also goes back. You want to go around the loop because not all of these will be stable. So if we're just looking for stable states, make sure you go all the way around the loop, around the feedback loop, and check. So this state is stable, but what if we had instead guessed Q equals one? What is P then? Zero. | [
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So if we're just looking for stable states, make sure you go all the way around the loop, around the feedback loop, and check. So this state is stable, but what if we had instead guessed Q equals one? What is P then? Zero. So in that case, if Q equals one, then the bottom gate is a NOR gate. It has a one coming in from Q. NOR gate has a one. It gives a zero output, so P is a zero. And then what is Q, given P? It's a one. | [
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] |
It gives a zero output, so P is a zero. And then what is Q, given P? It's a one | [
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] |
So this state is stable, but what if we had instead guessed Q equals one? What is P then? Zero. So in that case, if Q equals one, then the bottom gate is a NOR gate. It has a one coming in from Q. NOR gate has a one. It gives a zero output, so P is a zero. And then what is Q, given P? It's a one. Top gate now has two zeros going in. So those are stable states. So this thing will store a bit when R and S are both low. | [
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So those are stable states. So this thing will store a bit when R and S are both low. What if I set S to one? What can you tell me about Q and P? What's P? It's zero, right? Because that NOR gate on the bottom now has a one coming in from S, so P is zero. So what is Q? And that's why we call this S active high, right? It's setting Q to one when you have S input of one. What if R is equal to one and S is zero? So Q is zero, right? Because the upper NOR gate will output a zero. | [
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So what is Q? And that's why we call this S active high, right? It's setting Q to one when you have S input of one. What if R is equal to one and S is zero? So Q is zero, right? Because the upper NOR gate will output a zero. And then P is what? That's why we name it R, and it's active high. So if I set R to one, Q is reset to zero. And we'd want to avoid this state, just as in the latch on the left. | [
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So if I set R to one, Q is reset to zero. And we'd want to avoid this state, just as in the latch on the left. We said, well, we don't want to set them both to zero. Here we don't want to set them both to one. What output do we get if we do that? Zero, zero, right? Q and P are both zero. So we don't want to do that because then the actual value, once we stop trying to force both of them to zero, we can't depend which of these we lower first. | [
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What output do we get if we do that? Zero, zero, right? Q and P are both zero. So we don't want to do that because then the actual value, once we stop trying to force both of them to zero, we can't depend which of these we lower first. And so we don't know what the final answer will be unless we analyze the timing carefully, which is a pain. So we try not to use this one this way. No, it would be, so if you write this as next state table, then Q retains its state, right? It's able to hold a bit. | [
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So we try not to use this one this way. No, it would be, so if you write this as next state table, then Q retains its state, right? It's able to hold a bit. So you put Q plus equals Q in the notation of the homework, yeah. So if you wanted to rewrite this, you could say RS00, Q plus is Q, 01, Q plus is zero, I'm sorry, one, 10, Q plus is zero, and then 11, and then it's zero, but P is not Q bar, so you've got to be careful. | [
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So you put Q plus equals Q in the notation of the homework, yeah. So if you wanted to rewrite this, you could say RS00, Q plus is Q, 01, Q plus is zero, I'm sorry, one, 10, Q plus is zero, and then 11, and then it's zero, but P is not Q bar, so you've got to be careful. Okay. So the more complicated way that I showed you in class will also work. Usually we're asking you, though, for stable states, so you then have to go through your truth table and find the ones that are stable. | [
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So the more complicated way that I showed you in class will also work. Usually we're asking you, though, for stable states, so you then have to go through your truth table and find the ones that are stable. So remember that when we did it in class with a more complex circuit, we had to look for the rows that were stable by identifying the rows with Q equal to Q plus. One one is a stable state, but the outputs are both zero. | [
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So remember that when we did it in class with a more complex circuit, we had to look for the rows that were stable by identifying the rows with Q equal to Q plus. One one is a stable state, but the outputs are both zero. So the outputs are both zero, which means that when you let RS go down, the bits that's actually stored depends which one goes down first, right? And so unless you really understand the timing of your RNS signals, you won't know what bit is stored after you've raised both of them. | [
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One one is a stable state, but the outputs are both zero. So the outputs are both zero, which means that when you let RS go down, the bits that's actually stored depends which one goes down first, right? And so unless you really understand the timing of your RNS signals, you won't know what bit is stored after you've raised both of them. And worse, if you lower both of them at the same time, you can encounter metastable states where the thing actually oscillates after you've lowered RNS. | [
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So the outputs are both zero, which means that when you let RS go down, the bits that's actually stored depends which one goes down first, right? And so unless you really understand the timing of your RNS signals, you won't know what bit is stored after you've raised both of them. And worse, if you lower both of them at the same time, you can encounter metastable states where the thing actually oscillates after you've lowered RNS | [
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So the outputs are both zero, which means that when you let RS go down, the bits that's actually stored depends which one goes down first, right? And so unless you really understand the timing of your RNS signals, you won't know what bit is stored after you've raised both of them. And worse, if you lower both of them at the same time, you can encounter metastable states where the thing actually oscillates after you've lowered RNS. There is no unstable state in this design, but if you, for example, had, I think there was a, let's see, where did I put it? Here's XOR, and let's run that through, I don't know, a NAND gate or something. | [
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And worse, if you lower both of them at the same time, you can encounter metastable states where the thing actually oscillates after you've lowered RNS. There is no unstable state in this design, but if you, for example, had, I think there was a, let's see, where did I put it? Here's XOR, and let's run that through, I don't know, a NAND gate or something. If I do something like this, so it's unstable if this thing is a single inverter, right? And so since that's an XOR gate, if I put A equals zero, then that's basically a wire from its other input to its output, right? So if I say A equals zero, then that implies, let me call this, oh, this is P, then that implies that Q equals P, right, from that XOR gate. | [
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There is no unstable state in this design, but if you, for example, had, I think there was a, let's see, where did I put it? Here's XOR, and let's run that through, I don't know, a NAND gate or something. If I do something like this, so it's unstable if this thing is a single inverter, right? And so since that's an XOR gate, if I put A equals zero, then that's basically a wire from its other input to its output, right? So if I say A equals zero, then that implies, let me call this, oh, this is P, then that implies that Q equals P, right, from that XOR gate. But then looking at the other gate, if, let's see, if I put B equals zero, I'm sorry, B equals one down there, then P equals Q prime. | [
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If I do something like this, so it's unstable if this thing is a single inverter, right? And so since that's an XOR gate, if I put A equals zero, then that's basically a wire from its other input to its output, right? So if I say A equals zero, then that implies, let me call this, oh, this is P, then that implies that Q equals P, right, from that XOR gate. But then looking at the other gate, if, let's see, if I put B equals zero, I'm sorry, B equals one down there, then P equals Q prime. But I can't satisfy both of these at the same time, right? So if I look at A and B and I say zero, one, then this is unstable. | [
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But then looking at the other gate, if, let's see, if I put B equals zero, I'm sorry, B equals one down there, then P equals Q prime. But I can't satisfy both of these at the same time, right? So if I look at A and B and I say zero, one, then this is unstable. Does that make sense? Right, because if you look at these two inputs now, on the top one, that XOR gate is just forwarding its other input to its output, and the bottom one is inverting its input, so you have a loop of one inverter, effectively. | [
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But I can't satisfy both of these at the same time, right? So if I look at A and B and I say zero, one, then this is unstable. Does that make sense? Right, because if you look at these two inputs now, on the top one, that XOR gate is just forwarding its other input to its output, and the bottom one is inverting its input, so you have a loop of one inverter, effectively. If you have a loop of one inverter, it's just an oscillator. | [
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Does that make sense? Right, because if you look at these two inputs now, on the top one, that XOR gate is just forwarding its other input to its output, and the bottom one is inverting its input, so you have a loop of one inverter, effectively. If you have a loop of one inverter, it's just an oscillator. Yeah, this just keeps inverting the one value, and it keeps oscillating. Yeah, so that's why that kind of thing is unstable. | [
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Yeah, this just keeps inverting the one value, and it keeps oscillating. Yeah, so that's why that kind of thing is unstable. In the two latch designs, there is no unstable state. There's the possibility for metastable state, but that's not something that our analysis will show you. Yeah, mm-hmm. So we might ask you to identify the stable states, or we might ask you to identify which states oscillate or things like that. | [
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Yeah, mm-hmm. So we might ask you to identify the stable states, or we might ask you to identify which states oscillate or things like that. Yeah. No, no, no, you won't have to analyze the oscillator, but you might have to identify which states might oscillate. Right, so we don't use SR latches or R bar S bar latches. We don't use the two states where you both try to set and reset at the same time. | [
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Right, so we don't use SR latches or R bar S bar latches. We don't use the two states where you both try to set and reset at the same time. So we can build extra logic around that so as not to use those states. If you had something that had an oscillating set of inputs, you could build around it so as not to use that particular input combination. And these kind of things are not really that useful in practice, but they let you think about, well, how do these things work and identify oscillation. | [
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If you had something that had an oscillating set of inputs, you could build around it so as not to use that particular input combination. And these kind of things are not really that useful in practice, but they let you think about, well, how do these things work and identify oscillation. So we might ask you that kind of question. But we won't ask you to tell us what happens after a certain amount of time with oscillations, for example, because the detailed timing questions are beyond the course. | [
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] |
So we might ask you that kind of question. But we won't ask you to tell us what happens after a certain amount of time with oscillations, for example, because the detailed timing questions are beyond the course. But knowing that it'll oscillate, you should be able to do. Anything else on this one? All right. Okay, so here is our list | [
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And these kind of things are not really that useful in practice, but they let you think about, well, how do these things work and identify oscillation. So we might ask you that kind of question. But we won't ask you to tell us what happens after a certain amount of time with oscillations, for example, because the detailed timing questions are beyond the course. But knowing that it'll oscillate, you should be able to do. | [
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But we won't ask you to tell us what happens after a certain amount of time with oscillations, for example, because the detailed timing questions are beyond the course. But knowing that it'll oscillate, you should be able to do. Anything else on this one? All right. Okay, so here is our list. So we have decoders, timing diagrams, analyzing sequential feedback circuits. So implementing functions with MUXes. | [
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So we have decoders, timing diagrams, analyzing sequential feedback circuits. So implementing functions with MUXes. Oh, and so I guess we did have MUXes on there. Okay, so the general question I remember was how to implement a function with a MUX. So let's say that you have some function f of a, b, and c. Okay, so there's the truth table. So f will have some values. If I want to implement this with a 4-to-1 MUX, then, for example, I can put in a and b. | [
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So f will have some values. If I want to implement this with a 4-to-1 MUX, then, for example, I can put in a and b. So let's call those s1 and s0. And I want this output to be f. So the way I want you to think about this is, well, if a is 0 and b is 0, then I can simplify my truth table, right? So if a is 0 and b is 0, I'm going to pick the 0 input of the MUX, and I'm going to forward that wire to the output f. | [
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And I want this output to be f. So the way I want you to think about this is, well, if a is 0 and b is 0, then I can simplify my truth table, right? So if a is 0 and b is 0, I'm going to pick the 0 input of the MUX, and I'm going to forward that wire to the output f. But if a is 0 and b is 0, the only thing I need to look at is this part of the truth table right here. That needs to go into this input here. | [
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But if a is 0 and b is 0, the only thing I need to look at is this part of the truth table right here. That needs to go into this input here. And if a is 0 and b is 1, similarly, that needs to go there. And this part, where this is going to maybe get a little messy, needs to go here. And this part needs to go there. But really, all you need to do is figure out, well, for these four little mini truth tables, what functions do I need? Clearly, they're only functions of c. | [
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And this part needs to go there. But really, all you need to do is figure out, well, for these four little mini truth tables, what functions do I need? Clearly, they're only functions of c. But if I implement those four functions, and I put those four functions in as my four MUX values, that MUX will put together these four pieces of my bigger truth table into one big truth table. Right? So let me fill in some bits, and then we can figure out what functions to put there. | [
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But if I implement those four functions, and I put those four functions in as my four MUX values, that MUX will put together these four pieces of my bigger truth table into one big truth table. Right? So let me fill in some bits, and then we can figure out what functions to put there. Actually, it's only one bit input, right? How many functions are there again? How many different functions do I have to pick from? There are four, right? So let's put them all. | [
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Right? So let me fill in some bits, and then we can figure out what functions to put there. Actually, it's only one bit input, right? How many functions are there again? How many different functions do I have to pick from? There are four, right? So let's put them all. Here's one. Here's another. Here's another. Here's another. So what goes into the – what function is this? It's a zero. | [
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Here's another. So what goes into the – what function is this? It's a zero. What's this? What's this? Not c. What's that? Okay. So if I want that particular f, then up here, let's see, you said zero. What is this one again? c. Not c. 1. So I think there was something in the homework about what if I gave you one MUX and one inverter? This input two is where you would use that one inverter. | [
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1. So I think there was something in the homework about what if I gave you one MUX and one inverter? This input two is where you would use that one inverter. If you have a function of four variables and you want to implement it with a four-to-one MUX, then you also need a little bit of luck | [
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] |
So if I want that particular f, then up here, let's see, you said zero. What is this one again? c. Not c. 1. So I think there was something in the homework about what if I gave you one MUX and one inverter? This input two is where you would use that one inverter. If you have a function of four variables and you want to implement it with a four-to-one MUX, then you also need a little bit of luck. | [
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So I think there was something in the homework about what if I gave you one MUX and one inverter? This input two is where you would use that one inverter. If you have a function of four variables and you want to implement it with a four-to-one MUX, then you also need a little bit of luck. Oh, was it? So I've asked that question on homework, but of the form of can you do it and if not, prove that you can't. | [
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If you have a function of four variables and you want to implement it with a four-to-one MUX, then you also need a little bit of luck. Oh, was it? So I've asked that question on homework, but of the form of can you do it and if not, prove that you can't. And I think XOR is – if I remember correctly, XOR suffices to show that you can't do it, but I'd have to think about it a little more. | [
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Oh, was it? So I've asked that question on homework, but of the form of can you do it and if not, prove that you can't. And I think XOR is – if I remember correctly, XOR suffices to show that you can't do it, but I'd have to think about it a little more. Yeah, and generally you won't be able to. You have four values left based on two variables, and so generally you won't necessarily be able to do it. | [
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Yeah, and generally you won't be able to. You have four values left based on two variables, and so generally you won't necessarily be able to do it. You have a little bit of freedom in how you break up this truth table, right? So I just picked B and C. You can pick any two, right? So you've got three different choices here. With four variables, you have six different choices, right? So maybe one of those six choices you could actually implement, whereas the other five choices maybe you can't. | [
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You can pick any two, right? So you've got three different choices here. With four variables, you have six different choices, right? So maybe one of those six choices you could actually implement, whereas the other five choices maybe you can't. So you need to do a fair bit of thinking before you decide it's impossible to do this function. Whereas the reason I'm thinking XOR is because XOR is symmetric under that set of choices, right? So you just show that you can't do one of them and then you're done. | [
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So you need to do a fair bit of thinking before you decide it's impossible to do this function. Whereas the reason I'm thinking XOR is because XOR is symmetric under that set of choices, right? So you just show that you can't do one of them and then you're done. Really? Someone asked that on the exam? That's kind of mean. Right? Yeah, so bit-sliced adder, if you know the bits, plug it in. | [
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Really? Someone asked that on the exam? That's kind of mean. Right? Yeah, so bit-sliced adder, if you know the bits, plug it in. Yeah, wasn't that one on the homework? Okay. So maybe in the interest of time, I'll skip that one because you guys have homework solutions, right? Yeah. Anything else on this? Okay. All right. So let's go back to our list. And we have just done this one. So MOSFETs and networks for gates. | [
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And we have just done this one. So MOSFETs and networks for gates. So let's take a look. I think Mohamed, you asked this. Did you want to go over n-type, p-type functionality or just the networks? Okay. Yeah, yeah. All right. Okay. So generally, I mentioned this in class, but the n-type and p-type are duals. So we can read these off just as functions. The reason they need to be duals is so that we guarantee there's no shorts. | [
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So we can read these off just as functions. The reason they need to be duals is so that we guarantee there's no shorts. So at least in a gate that will always produce either a 0 or a 1, they will be dual networks. So let's talk about why. So if F is equal to some function of uncomplemented literals, then primed. So why do I say uncomplemented literals? Remember that the n-type MOSFET, what we use it for is to connect to ground. | [
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] |
So if F is equal to some function of uncomplemented literals, then primed. So why do I say uncomplemented literals? Remember that the n-type MOSFET, what we use it for is to connect to ground. So either it's directly to ground or it's through some other n-type MOSFETs going to ground. The reason is that in order for this thing to turn on, we need a voltage between the input and this side. | [
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] |
So either it's directly to ground or it's through some other n-type MOSFETs going to ground. The reason is that in order for this thing to turn on, we need a voltage between the input and this side. And so by connecting this side of the circuit to ground, we guarantee that this thing will actually stay on until it brings the other side down to ground | [
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So either it's directly to ground or it's through some other n-type MOSFETs going to ground. The reason is that in order for this thing to turn on, we need a voltage between the input and this side. And so by connecting this side of the circuit to ground, we guarantee that this thing will actually stay on until it brings the other side down to ground. If we try to use this connecting it to VDD, then the transistor will actually turn itself off before it pulls the output up to high voltage. | [
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And so by connecting this side of the circuit to ground, we guarantee that this thing will actually stay on until it brings the other side down to ground. If we try to use this connecting it to VDD, then the transistor will actually turn itself off before it pulls the output up to high voltage. So we only use n-types to connect to ground. What that means is that if we're feeding our input variables into the n-types, well, that means in our function, they will not be complemented. | [
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So we only use n-types to connect to ground. What that means is that if we're feeding our input variables into the n-types, well, that means in our function, they will not be complemented. So whatever we built here, unless we've got inverters in front of it, whatever we built here will be our uncomplemented literals. And then if that function holds, output will have a path down to ground, which means that f, the function, is then complemented. | [
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So whatever we built here, unless we've got inverters in front of it, whatever we built here will be our uncomplemented literals. And then if that function holds, output will have a path down to ground, which means that f, the function, is then complemented. So the kind of things we can build would be the ones we looked at. So if you want an inverter, you would just put one of these. | [
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So the kind of things we can build would be the ones we looked at. So if you want an inverter, you would just put one of these. So if this is your output, for example, and this was, say, f, then this would be f equals a prime. You could put more than one. So you could say, OK, let me do two of them. And this would be f equals a or b prime. Because here, if I turn on, sorry, I left my connection out there for b. | [
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And this would be f equals a or b prime. Because here, if I turn on, sorry, I left my connection out there for b. But if I turn on a, well, then f has a path to ground. If I turn on b, f has a path to ground. So either a or b, f has a path to ground. In that case, f equals zero. Did I do that right? Yeah. Yes. So let me get to the p-type. So I just want to make sure people understand the relationship here. | [
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So let me get to the p-type. So I just want to make sure people understand the relationship here. So the way we make constructs here is and constructs, down here, which would look like this. So this would be f equals a and b complemented. Because in order to get to ground, we have to turn on a and turn on b. But in all of them, they're uncomplemented literals. So what about the p-type stuff? So the p-type stuff, f is going to be some function of only complemented literals. | [
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But in all of them, they're uncomplemented literals. So what about the p-type stuff? So the p-type stuff, f is going to be some function of only complemented literals. And why is that? Well, for a p-type, remember that we're connecting it to high voltage. And so if we connect a p-type to high voltage, that means that it can actually, it's looking for a voltage from the side on the right, those two terminals, the source and the drain, to the gate. | [
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And why is that? Well, for a p-type, remember that we're connecting it to high voltage. And so if we connect a p-type to high voltage, that means that it can actually, it's looking for a voltage from the side on the right, those two terminals, the source and the drain, to the gate. And so by connecting this to high voltage, that means the transistor will stay on until it pulls the other terminal up to high voltage. | [
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And so if we connect a p-type to high voltage, that means that it can actually, it's looking for a voltage from the side on the right, those two terminals, the source and the drain, to the gate. And so by connecting this to high voltage, that means the transistor will stay on until it pulls the other terminal up to high voltage. And so that transistor will turn on when a equals zero. | [
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And so by connecting this to high voltage, that means the transistor will stay on until it pulls the other terminal up to high voltage. And so that transistor will turn on when a equals zero. And so that's why in our function written from the top network, all of our literals will be complemented. So that one I've drawn with just the one p-type is, of course, the inverter. And so in that case, we have f equals a prime, which is the same here and here, these two. | [
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So that one I've drawn with just the one p-type is, of course, the inverter. And so in that case, we have f equals a prime, which is the same here and here, these two. And if I want to draw the complement of this one, then I would draw it as this way. And then if I were to derive that, I would get f equals a prime b prime | [
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So that one I've drawn with just the one p-type is, of course, the inverter. And so in that case, we have f equals a prime, which is the same here and here, these two. And if I want to draw the complement of this one, then I would draw it as this way. And then if I were to derive that, I would get f equals a prime b prime. So if I derive that from the top network, I get f equals a prime and b prime. | [
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And then if I were to derive that, I would get f equals a prime b prime. So if I derive that from the top network, I get f equals a prime and b prime. And that gives me f. If I derive it from the bottom network, I get f equals a or b value complement. And if you look at those two expressions and apply generalized De Morgan's law, you see that they're equal. So this one is equal to that one by De Morgan's law. | [
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And if you look at those two expressions and apply generalized De Morgan's law, you see that they're equal. So this one is equal to that one by De Morgan's law. So I can do the same thing for the other gate. But more generally, these two expressions need to be equal. So the p-type expression and the n-type expression must be equal. So if I write the n-type expression, remember, I get some uncomplemented literals with a prime on the end. | [
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] |
So the p-type expression and the n-type expression must be equal. So if I write the n-type expression, remember, I get some uncomplemented literals with a prime on the end. If I then apply generalized De Morgan's and I push that prime through all the way down to the literal level, then I've swapped all my ends with ors. And instead of uncomplemented literals, all of my literals will be complemented. | [
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If I then apply generalized De Morgan's and I push that prime through all the way down to the literal level, then I've swapped all my ends with ors. And instead of uncomplemented literals, all of my literals will be complemented. So if you remember in class, we did that as a two-step process, which is to apply generalized De Morgan's, you take the dual. And then you apply, you replace, or maybe I should say swap, literals and uncomplemented literals. | [
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So if you remember in class, we did that as a two-step process, which is to apply generalized De Morgan's, you take the dual. And then you apply, you replace, or maybe I should say swap, literals and uncomplemented literals. So in this case, this duality is in the structure of the network. And the swapping comes from the use of n-type and p-type. So the n-type on the bottom turn on when the input variable is high. | [
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And the swapping comes from the use of n-type and p-type. So the n-type on the bottom turn on when the input variable is high. The p-type on the top turn on when the input variable is low. And so in the gate structures, the dual structure is in the two networks. And so in order for these two equations to be equal, I need this relationship. But I can get that relationship algebraically by applying generalized De Morgan's. | [
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And so in order for these two equations to be equal, I need this relationship. But I can get that relationship algebraically by applying generalized De Morgan's. And again, the dual structure is inherent in the topology of the network. So parallel and serial are duals of one another. But you can generalize that to any function and use the principle of duality to get the structure. And then the swapping is because the n-type and p-type. | [
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But you can generalize that to any function and use the principle of duality to get the structure. And then the swapping is because the n-type and p-type. Let me do an example. That will be specific to the problem, but I want to stick to this one until we finish it. All right, so let's do an example. So let's say that you have something like this on top, let's say. Sorry, let me start over. | [
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So let's say that you have something like this on top, let's say. Sorry, let me start over. Okay, so what is F here? Yeah, so we have to have the top one on, right? So that's A0. And then what about that middle part? Yeah, B0 or C0. And then the bottom, D0, right? Okay, so we can just write it down. Now I claim you can just take the dual. So the dual here, remember how to take the dual? Swap and and or, swap zeros and ones. | [
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Now I claim you can just take the dual. So the dual here, remember how to take the dual? Swap and and or, swap zeros and ones. There's no zeros and ones, right? So we just swap and and or. So I'm going to change the ands to ors and change the or to and. What that'll give me is A | [
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And then the bottom, D0, right? Okay, so we can just write it down. Now I claim you can just take the dual. So the dual here, remember how to take the dual? Swap and and or, swap zeros and ones. There's no zeros and ones, right? So we just swap and and or. So I'm going to change the ands to ors and change the or to and. What that'll give me is A. I'm going to leave off the complement. | [
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What that'll give me is A. I'm going to leave off the complement. So A or B, C or D, right? And I'll put the complement outside. Okay, so this dual form gives us then the structure of the bottom network. So it says A or B and C or D. So the ors are represented as parallel structures. So there's A. There's B, C. There's D. And I claim that if you go fill in the 16 row truth table that you'll see that this is compatible. | [
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There's D. And I claim that if you go fill in the 16 row truth table that you'll see that this is compatible. There's always exactly one path from out either to high voltage or down to ground. So all I did is I took the dual and then I built the dual structure down below. Yeah. Yes. Yeah, so generalized DeMorgan's, remember, you take the dual and then you flip all the literals and complemented literals, which you can see here at the top. | [
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Yes. Yeah, so generalized DeMorgan's, remember, you take the dual and then you flip all the literals and complemented literals, which you can see here at the top. I already did that by hand. Yeah, Daniel? The second expression is the complement of the dual because only because I added the prime at the end. The dual, technically the dual expression has all of the literals the same way. | [
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Yeah, Daniel? The second expression is the complement of the dual because only because I added the prime at the end. The dual, technically the dual expression has all of the literals the same way. So the actual dual is this. The dual structure down there tells us the network structure. The bottom two expressions are equal. And they're also the two values. So if you read this one off, you'll get the bottom expression. | [
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And they're also the two values. So if you read this one off, you'll get the bottom expression. So of course you want those to be equal. Yeah. The bottom two are exactly the same. They're both equal. And if you read the top network, that's where we got the middle one. And the bottom one is what you would get if you read this one off. So the prime comes from using the n-type and the dual nature comes from construction of the dual networks. | [
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And the bottom one is what you would get if you read this one off. So the prime comes from using the n-type and the dual nature comes from construction of the dual networks. You get one cheat sheet just like last time. What did the person who wanted to talk about DeMorgan's Law, I mean I would write them down if it's hard to remember them, I guess. Which is those. Yeah, and we also, I'm sorry, we actually do give you the properties list. | [
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] |
Which is those. Yeah, and we also, I'm sorry, we actually do give you the properties list. So these will be there. So you don't even need to read it. Write it on your cheat sheet. Very little. So I mean you need to know how to do K-maps and read SOP, POS, stuff like that. But actually manipulating Boolean algebra is not a skill that's useful these days. Yeah. Yeah. So two things about that. | [
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Yeah. So two things about that. Perfect induction is the fancy name for brute force. So it means write a truth table, fill it in, 0, 0, 0, 0, 1, 1. Good. Yeah, that's perfect induction. Yeah, I don't know who made that name up. Do you need to know how to simplify Boolean expressions with Boolean identities? No, not for our class. Yeah, Mohamed? On the p-types, yeah, because remember that they're always connected to high voltage. | [
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] |
Do you need to know how to simplify Boolean expressions with Boolean identities? No, not for our class. Yeah, Mohamed? On the p-types, yeah, because remember that they're always connected to high voltage. The p-types are always used in the upper half of the network. And basically if the input is low, then they'll turn on | [
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Good. Yeah, that's perfect induction. Yeah, I don't know who made that name up. Do you need to know how to simplify Boolean expressions with Boolean identities? No, not for our class. Yeah, Mohamed? On the p-types, yeah, because remember that they're always connected to high voltage. The p-types are always used in the upper half of the network. And basically if the input is low, then they'll turn on. | [
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] |
The p-types are always used in the upper half of the network. And basically if the input is low, then they'll turn on. So you can always put an inverter, but I mean then logically that's an inverter. Yeah, yeah. So just think of the way the current has to flow. So if you can get a wire, if they're in series, that's an AND. If they're in parallel, that's an OR. Yeah. In this setting. | [
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Yeah. In this setting. I'm sorry? Sorry, I still can't hear you. Yeah. Taking the dual and then swapping literals and complement is generalized to Morgan's law. So if you want to take the complement of a big nasty expression, you can just take the dual and then swap also literals and complement literals, and that'll be the complement of the whole thing. | [
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Okay. So I think it's three. Hope everyone enjoyed the midterm. Sorry, I didn't ask for your help. Yeah, mine were problems four and five. So you've got fun with multiple choice and fun with LC3. So, all right. So I want to finish up. So we'll do a little more review since it's been, I guess, five days since we looked at this program. So we'll go through the last few slides that we saw and then finish up the coding of the letter frequency code. | [
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So we'll do a little more review since it's been, I guess, five days since we looked at this program. So we'll go through the last few slides that we saw and then finish up the coding of the letter frequency code. So that'll be our last program. I'm not even going to change it into bits for you. You can find the bits on the website if you really want to see the bits. But that program technically we sort of did in binary. | [
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You can find the bits on the website if you really want to see the bits. But that program technically we sort of did in binary. Then we're going to talk about assembly language and then what you can do in assembly language and things like that. And I'll talk about assemblers and how they work. That may take all of today. I have this think-pair-share exercise. Originally I had it above the assemblers, but then I thought, really, do I want to see binary code? Probably not. | [
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I have this think-pair-share exercise. Originally I had it above the assemblers, but then I thought, really, do I want to see binary code? Probably not. I'm tired of it. So I'll put it down here. And if we get to it, we'll do it in assembly and that'll be slightly easier. All right. So this was the problem, just to remind you what we're trying to solve. So we said, OK, what if we have a string? String is a bunch of ASCII characters terminated by a null. | [
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So this was the problem, just to remind you what we're trying to solve. So we said, OK, what if we have a string? String is a bunch of ASCII characters terminated by a null. This is a zero in ASCII. So we count the occurrences of each letter regardless of case and count the number of alphabetic characters. And then we decided on this high-level algorithm where we go initialize 27 bins to zero for a histogram. | [
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So we count the occurrences of each letter regardless of case and count the number of alphabetic characters. And then we decided on this high-level algorithm where we go initialize 27 bins to zero for a histogram. And then we go through the string once. We look at each character in the string and figure out, well, is it a letter, uppercase or lowercase, or is it a not letter? And we go find the right bin and increment it. | [
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And then we go through the string once. We look at each character in the string and figure out, well, is it a letter, uppercase or lowercase, or is it a not letter? And we go find the right bin and increment it. So we did a breakdown. I didn't keep the flowchart around or anything like that. It's available online if you want to see it. So we had decided, well, the string will stick at 4,000 hex. | [
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] |
It's available online if you want to see it. So we had decided, well, the string will stick at 4,000 hex. The start of our code will be 3,000 hex. And then the histogram will run from 3,100 hex up to 3,11a. This is 1a is 26. So 1 to 26, a to z. And then the non-alpha bin at 3,100. And then we had these registers assigned during the counting part of the code. So r0 has a pointer to the histogram, meaning this value. | [
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And then we had these registers assigned during the counting part of the code. So r0 has a pointer to the histogram, meaning this value. r1 is a string pointer that goes from character to character in the string, starts at 4,000, goes until we find the end of the string. And that lets us load individual memory elements, the characters in the string, into r2. r3, r4, and r5, you remember we needed to identify the boundaries in the ASCII table where the letters start and end. | [
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] |
And that lets us load individual memory elements, the characters in the string, into r2. r3, r4, and r5, you remember we needed to identify the boundaries in the ASCII table where the letters start and end. And so we had ASCII constants in here that we've mentioned as we went through what they are. And then we had r6 as a temporary register. And so those were our register assignments. | [
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And then we had r6 as a temporary register. And so those were our register assignments. Those you would put in comments. So if you look at either of the versions in the website, you'll see this is all listed out in comments. So you don't have to try to remember it. And this is where we had gotten to | [
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] |
And so we had ASCII constants in here that we've mentioned as we went through what they are. And then we had r6 as a temporary register. And so those were our register assignments. Those you would put in comments. So if you look at either of the versions in the website, you'll see this is all listed out in comments. So you don't have to try to remember it. And this is where we had gotten to. | [
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So you don't have to try to remember it. And this is where we had gotten to. We had actually done a little bit more of this. But this was our last comparison to say, well, is the character we're looking at, which we had converted or were about to convert into a number from 1 to 26 for the lowercase letters, is that character greater than capital Z? So is it a letter or not? I guess actually we'd already converted it. | [
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