In the previous two units in this week, we talked about Boolean functions, and most of this discussion has been theoretical. Starting with this unit we are going to talk about how we actually implement these Boolean functions using hardware. In particular, we're going to talk about a general technique called Gate Logic, which, you know, loosely speaking, it's a technique for implementing Boolean functions using logic gates. Now, what is a logic gate? Well, a logic gate is a stand alone chip or a, you know a very simple chip or an elementary chip which is designed to deliver a well-defined functionality. You know, something like NaN functionality, and functionality or, and so on. What is a composite logic gate? A composite logic gate is one which is made up from elementary logic gate and other composite logic gates. Or simply, it's, it's a more complex gate than the elementary ones. And in this course, we'll develop things like multiplexes and adders. And, obviously, most of you don't what the, what these terms mean. Don't worry about it. You will get acquainted with them and actually build them in the next few weeks of the course. All right, so let us begin with the most fundamental logic gate that we use in the course called, Nand. Here is the definition of a Nand. You know, this is actually the standard diagram that we use to describe a Nand gate. It has two inputs and a single output. Everything here is binary so a, b and out are either 0 and or 1. Here is the functional description of this of this gate. If both inputs are 1, we output 0. Under any other circumstance we output 1. And here is also a truth table description of the same functional specification. Any one of these descriptions is fine. Taken together, what we have here is an obstruction of the Nand gate. We didn't say a word about how this thing is actually working. We just describe what kind of functionality we can expect it to deliver. Okay, moving forward, here are, three classical gates that, pop up in every, digital design, project. We have an And gate which outputs 1, when both its inputs are 1 and 0 elsewhere. We, we have an Or gate which outputs 1 when either one of its inputs are 1 and 0 elsewhere. And we have a, a Not gate that operates as a converter. And we can put these gates together to create more complex functionality as we'll illustrate later on. In particular, we can use these gates to composite composite ones. So for example, here's a three-way And, which is an extension, if you will, of the of the simple two-way And. And one way to build it is to use this trick here. We can take the output of one And and feed it into one of the inputs of another And gate. And if we wire everything correctly this gate is supposed to deliver the required functionality. What we see in the dashed line rectangle around the implementation is a documentation of the interface of this chip. So the user lives outside the dashed rectangle. So, and, and so the user sees only the three inputs and the output, as you see in the left-hand side of of the gate description. If you want to go in, or to sort of open up the black box you have to see what is documented and done inside the dashed rectangle. So with that, I would like to say a few words about the notion of interface and implementation. The gate interface is the gate obstruction. You know, this is how the user thinks about what the gate is supposed to do. Interface answers the question, what. At the same time, if you want to understand how the chip is doing, what it's doing, you have to, go deeper and go int, well I'm not sure that deeper is the right word. But you have to go to another level of detail in which you actually, the black box opens up and you see how the chip is actually constructed. Or you do it yourself, you know, if, if your job is to be the person who, who is actually realizing this, obstruction. Now, the interface of the gate is unique. There's only one way to describe the gate's, functionality. Otherwise, you know, if there's more than one way, either you're not describing it well or you're confusing the user because, you know, there should be only one unique way to describe what the gate is supposed to do. At the same time, there may be several different implementations that realize the same obstruction. And some implementations may be, you know, more elegant, they may use less energy, they maybe less or more expensive, and so on. So one obstruction, many different implementations. This is very typical in in computer science, whenever you build a large system, you have this this duality. All right. Now moving along, so we talked about the gate interface and the gate implementation. Let's talk about something known as circuit implementations. If I want, I can realize these gates using hardwired circuits like, like this one. And there's a certain sort of graphical language here that describes what's going on. When we want to represent the fact that the gate outputs 1, we assume that a light bulb will be turned on. And when the gate delivers output 0, the light bulb will be off. So if we look at this implementation of an And circuit, we see that because of the architecture of this circuit the light bulb will will be on only if the two relays, both relays are latched. Under any other circumstance the light bulb will go off. And, that's, that's exactly what I want to do when I represent the And logic. What about Or logic? Well, once again, if I'm, if I have to build circuit, circuit implementation of Or, I can realize it using this particular architecture. In here it takes only one latch to close the circuit and turn on the light bulb. And once again, this is consistent with the desired obstruction of an Or circuit. Moving along, here is a circuit implement, implementation of a three-way And. And you can easily, I think, convince yourself that it will deliver the the required functionality. And yet I want to remind you that earlier in this unit we also presented this implementation of, of an And gate if you recall. So I'd like to say a few words about these two, different, approaches to, designing hardware. And, I want to start by saying that, in this particular course, we don't deal with physical implementations. And therefore, this whole discussion of circuits, transistors, relays, and so on, you know, all this stuff, and what you see in the top left corner of the screen here, this is electrical engineering. It's not computer science. And the people who build these gates using circuits and, and similar technologies, some of them are far more advanced and sophisticated. Well, these people are called electrical engineers, God bless their souls. You know, we have problems in computer science to worry about. So we are not going to worry at all about physical implementations, and all the designs that we are going to use in our course are going to be similar to what we did here at the bottom right of the screen. We are going to take existing logic gates beginning with Nand, and, and beginning with Nand only, and we're going to piece them together in some clever ways in order to generate and, and produce the required functionality. So this has been a unit in which we we gave you a first brief introduction of logic gates. And in the next unit we are going to introduce to you a language some sort of a programming language called HDL or Hardware Description Language, with which you can actually build logic gates like the ones you saw in this unit, and the far more complex logic gates and chips down the road.
