WEBVTT 00:00:00.000 --> 00:00:03.020 You know, there's a phrase that I think pretty 00:00:03.020 --> 00:00:05.259 much every single one of us has used at some 00:00:05.259 --> 00:00:07.179 point. Oh, absolutely. Usually it's when you 00:00:07.179 --> 00:00:09.240 are a teenager rolling your eyes at your parents. 00:00:09.419 --> 00:00:11.880 Or maybe it's just like, you know, 5 p .m. on 00:00:11.880 --> 00:00:15.480 a Friday and you are completely burned out. Yeah, 00:00:15.599 --> 00:00:17.800 I know exactly the feeling you're talking about. 00:00:17.960 --> 00:00:20.739 That phrase is, I don't care. It is a classic. 00:00:20.800 --> 00:00:24.059 It's the universal signal for apathy. Usually 00:00:24.059 --> 00:00:27.620 implies... laziness or checking out or just, 00:00:27.679 --> 00:00:29.420 you know, giving up on whatever the problem is. 00:00:29.480 --> 00:00:31.339 Exactly. It sounds like you're quitting. But 00:00:31.339 --> 00:00:33.740 and this is really why I love digging into these 00:00:33.740 --> 00:00:35.780 technical stacks we get for the show in the world 00:00:35.780 --> 00:00:38.479 of computer engineering and digital logic saying, 00:00:38.560 --> 00:00:41.460 I don't care, isn't lazy at all. Not even a little 00:00:41.460 --> 00:00:44.119 bit. It's actually a superpower. It is a strategic 00:00:44.119 --> 00:00:47.439 weapon. It really is. It's a fundamental pillar 00:00:47.439 --> 00:00:50.219 of efficiency. Because if computers cared about 00:00:50.219 --> 00:00:53.159 every single theoretical possibility, they would 00:00:53.159 --> 00:00:56.780 be incredibly slow, they'd be massive, and they'd 00:00:56.780 --> 00:00:59.600 probably overheat in about 10 seconds. So today 00:00:59.600 --> 00:01:02.700 we are doing a deep dive into don't care terms. 00:01:02.859 --> 00:01:05.379 And no, for you listening, that isn't a list 00:01:05.379 --> 00:01:07.700 of people who ghosted me. Let's hope not. It 00:01:07.700 --> 00:01:10.859 is a very specific engineering concept that basically 00:01:10.859 --> 00:01:13.400 allows our devices to function at the speeds 00:01:13.400 --> 00:01:17.159 we expect. But, and here's the twist we are going 00:01:17.159 --> 00:01:19.560 to get to a bit later, if you aren't careful 00:01:19.560 --> 00:01:22.299 with the things you don't care about, they can 00:01:22.299 --> 00:01:23.799 actually come back to haunt you in some pretty 00:01:23.799 --> 00:01:26.079 catastrophic ways. That's the core tension of 00:01:26.079 --> 00:01:29.239 this entire field. It's about knowing what matters 00:01:29.239 --> 00:01:32.459 and strategically ignoring what doesn't so you 00:01:32.459 --> 00:01:35.019 can build a better system. But if you ignore 00:01:35.019 --> 00:01:37.319 the wrong thing, well, you end up in what the 00:01:37.319 --> 00:01:39.219 documentation literally calls a walled garden 00:01:39.219 --> 00:01:42.680 of error states. Which, you know. Sounds lovely, 00:01:42.879 --> 00:01:45.000 but is actually a total nightmare. A complete 00:01:45.000 --> 00:01:47.640 nightmare. Yes. So we have a great stack of source 00:01:47.640 --> 00:01:49.920 material today. We've got technical documentation, 00:01:50.319 --> 00:01:52.799 some encyclopedic records on digital logic design, 00:01:53.000 --> 00:01:55.340 and a bunch of files specifically focusing on 00:01:55.340 --> 00:01:58.500 Boolean functions and circuit minimization. It's 00:01:58.500 --> 00:02:00.900 a really comprehensive pile of sources. It is. 00:02:01.019 --> 00:02:03.819 And our mission today is to figure out how engineers 00:02:03.819 --> 00:02:06.120 decide what to ignore to make things faster, 00:02:06.200 --> 00:02:08.879 cheaper, and smaller, and what happens when it 00:02:08.879 --> 00:02:11.060 all goes wrong. Sounds like a plan. Let's start. 00:02:11.210 --> 00:02:13.830 at the beginning with the definition when a logic 00:02:13.830 --> 00:02:16.650 designer says don't care what are they actually 00:02:16.650 --> 00:02:18.990 looking at on their screen so in the strictest 00:02:18.990 --> 00:02:21.210 definition from the encyclopedic records we are 00:02:21.210 --> 00:02:23.349 looking at a boolean function right basically 00:02:23.349 --> 00:02:25.430 the math of logic the zeros and the ones right 00:02:25.430 --> 00:02:27.870 a don't care term which is often abbreviated 00:02:27.870 --> 00:02:32.129 just as dc is an input sequence a specific series 00:02:32.129 --> 00:02:35.409 of bits for a boolean function where the output 00:02:36.039 --> 00:02:37.840 effectively does not matter okay let's make that 00:02:37.840 --> 00:02:40.240 concrete for the listener the system receives 00:02:40.240 --> 00:02:43.379 a message like a specific pattern of bits and 00:02:43.379 --> 00:02:45.599 the designer basically says if you hear the specific 00:02:45.599 --> 00:02:47.860 sequence i really don't care what you do precisely 00:02:48.479 --> 00:02:51.099 But they don't say that out of laziness. It's 00:02:51.099 --> 00:02:53.139 usually because that specific input sequence 00:02:53.139 --> 00:02:55.379 is totally irrelevant to the function at hand. 00:02:55.520 --> 00:02:57.599 Right. The history of this is actually great 00:02:57.599 --> 00:03:00.780 because the names for this concept in the older 00:03:00.780 --> 00:03:03.400 literature are incredibly descriptive. You will 00:03:03.400 --> 00:03:06.139 see them called redundancies or irrelevancies. 00:03:06.620 --> 00:03:08.620 I actually found a few more colorful ones in 00:03:08.620 --> 00:03:10.960 the notes that I really liked. Vacuous combinations. 00:03:11.479 --> 00:03:14.080 Which sounds a bit harsh, honestly. Like an insult 00:03:14.080 --> 00:03:16.360 you would hear in a Victorian novel. You vacuous 00:03:16.360 --> 00:03:18.819 combinations. Exactly. Or they call them forbidden 00:03:18.819 --> 00:03:21.719 combinations. And my personal favorite, logical 00:03:21.719 --> 00:03:23.699 remainders. It sounds like a progressive rock 00:03:23.699 --> 00:03:27.539 band. It really does. But forbidden is actually 00:03:27.539 --> 00:03:30.580 the best hint at what's going on here. The reason 00:03:30.580 --> 00:03:33.819 the designer doesn't care. about the output for 00:03:33.819 --> 00:03:36.240 these terms is that they have the freedom to 00:03:36.240 --> 00:03:38.620 choose it arbitrarily. Now explain that freedom, 00:03:38.740 --> 00:03:41.360 because usually in engineering, precision is 00:03:41.360 --> 00:03:43.560 everything, right? You don't just get to choose 00:03:43.560 --> 00:03:47.400 your output on a whim. If I press A on my keyboard, 00:03:47.620 --> 00:03:50.500 I want an A to pop up, not a Q, just because 00:03:50.500 --> 00:03:52.460 the engineer felt like it. Well, think of it 00:03:52.460 --> 00:03:55.039 this way. If I tell the circuit, I don't care 00:03:55.039 --> 00:03:57.259 if the output is a 1 or a 0 for this specific 00:03:57.259 --> 00:04:00.110 input. That gives the mathematical model flexibility. 00:04:00.509 --> 00:04:03.729 I can assign it a 1 or I can assign it a 0. And 00:04:03.729 --> 00:04:05.750 I get to choose whichever one makes the overall 00:04:05.750 --> 00:04:08.669 math equation simple. Oh, I see. So it isn't 00:04:08.669 --> 00:04:11.050 just ignoring it. It's using it as a tool. It 00:04:11.050 --> 00:04:13.610 acts like a wild card. Exactly. It is a wild 00:04:13.610 --> 00:04:16.449 card. By picking the most convenient output for 00:04:16.449 --> 00:04:19.449 those don't care spots, we achieve minimization. 00:04:20.050 --> 00:04:21.870 Minimization. Right. We can make the circuit 00:04:21.870 --> 00:04:24.829 simplest, smallest, fastest. We can make it cheapest 00:04:24.829 --> 00:04:27.810 to manufacture, or we can use it to reduce power 00:04:27.810 --> 00:04:31.149 consumption. So if assigning a 1 to that don't 00:04:31.149 --> 00:04:33.310 care spot allows me to group a bunch of other 00:04:33.310 --> 00:04:35.850 signals together and maybe, I don't know, delete 00:04:35.850 --> 00:04:38.970 a physical transistor or remove a wire entirely, 00:04:39.430 --> 00:04:42.199 I just do it. You do it immediately. In the old 00:04:42.199 --> 00:04:45.240 days, this saved actual physical bulky components. 00:04:45.620 --> 00:04:48.740 Today, inside a microscopic silicon chip, it 00:04:48.740 --> 00:04:51.500 saves space and energy. Every single logic gate 00:04:51.500 --> 00:04:54.500 you remove means less heat generated by the processor. 00:04:54.839 --> 00:04:56.959 OK, let's unpack this minimization concept a 00:04:56.959 --> 00:04:59.120 bit more, because I think when people hear minimization, 00:04:59.120 --> 00:05:01.079 they might think cutting corners like, oh, we 00:05:01.079 --> 00:05:03.300 didn't bolt the bolts on the bridge to save money. 00:05:03.379 --> 00:05:05.720 But you're saying it's really about pure optimization. 00:05:05.879 --> 00:05:08.899 Right. It's elegance, not laziness. And to truly 00:05:08.899 --> 00:05:10.680 understand that, we have to look at the logic. 00:05:10.920 --> 00:05:14.439 of ignoring things. There is a subtle but absolutely 00:05:14.439 --> 00:05:17.459 critical distinction in the source material between 00:05:17.459 --> 00:05:21.800 a pure don't care term and what is called a can't 00:05:21.800 --> 00:05:24.399 happen term. A can't happen term. That sounds 00:05:24.399 --> 00:05:27.120 pretty definitive. It is definitive. A can't 00:05:27.120 --> 00:05:30.120 happen term is an input sequence that we know, 00:05:30.139 --> 00:05:33.199 either physically or logically, will simply never 00:05:33.199 --> 00:05:35.720 occur in the normal operation of the system. 00:05:36.009 --> 00:05:38.129 Give me a real -world analogy for that because 00:05:38.129 --> 00:05:40.529 input sequence can feel a little abstract if 00:05:40.529 --> 00:05:42.550 you aren't staring at code all day. Okay, think 00:05:42.550 --> 00:05:44.529 about a standard digital calendar. Okay. You 00:05:44.529 --> 00:05:46.050 have inputs for the month and inputs for the 00:05:46.050 --> 00:05:48.230 day. Right. The input for the month is February 00:05:48.230 --> 00:05:51.290 and the input for the day is the 30th. That is 00:05:51.290 --> 00:05:53.769 a can't happen term. Oh, because February 30th 00:05:53.769 --> 00:05:56.250 doesn't exist. Exactly. So if you are designing 00:05:56.250 --> 00:05:58.069 the logic for what that calendar application 00:05:58.069 --> 00:06:00.790 should do, you do not need to write a specific 00:06:00.790 --> 00:06:03.389 complex rule for February 30th. You don't need 00:06:03.389 --> 00:06:06.430 to say if Feb 30, then display a giant error 00:06:06.430 --> 00:06:08.689 message. You just don't care. Because the user 00:06:08.689 --> 00:06:10.790 physically can't select it from the dropdown 00:06:10.790 --> 00:06:14.579 anyway? Ideally, yes. So seeing February 30th 00:06:14.579 --> 00:06:18.180 is a can't happen event. Okay, so if I'm the 00:06:18.180 --> 00:06:20.939 designer and I have this wildcard freedom, do 00:06:20.939 --> 00:06:23.379 I just let the system crash if it somehow happens? 00:06:23.800 --> 00:06:26.939 Or do I actively use it? Well, here is the key 00:06:26.939 --> 00:06:29.899 takeaway from the encyclopedias. To a logic designer, 00:06:30.199 --> 00:06:33.800 a can't happen term and a don't care term are 00:06:33.800 --> 00:06:36.779 treated the exact same way. They are. Yes. They 00:06:36.779 --> 00:06:38.740 were both just opportunities for optimization. 00:06:39.120 --> 00:06:41.680 If you know an input can't happen, you don't 00:06:41.680 --> 00:06:43.459 care what the output would be if it did happen. 00:06:43.600 --> 00:06:45.680 Because theoretically, it is never, ever going 00:06:45.680 --> 00:06:47.819 to happen. In theory. And that's exactly where 00:06:47.819 --> 00:06:50.180 the optimization comes in. Engineers use specific 00:06:50.180 --> 00:06:52.699 tools to visualize all this. The sources mention 00:06:52.699 --> 00:06:55.800 methods like Carnivage maps. Ah, K -maps. I saw 00:06:55.800 --> 00:06:57.480 these in the documentation. They look like little 00:06:57.480 --> 00:06:59.839 grids of squares. It honestly looks like a Sudoku 00:06:59.839 --> 00:07:02.300 puzzle. It's a bit like Sudoku, or maybe Tetris. 00:07:02.620 --> 00:07:04.860 You map out your ones and zeros on this grid. 00:07:05.290 --> 00:07:07.329 Each square represents a combination of inputs. 00:07:07.750 --> 00:07:09.730 And then you have these blank spots on the board. 00:07:10.230 --> 00:07:12.670 Those are the don't cares. And the goal is to 00:07:12.670 --> 00:07:15.589 circle groups of them, right? Yes. The goal of 00:07:15.589 --> 00:07:18.910 the KMAP is to draw boxes around groups of ones 00:07:18.910 --> 00:07:22.589 to simplify the logic. And the mathematical rule 00:07:22.589 --> 00:07:25.449 is the bigger the box you can draw, the simpler 00:07:25.449 --> 00:07:27.550 the resulting circuit becomes. So if I have a 00:07:27.550 --> 00:07:29.689 don't care square sitting right next to a bunch 00:07:29.689 --> 00:07:32.069 of one squares. You should pretend the don't 00:07:32.069 --> 00:07:35.670 care is a one. Now your box is bigger. Your logic 00:07:35.670 --> 00:07:38.230 is simpler. You have just saved a logic gate. 00:07:38.670 --> 00:07:41.750 That is incredibly satisfying. It's like connecting 00:07:41.750 --> 00:07:43.930 the dots, but you get to draw extra dots wherever 00:07:43.930 --> 00:07:45.910 you want just to make the picture look better. 00:07:46.029 --> 00:07:48.250 That is a perfect way to describe it. There is 00:07:48.250 --> 00:07:50.269 also the Quine -McCluskey algorithm, which is 00:07:50.269 --> 00:07:52.110 heavily featured in the notes. That's essentially 00:07:52.110 --> 00:07:54.689 just an algebraic way to do the exact same thing 00:07:54.689 --> 00:07:57.230 for much more complex systems where a grid is 00:07:57.230 --> 00:07:59.519 just way too hard to draw by hand. I do want 00:07:59.519 --> 00:08:02.060 to touch on a bit of the historical context here 00:08:02.060 --> 00:08:04.439 because the notes mention a guy named Seymour 00:08:04.439 --> 00:08:07.899 Ginsburg back in 1958. This seemed like a massive 00:08:07.899 --> 00:08:10.180 turning point in how we actually understand this 00:08:10.180 --> 00:08:13.959 efficiency. Yes, Ginsburg. This part is fascinating 00:08:13.959 --> 00:08:16.379 because it really highlights the physical limits 00:08:16.379 --> 00:08:20.199 of early computing. In 1958, they simply didn't 00:08:20.199 --> 00:08:22.439 have the processing power we have now to design 00:08:22.439 --> 00:08:25.220 chips. They were working with much clearer constraints. 00:08:25.620 --> 00:08:27.939 Right, because today we just assume computers 00:08:27.939 --> 00:08:30.839 can optimize anything instantly. You type code, 00:08:30.939 --> 00:08:34.019 and the compiler fixes your mess behind the scenes. 00:08:34.639 --> 00:08:37.100 Back then, Ginsburg proved something that was 00:08:37.100 --> 00:08:39.620 actually quite counterintuitive. He was looking 00:08:39.620 --> 00:08:42.580 at finite state machines. Those are basically 00:08:42.580 --> 00:08:45.679 systems that move step by step from state A to 00:08:45.679 --> 00:08:48.019 state B. Okay. He showed that just because you 00:08:48.019 --> 00:08:50.639 minimize the states, like reducing the number 00:08:50.639 --> 00:08:53.080 of actual steps it takes to do a job, it doesn't 00:08:53.080 --> 00:08:54.779 necessarily mean you have minimized the logic 00:08:54.779 --> 00:08:57.080 elements. Wait, that sounds completely backwards. 00:08:57.340 --> 00:08:59.539 If I have fewer steps, shouldn't I automatically 00:08:59.539 --> 00:09:01.879 have less hardware? If I build a machine with 00:09:01.879 --> 00:09:03.919 three gears instead of four, it should be a smaller 00:09:03.919 --> 00:09:07.100 machine. In mechanics, yes. But in logic, not 00:09:07.100 --> 00:09:10.419 always. Sometimes, describing a shortcut takes 00:09:10.419 --> 00:09:12.659 a much more complex vocabulary than just taking 00:09:12.659 --> 00:09:15.139 the long way around. Oh, I get it. Imagine trying 00:09:15.139 --> 00:09:18.120 to explain a shortcut through a city that involves 00:09:18.120 --> 00:09:22.059 six very specific left turns through back alleys 00:09:22.059 --> 00:09:25.139 versus just saying, stay on the highway. The 00:09:25.139 --> 00:09:27.080 highway is technically longer, so it has more 00:09:27.080 --> 00:09:29.200 states, but the instruction is much simpler, 00:09:29.320 --> 00:09:32.610 less logic. That makes perfect sense. So Ginsburg 00:09:32.610 --> 00:09:34.950 proved mathematically that simple behavior doesn't 00:09:34.950 --> 00:09:38.250 always equal simple wiring. Exactly. And he pointed 00:09:38.250 --> 00:09:41.750 out a major historical limitation. Calculating 00:09:41.750 --> 00:09:44.789 the absolute perfect direct minimization using 00:09:44.789 --> 00:09:47.090 these don't care conditions for large systems 00:09:47.090 --> 00:09:51.500 was computationally impractical in 1958. They 00:09:51.500 --> 00:09:53.500 fully understood the theory, but they couldn't 00:09:53.500 --> 00:09:55.879 always brute force the perfect solution. So they 00:09:55.879 --> 00:09:57.919 basically had to make trade -offs. They knew 00:09:57.919 --> 00:09:59.779 they were leaving some efficiency on the table 00:09:59.779 --> 00:10:01.860 because they just didn't have the computers to 00:10:01.860 --> 00:10:03.960 calculate how to build better computers. Which 00:10:03.960 --> 00:10:05.899 is humbling, honestly. It shows that even the 00:10:05.899 --> 00:10:08.679 concept of strategically ignoring things requires 00:10:08.679 --> 00:10:10.720 a massive amount of calculation to get right. 00:10:11.100 --> 00:10:13.039 Let's try to really ground this for the listener. 00:10:13.200 --> 00:10:15.940 We've been talking a lot about logic and sequences 00:10:15.940 --> 00:10:18.379 and algorithms. Can we give a concrete example 00:10:18.379 --> 00:10:21.240 of where a don't care term actually lives inside 00:10:21.240 --> 00:10:24.039 a gadget that I might recognize? The absolute 00:10:24.039 --> 00:10:26.220 classic example and the one the source material 00:10:26.220 --> 00:10:30.200 highlights heavily is BCD, binary coded decimal. 00:10:30.399 --> 00:10:32.379 Binary coded decimal. Break that down for us. 00:10:32.480 --> 00:10:34.639 So ideally, computers just like binary, pure 00:10:34.639 --> 00:10:38.440 ones and zeros. But humans, we like. decimal. 00:10:38.500 --> 00:10:41.840 We like our digits 0 through 9. BCD is a specific 00:10:41.840 --> 00:10:43.940 way of encoding those human -friendly digits 00:10:43.940 --> 00:10:47.830 using exactly four bits of binary code. Okay, 00:10:47.889 --> 00:10:50.529 four bits. So that's 0, 0, 0, 0, 0, 0, 0, 1, 00:10:50.690 --> 00:10:53.070 all the way up to weight. Four bits. That's 2 00:10:53.070 --> 00:10:56.190 to the power of 4. That is 16 combinations. You 00:10:56.190 --> 00:10:58.690 got it. Four bits can count from 0 up to 15. 00:10:58.809 --> 00:11:02.409 That's 16 possible combinations. But for BCD, 00:11:02.509 --> 00:11:04.710 which represents just a single decimal digit, 00:11:04.950 --> 00:11:07.889 we only actually need 0 through 9. So what happens 00:11:07.889 --> 00:11:11.570 to the binary codes for 10, 11, 12, 13, 14, and 00:11:11.570 --> 00:11:14.070 15? They are the don't care terms. The sources 00:11:14.070 --> 00:11:17.250 refer to them as pseudotetrates. Pseudotetrades. 00:11:17.309 --> 00:11:20.789 That sounds like a dinosaur. Or maybe a very 00:11:20.789 --> 00:11:24.710 specific weird geometric shape. Look out! A wild 00:11:24.710 --> 00:11:27.269 pseudotetrade appeared. It does sound ancient. 00:11:27.769 --> 00:11:29.730 But it simply means they are combinations of 00:11:29.730 --> 00:11:32.750 four bits that in the specific language of BCD 00:11:32.750 --> 00:11:34.590 are complete nonsense. They are totally unused. 00:11:34.830 --> 00:11:37.549 So if I'm building a circuit for, say, an old 00:11:37.549 --> 00:11:40.929 school alarm clock with those glowing red seven 00:11:40.929 --> 00:11:43.549 segment displays. Perfect example. Everyone knows 00:11:43.549 --> 00:11:46.070 those clocks. The displays have seven little 00:11:46.070 --> 00:11:48.789 LED light bars arranged in a figure eight pattern. 00:11:49.230 --> 00:11:51.009 To make the number one, you light up two bars 00:11:51.009 --> 00:11:52.990 on the right. To make the number eight, you light 00:11:52.990 --> 00:11:56.370 up all seven bars. Right. Now imagine you are 00:11:56.370 --> 00:11:58.649 the engineer designing the circuit that controls 00:11:58.649 --> 00:12:01.529 just the lower left vertical bar of that display, 00:12:01.710 --> 00:12:04.029 just that one little line. Okay. You need to 00:12:04.029 --> 00:12:06.070 tell that bar exactly when to turn on. It turns 00:12:06.070 --> 00:12:07.769 on for the number zero, the number two, the number 00:12:07.769 --> 00:12:09.769 six, the number eight. And it stays off for one, 00:12:09.889 --> 00:12:12.629 three, four, five, seven, and nine. Correct. 00:12:12.850 --> 00:12:16.330 But what about inputs 10 through 15? Well, the 00:12:16.330 --> 00:12:18.529 clock is only ever supposed to show zero to nine. 00:12:19.070 --> 00:12:21.730 If my bedside clock shows a 12 in a single digit 00:12:21.730 --> 00:12:24.070 spot, something is very physically wrong with 00:12:24.070 --> 00:12:27.169 my bedroom. So the chip should never, ever receive 00:12:27.169 --> 00:12:30.610 a 12. Exactly. It's a can't happen term. So for 00:12:30.610 --> 00:12:33.950 the inputs 10 through 15, which is binary, 1010 00:12:33.950 --> 00:12:37.409 through 11111, the designer just says, I don't 00:12:37.409 --> 00:12:41.100 care. If the chip did somehow receive a 12, it 00:12:41.100 --> 00:12:43.139 literally does not matter if that lower left 00:12:43.139 --> 00:12:45.379 bar lights up or not because the user is never 00:12:45.379 --> 00:12:47.120 supposed to be in a situation to see it. And 00:12:47.120 --> 00:12:49.539 by saying I don't care, the math for that lower 00:12:49.539 --> 00:12:53.269 left bar gets easier. So much easier. The Boolean 00:12:53.269 --> 00:12:56.009 function can be simplified significantly. Without 00:12:56.009 --> 00:12:57.950 the don't cares, the logic equation might be 00:12:57.950 --> 00:12:59.889 huge because you have to explicitly tell it to 00:12:59.889 --> 00:13:03.830 be 0 off f for 10, 0 off for 11, 0 off for 12. 00:13:04.250 --> 00:13:07.330 But if we treat those unused numbers as wildcards, 00:13:07.629 --> 00:13:10.330 we can simplify the equation down to practically 00:13:10.330 --> 00:13:13.409 nothing. You save actual transistors on the chip. 00:13:13.629 --> 00:13:15.389 That is really cool. It's kind of like cleaning 00:13:15.389 --> 00:13:17.230 your room by shoving everything into the closet, 00:13:17.330 --> 00:13:19.110 but because nobody is ever allowed to open the 00:13:19.110 --> 00:13:21.509 closet, the room is technically clean? In a way, 00:13:21.610 --> 00:13:25.389 yes. You're optimizing the visible space by utilizing 00:13:25.389 --> 00:13:28.049 the forbidden space. And there is actually another 00:13:28.049 --> 00:13:31.289 super quirky example of this in vintage hardware. 00:13:31.830 --> 00:13:34.850 Write -only registers. I saw this in the notes 00:13:34.850 --> 00:13:37.610 and it confused me. A register is basically a 00:13:37.610 --> 00:13:40.029 memory slot. So this is a slot that you can write 00:13:40.029 --> 00:13:42.610 data to, but you literally cannot read data from. 00:13:42.649 --> 00:13:44.730 That sounds useless. How do you know what's in 00:13:44.730 --> 00:13:47.210 it? You don't. It is strictly one -way communication. 00:13:47.789 --> 00:13:51.009 In older hardware... Reading data back out required 00:13:51.009 --> 00:13:54.210 extra logic gates. It needed extra wires, extra 00:13:54.210 --> 00:13:56.929 multiplexers. If the designer desperately needed 00:13:56.929 --> 00:13:59.549 to save space or cost and the software didn't 00:13:59.549 --> 00:14:01.950 strictly need to check what it just wrote, they 00:14:01.950 --> 00:14:04.470 would just completely leave out the read circuitry. 00:14:04.549 --> 00:14:06.450 So what happens if a programmer makes a mistake 00:14:06.450 --> 00:14:08.289 and tries to read it anyway? Like, hey, computer, 00:14:08.389 --> 00:14:10.149 what number's in that box? You get a don't care 00:14:10.149 --> 00:14:12.850 result. You just get garbage data. So the computer 00:14:12.850 --> 00:14:14.509 basically just shrugs and hands you whatever 00:14:14.509 --> 00:14:17.159 random electricity happened to be nearby. Basically, 00:14:17.240 --> 00:14:20.059 yeah. It might return all zeros, all ones, or 00:14:20.059 --> 00:14:23.279 just pure static noise. It is a direct consequence 00:14:23.279 --> 00:14:26.659 of don't care optimizations. The engineers actively 00:14:26.659 --> 00:14:29.200 traded functionality, the ability to read for 00:14:29.200 --> 00:14:31.879 hardware size. The output for read was explicitly 00:14:31.879 --> 00:14:35.539 defined as a don't care. Okay, so we've got a 00:14:35.539 --> 00:14:37.279 good handle on the hardware side, but I want 00:14:37.279 --> 00:14:39.200 to transition to the software side, or at least 00:14:39.200 --> 00:14:41.659 the simulation side of things, because the technical 00:14:41.659 --> 00:14:44.419 docs mention this somewhat mysterious X value. 00:14:44.720 --> 00:14:47.600 Ah, yes, the X factor. This is where things get 00:14:47.600 --> 00:14:49.840 really interesting, and honestly, where the critical 00:14:49.840 --> 00:14:52.620 distinction between simulation and physical reality 00:14:52.620 --> 00:14:55.019 becomes so important. So in hardware description 00:14:55.019 --> 00:14:58.440 languages, things like Verilog or VHDL, which 00:14:58.440 --> 00:15:00.720 engineers use to code the hardware before it's 00:15:00.720 --> 00:15:02.919 actually built in a factory, you have zero, you 00:15:02.919 --> 00:15:05.919 have one, and you have... X. Right. In multivalued 00:15:05.919 --> 00:15:08.799 logic for Verilog, it's literally just X. In 00:15:08.799 --> 00:15:12.059 VHDL, it can be X for a forced unknown or sometimes 00:15:12.059 --> 00:15:15.139 a W for a weak unknown. A weak unknown. That 00:15:15.139 --> 00:15:16.740 sounds like me trying to remember where I parked 00:15:16.740 --> 00:15:18.799 at the grocery store. But what does X actually 00:15:18.799 --> 00:15:20.799 mean when you're running a simulation on a screen? 00:15:21.059 --> 00:15:24.539 In a computer simulation of a chip, an X usually 00:15:24.539 --> 00:15:27.600 represents a problem or a transition state. It 00:15:27.600 --> 00:15:29.299 might mean you have a flip -flop, a little memory 00:15:29.299 --> 00:15:31.399 storage unit that simply hasn't settled yet. 00:15:31.419 --> 00:15:34.590 It's in flux. Okay. Or, and this is worse, it 00:15:34.590 --> 00:15:36.389 could mean you have two different sources trying 00:15:36.389 --> 00:15:38.490 to drive the exact same signal at the same time. 00:15:39.110 --> 00:15:41.330 One wire is trying to say zero and the other 00:15:41.330 --> 00:15:43.090 wire is trying to say one. So it's a conflict. 00:15:43.230 --> 00:15:46.190 A massive conflict. The simulator basically throws 00:15:46.190 --> 00:15:48.789 up its hands and says, X, I don't know who wins. 00:15:49.509 --> 00:15:51.629 It's a very useful way for the simulation to 00:15:51.629 --> 00:15:54.289 tell the engineer, hey, your logic is ambiguous 00:15:54.289 --> 00:15:56.970 right here. And I think here is where it gets 00:15:56.970 --> 00:15:59.029 really interesting for anyone trying to wrap 00:15:59.029 --> 00:16:01.850 their head around this. You can't have an X in 00:16:01.850 --> 00:16:04.649 real life, can you? If I took a physical voltmeter 00:16:04.649 --> 00:16:08.250 to a physical wire inside my laptop motherboard, 00:16:08.690 --> 00:16:13.049 the dial wouldn't say X. No, you cannot measure 00:16:13.049 --> 00:16:15.149 an X. And this is a concept that actually trips 00:16:15.149 --> 00:16:17.330 up a lot of students and even junior engineers. 00:16:17.509 --> 00:16:20.649 An X value simply does not exist in synthesized 00:16:20.649 --> 00:16:23.429 hardware in reality. Because physics is physics. 00:16:23.629 --> 00:16:26.070 There is either a voltage on the wire or there 00:16:26.070 --> 00:16:28.309 isn't. It's not magic. It's not Schrodinger's 00:16:28.309 --> 00:16:31.210 cat. Exactly. The voltage is always either a 00:16:31.210 --> 00:16:33.629 zero or a one, assuming it's within the logic 00:16:33.629 --> 00:16:36.769 thresholds. The real problem is, because of that 00:16:36.769 --> 00:16:39.029 conflict or that unsettled state we saw in the 00:16:39.029 --> 00:16:41.570 simulation, we as the observers don't actually 00:16:41.570 --> 00:16:44.389 know which one it is. It's just a ghost in the 00:16:44.389 --> 00:16:47.509 machine. It truly is. The specific zero or one 00:16:47.509 --> 00:16:51.169 is not determinable from the inputs. So in the 00:16:51.169 --> 00:16:53.730 real physical world, the chip isn't outputting 00:16:53.730 --> 00:16:56.860 a magical X state. It is outputting a 1 or a 00:16:56.860 --> 00:16:59.440 0, but it might be flickering rapidly or it might 00:16:59.440 --> 00:17:01.679 be stuck and we just can't predict it based on 00:17:01.679 --> 00:17:03.960 the logic rules we wrote. That feels a little 00:17:03.960 --> 00:17:06.339 unsettling, to be honest. We build these machines 00:17:06.339 --> 00:17:09.400 to be these precise, perfectly logical beasts, 00:17:09.559 --> 00:17:12.480 but deep down there are these pockets of we just 00:17:12.480 --> 00:17:14.680 don't know. And that leads us directly to the 00:17:14.680 --> 00:17:17.339 truly dangerous part of don't care terms. Because 00:17:17.339 --> 00:17:20.180 sometimes what we think implies I don't care 00:17:20.180 --> 00:17:22.900 actually turns into I really should have cared. 00:17:23.059 --> 00:17:25.609 Right. Let's get into the danger zone. So we 00:17:25.609 --> 00:17:27.569 established earlier that can't happen terms are 00:17:27.569 --> 00:17:30.009 treated as don't cares. We optimize the chip, 00:17:30.049 --> 00:17:33.089 assuming they will never, ever occur. But what 00:17:33.089 --> 00:17:35.130 if a can't happen condition actually does happen? 00:17:35.329 --> 00:17:38.670 This is the absolute nightmare scenario for reliability 00:17:38.670 --> 00:17:42.069 engineers. And this is especially risky in circuits 00:17:42.069 --> 00:17:44.569 that involve feedback. Things like state machines, 00:17:44.950 --> 00:17:46.950 where the next step the machine takes depends 00:17:46.950 --> 00:17:49.960 entirely on the previous step it just took. Okay, 00:17:50.019 --> 00:17:52.420 so let's visualize this for the listener. Normally, 00:17:52.480 --> 00:17:54.539 the machine bounces happily between state A, 00:17:54.619 --> 00:17:57.500 state B, and state C. It is never supposed to 00:17:57.500 --> 00:18:00.500 go to state D. So the engineers optimized state 00:18:00.500 --> 00:18:03.079 D completely out of existence. They treated it 00:18:03.079 --> 00:18:05.599 as a don't care. Correct. We didn't build a logic 00:18:05.599 --> 00:18:07.759 path out of state D because we confidently assumed 00:18:07.759 --> 00:18:10.480 we would never get into state D. But how does 00:18:10.480 --> 00:18:13.319 the physical machine get to state D if it's supposed 00:18:13.319 --> 00:18:15.980 to be mathematically impossible? The source material 00:18:15.980 --> 00:18:19.589 lists a few major culprits. Power up is a really 00:18:19.589 --> 00:18:21.809 big one. When you first turn a device on, for 00:18:21.809 --> 00:18:24.390 just a split second, all the voltage levels across 00:18:24.390 --> 00:18:26.829 the board are stabilizing. The flip -flops might 00:18:26.829 --> 00:18:28.910 wake up in a completely random state. So you 00:18:28.910 --> 00:18:31.049 turn on your smart toaster, and for one tiny 00:18:31.049 --> 00:18:33.789 millisecond, it wakes up and thinks it is in 00:18:33.789 --> 00:18:36.009 a state that doesn't logically exist. Exactly. 00:18:36.049 --> 00:18:39.210 It wakes up in that forbidden state D. Another 00:18:39.210 --> 00:18:41.990 cause could be that the power supply fluctuated. 00:18:42.509 --> 00:18:46.140 And the other causes are even more exotic. Things 00:18:46.140 --> 00:18:48.819 like electrical noise, extreme heat, or even 00:18:48.819 --> 00:18:52.039 cosmic radiation. Cosmic radiation. I love that 00:18:52.039 --> 00:18:54.819 actual space particles can fly through the atmosphere, 00:18:55.019 --> 00:18:57.119 flip a single bit in my computer, and cause a 00:18:57.119 --> 00:18:59.440 don't -care error. It sounds like pure science 00:18:59.440 --> 00:19:02.200 fiction. It absolutely happens. In the industry, 00:19:02.339 --> 00:19:04.980 it's called a soft error. A high -energy particle 00:19:04.980 --> 00:19:07.900 hits a microscopic transistor and literally flips 00:19:07.900 --> 00:19:11.220 a zero to a one. And here's the dire consequence. 00:19:11.599 --> 00:19:14.420 If that physical flip pushes the machine into 00:19:14.420 --> 00:19:16.259 a state it was never supposed to be in a forbidden 00:19:16.259 --> 00:19:19.200 input, it might get entirely stuck. The source 00:19:19.200 --> 00:19:21.839 text uses a very poetic phrase for this specific 00:19:21.839 --> 00:19:24.519 lockup. It calls it a walled garden of states. 00:19:24.819 --> 00:19:27.740 It is a beautiful image, isn't it? But a terrible, 00:19:27.779 --> 00:19:30.599 terrible situation for a circuit. Imagine the 00:19:30.599 --> 00:19:32.819 machine accidentally lands in one of those can't 00:19:32.819 --> 00:19:35.769 -happen states. Because we treated it as a don't 00:19:35.769 --> 00:19:38.309 care to save space, we didn't program a way out. 00:19:38.349 --> 00:19:40.130 We didn't give it an instruction that says, if 00:19:40.130 --> 00:19:42.289 you find yourself here, go back to start. We 00:19:42.289 --> 00:19:44.829 just left the logic completely undefined. Yes. 00:19:45.369 --> 00:19:47.609 So the machine might look at its internal logic 00:19:47.609 --> 00:19:50.869 table, see that the next step from this impossible 00:19:50.869 --> 00:19:54.109 state is just another impossible state. And it 00:19:54.109 --> 00:19:56.869 just bounces around in this walled garden of 00:19:56.869 --> 00:19:59.769 invalid states, unable to ever get back to normal 00:19:59.769 --> 00:20:02.529 operation. That's a hardware lockup. That is 00:20:02.529 --> 00:20:04.730 your computer freezing completely and needing 00:20:04.730 --> 00:20:07.630 you to pull the plug for a hard reboot. It is 00:20:07.630 --> 00:20:10.950 literally lost in a logic maze that has no physical 00:20:10.950 --> 00:20:13.950 exit. That is terrifying for anything mission 00:20:13.950 --> 00:20:16.910 critical. I mean, if my smart toaster freezes 00:20:16.910 --> 00:20:18.970 in a walled garden, fine, I get bumped toast. 00:20:19.329 --> 00:20:22.730 But what if it's a pacemaker or a communications 00:20:22.730 --> 00:20:25.349 satellite or the anti -lock breaks on a car? 00:20:25.549 --> 00:20:27.890 Exactly. That is where the stakes completely 00:20:27.890 --> 00:20:30.549 change. So responsible designers have to ensure 00:20:30.549 --> 00:20:32.269 that can't -happen states are truly impossible. 00:20:32.849 --> 00:20:35.950 Or, if the safety is critical, they simply cannot 00:20:35.950 --> 00:20:38.549 use those states for optimization. They have 00:20:38.549 --> 00:20:40.470 to leave the efficiency on the table. So you 00:20:40.470 --> 00:20:42.529 have to sacrifice the smaller chip size. You 00:20:42.529 --> 00:20:44.289 have to leave the extra logic gate on the board 00:20:44.289 --> 00:20:46.769 just in case. You have to. You design what the 00:20:46.769 --> 00:20:48.829 documents call don't -care alarms. An alarm, 00:20:48.950 --> 00:20:50.869 like a physical siren goes off. Essentially, 00:20:50.970 --> 00:20:53.509 it's logic that detects if the machine enters 00:20:53.509 --> 00:20:56.380 a forbidden state. It flags an emergency signal 00:20:56.380 --> 00:20:59.299 to the main processor. Or, even more commonly, 00:20:59.500 --> 00:21:01.859 you design the logic so that all those unused 00:21:01.859 --> 00:21:04.539 states automatically transition back to a known, 00:21:04.720 --> 00:21:07.660 safe, normal state. These are called transitory 00:21:07.660 --> 00:21:11.339 states. Oh, like self -healing logic. Yes. It 00:21:11.339 --> 00:21:13.660 assumes the impossible might actually happen 00:21:13.660 --> 00:21:17.519 due to a cosmic ray or a power surge, and it 00:21:17.519 --> 00:21:21.089 deliberately provides a path home. It explicitly 00:21:21.089 --> 00:21:23.630 says, if you wake up in state D, go immediately 00:21:23.630 --> 00:21:26.809 to state A. It costs more logic, it makes the 00:21:26.809 --> 00:21:28.589 physical chip slightly bigger and more power 00:21:28.589 --> 00:21:30.710 -hungry, but it prevents the walled garden scenario 00:21:30.710 --> 00:21:32.769 entirely. So it really just comes down to, how 00:21:32.769 --> 00:21:35.109 much do you trust the universe not to mess with 00:21:35.109 --> 00:21:37.990 your voltages today? Pretty much. If you are 00:21:37.990 --> 00:21:40.190 building a cheap plastic toy that sings when 00:21:40.190 --> 00:21:41.829 you press a button, you optimize everything, 00:21:42.069 --> 00:21:44.029 use all the don't cares, and you risk the lockup. 00:21:44.269 --> 00:21:46.410 If you are building the navigation system for 00:21:46.410 --> 00:21:48.759 an airplane, you build the path home. This has 00:21:48.759 --> 00:21:50.400 been a really enlightening look at something 00:21:50.400 --> 00:21:52.359 that sounds so dismissive on the surface, don't 00:21:52.359 --> 00:21:54.700 care. But really, it is entirely about efficiency. 00:21:54.980 --> 00:21:57.400 It is. It's about recognizing that not every 00:21:57.400 --> 00:22:00.339 single input requires a specific, thoughtful 00:22:00.339 --> 00:22:04.160 response. Strategically ignoring the impossible 00:22:04.160 --> 00:22:07.519 allows the possible to run smoother, faster, 00:22:07.759 --> 00:22:10.900 and cheaper. It's focusing purely on what matters. 00:22:11.160 --> 00:22:13.519 Exactly. But as we learned with the walled gardens, 00:22:13.660 --> 00:22:15.740 you have to be careful because if you ignore 00:22:15.740 --> 00:22:19.200 the wrong thing or if the universe decides to 00:22:19.200 --> 00:22:22.039 force an impossible input on you anyway, you 00:22:22.039 --> 00:22:25.000 end up scrapped. It is the absolute classic engineering 00:22:25.000 --> 00:22:28.470 tension. Optimization versus robustness. You 00:22:28.470 --> 00:22:30.869 can strip everything down to the bare metal to 00:22:30.869 --> 00:22:33.750 make it blazing fast, but you lose that critical 00:22:33.750 --> 00:22:35.930 buffer against the unexpected. Before we go, 00:22:35.990 --> 00:22:37.769 I want to leave you, the listener, with a thought 00:22:37.769 --> 00:22:40.049 about that X value we discussed. You mentioned 00:22:40.049 --> 00:22:42.250 that in reality, X doesn't exist. It is always 00:22:42.250 --> 00:22:43.970 a zero or a one. We just don't know which one 00:22:43.970 --> 00:22:47.359 it is. Right. The physics demands value. The 00:22:47.359 --> 00:22:49.640 universe doesn't do undefined. I feel like there's 00:22:49.640 --> 00:22:52.299 a real life philosophy buried in there. In the 00:22:52.299 --> 00:22:54.660 simulation of our lives, our plans, our daily 00:22:54.660 --> 00:22:57.680 worries, we have all these unknowns. These X 00:22:57.680 --> 00:23:00.259 values where we feel like we are in conflict 00:23:00.259 --> 00:23:02.619 or totally unsettled. We don't know the outcome. 00:23:02.819 --> 00:23:05.240 But in the physical reality of the moment, everything 00:23:05.240 --> 00:23:08.160 already has a value. We are where we are. The 00:23:08.160 --> 00:23:11.039 outcome is already unfolding. We just might not 00:23:11.039 --> 00:23:12.940 have the inputs to determine what it is yet. 00:23:13.059 --> 00:23:15.670 That is... Surprisingly deep for a technical 00:23:15.670 --> 00:23:18.230 discussion about logic gates. But it's true. 00:23:18.410 --> 00:23:20.630 The value is always there, even if the simulator 00:23:20.630 --> 00:23:22.930 in our heads can't quite see it yet. Just don't 00:23:22.930 --> 00:23:25.130 let a cosmic ray knock you into a walled garden 00:23:25.130 --> 00:23:27.769 today. Thanks for joining us for this deep dive. 00:23:27.890 --> 00:23:29.150 Stay curious. We will see you downtime.