You know, there's a phrase that I think pretty
much every single one of us has used at some
point. Oh, absolutely. Usually it's when you
are a teenager rolling your eyes at your parents.
Or maybe it's just like, you know, 5 p .m. on
a Friday and you are completely burned out. Yeah,
I know exactly the feeling you're talking about.
That phrase is, I don't care. It is a classic.
It's the universal signal for apathy. Usually
implies... laziness or checking out or just,
you know, giving up on whatever the problem is.
Exactly. It sounds like you're quitting. But
and this is really why I love digging into these
technical stacks we get for the show in the world
of computer engineering and digital logic saying,
I don't care, isn't lazy at all. Not even a little
bit. It's actually a superpower. It is a strategic
weapon. It really is. It's a fundamental pillar
of efficiency. Because if computers cared about
every single theoretical possibility, they would
be incredibly slow, they'd be massive, and they'd
probably overheat in about 10 seconds. So today
we are doing a deep dive into don't care terms.
And no, for you listening, that isn't a list
of people who ghosted me. Let's hope not. It
is a very specific engineering concept that basically
allows our devices to function at the speeds
we expect. But, and here's the twist we are going
to get to a bit later, if you aren't careful
with the things you don't care about, they can
actually come back to haunt you in some pretty
catastrophic ways. That's the core tension of
this entire field. It's about knowing what matters
and strategically ignoring what doesn't so you
can build a better system. But if you ignore
the wrong thing, well, you end up in what the
documentation literally calls a walled garden
of error states. Which, you know. Sounds lovely,
but is actually a total nightmare. A complete
nightmare. Yes. So we have a great stack of source
material today. We've got technical documentation,
some encyclopedic records on digital logic design,
and a bunch of files specifically focusing on
Boolean functions and circuit minimization. It's
a really comprehensive pile of sources. It is.
And our mission today is to figure out how engineers
decide what to ignore to make things faster,
cheaper, and smaller, and what happens when it
all goes wrong. Sounds like a plan. Let's start.
at the beginning with the definition when a logic
designer says don't care what are they actually
looking at on their screen so in the strictest
definition from the encyclopedic records we are
looking at a boolean function right basically
the math of logic the zeros and the ones right
a don't care term which is often abbreviated
just as dc is an input sequence a specific series
of bits for a boolean function where the output
effectively does not matter okay let's make that
concrete for the listener the system receives
a message like a specific pattern of bits and
the designer basically says if you hear the specific
sequence i really don't care what you do precisely
But they don't say that out of laziness. It's
usually because that specific input sequence
is totally irrelevant to the function at hand.
Right. The history of this is actually great
because the names for this concept in the older
literature are incredibly descriptive. You will
see them called redundancies or irrelevancies.
I actually found a few more colorful ones in
the notes that I really liked. Vacuous combinations.
Which sounds a bit harsh, honestly. Like an insult
you would hear in a Victorian novel. You vacuous
combinations. Exactly. Or they call them forbidden
combinations. And my personal favorite, logical
remainders. It sounds like a progressive rock
band. It really does. But forbidden is actually
the best hint at what's going on here. The reason
the designer doesn't care. about the output for
these terms is that they have the freedom to
choose it arbitrarily. Now explain that freedom,
because usually in engineering, precision is
everything, right? You don't just get to choose
your output on a whim. If I press A on my keyboard,
I want an A to pop up, not a Q, just because
the engineer felt like it. Well, think of it
this way. If I tell the circuit, I don't care
if the output is a 1 or a 0 for this specific
input. That gives the mathematical model flexibility.
I can assign it a 1 or I can assign it a 0. And
I get to choose whichever one makes the overall
math equation simple. Oh, I see. So it isn't
just ignoring it. It's using it as a tool. It
acts like a wild card. Exactly. It is a wild
card. By picking the most convenient output for
those don't care spots, we achieve minimization.
Minimization. Right. We can make the circuit
simplest, smallest, fastest. We can make it cheapest
to manufacture, or we can use it to reduce power
consumption. So if assigning a 1 to that don't
care spot allows me to group a bunch of other
signals together and maybe, I don't know, delete
a physical transistor or remove a wire entirely,
I just do it. You do it immediately. In the old
days, this saved actual physical bulky components.
Today, inside a microscopic silicon chip, it
saves space and energy. Every single logic gate
you remove means less heat generated by the processor.
OK, let's unpack this minimization concept a
bit more, because I think when people hear minimization,
they might think cutting corners like, oh, we
didn't bolt the bolts on the bridge to save money.
But you're saying it's really about pure optimization.
Right. It's elegance, not laziness. And to truly
understand that, we have to look at the logic.
of ignoring things. There is a subtle but absolutely
critical distinction in the source material between
a pure don't care term and what is called a can't
happen term. A can't happen term. That sounds
pretty definitive. It is definitive. A can't
happen term is an input sequence that we know,
either physically or logically, will simply never
occur in the normal operation of the system.
Give me a real -world analogy for that because
input sequence can feel a little abstract if
you aren't staring at code all day. Okay, think
about a standard digital calendar. Okay. You
have inputs for the month and inputs for the
day. Right. The input for the month is February
and the input for the day is the 30th. That is
a can't happen term. Oh, because February 30th
doesn't exist. Exactly. So if you are designing
the logic for what that calendar application
should do, you do not need to write a specific
complex rule for February 30th. You don't need
to say if Feb 30, then display a giant error
message. You just don't care. Because the user
physically can't select it from the dropdown
anyway? Ideally, yes. So seeing February 30th
is a can't happen event. Okay, so if I'm the
designer and I have this wildcard freedom, do
I just let the system crash if it somehow happens?
Or do I actively use it? Well, here is the key
takeaway from the encyclopedias. To a logic designer,
a can't happen term and a don't care term are
treated the exact same way. They are. Yes. They
were both just opportunities for optimization.
If you know an input can't happen, you don't
care what the output would be if it did happen.
Because theoretically, it is never, ever going
to happen. In theory. And that's exactly where
the optimization comes in. Engineers use specific
tools to visualize all this. The sources mention
methods like Carnivage maps. Ah, K -maps. I saw
these in the documentation. They look like little
grids of squares. It honestly looks like a Sudoku
puzzle. It's a bit like Sudoku, or maybe Tetris.
You map out your ones and zeros on this grid.
Each square represents a combination of inputs.
And then you have these blank spots on the board.
Those are the don't cares. And the goal is to
circle groups of them, right? Yes. The goal of
the KMAP is to draw boxes around groups of ones
to simplify the logic. And the mathematical rule
is the bigger the box you can draw, the simpler
the resulting circuit becomes. So if I have a
don't care square sitting right next to a bunch
of one squares. You should pretend the don't
care is a one. Now your box is bigger. Your logic
is simpler. You have just saved a logic gate.
That is incredibly satisfying. It's like connecting
the dots, but you get to draw extra dots wherever
you want just to make the picture look better.
That is a perfect way to describe it. There is
also the Quine -McCluskey algorithm, which is
heavily featured in the notes. That's essentially
just an algebraic way to do the exact same thing
for much more complex systems where a grid is
just way too hard to draw by hand. I do want
to touch on a bit of the historical context here
because the notes mention a guy named Seymour
Ginsburg back in 1958. This seemed like a massive
turning point in how we actually understand this
efficiency. Yes, Ginsburg. This part is fascinating
because it really highlights the physical limits
of early computing. In 1958, they simply didn't
have the processing power we have now to design
chips. They were working with much clearer constraints.
Right, because today we just assume computers
can optimize anything instantly. You type code,
and the compiler fixes your mess behind the scenes.
Back then, Ginsburg proved something that was
actually quite counterintuitive. He was looking
at finite state machines. Those are basically
systems that move step by step from state A to
state B. Okay. He showed that just because you
minimize the states, like reducing the number
of actual steps it takes to do a job, it doesn't
necessarily mean you have minimized the logic
elements. Wait, that sounds completely backwards.
If I have fewer steps, shouldn't I automatically
have less hardware? If I build a machine with
three gears instead of four, it should be a smaller
machine. In mechanics, yes. But in logic, not
always. Sometimes, describing a shortcut takes
a much more complex vocabulary than just taking
the long way around. Oh, I get it. Imagine trying
to explain a shortcut through a city that involves
six very specific left turns through back alleys
versus just saying, stay on the highway. The
highway is technically longer, so it has more
states, but the instruction is much simpler,
less logic. That makes perfect sense. So Ginsburg
proved mathematically that simple behavior doesn't
always equal simple wiring. Exactly. And he pointed
out a major historical limitation. Calculating
the absolute perfect direct minimization using
these don't care conditions for large systems
was computationally impractical in 1958. They
fully understood the theory, but they couldn't
always brute force the perfect solution. So they
basically had to make trade -offs. They knew
they were leaving some efficiency on the table
because they just didn't have the computers to
calculate how to build better computers. Which
is humbling, honestly. It shows that even the
concept of strategically ignoring things requires
a massive amount of calculation to get right.
Let's try to really ground this for the listener.
We've been talking a lot about logic and sequences
and algorithms. Can we give a concrete example
of where a don't care term actually lives inside
a gadget that I might recognize? The absolute
classic example and the one the source material
highlights heavily is BCD, binary coded decimal.
Binary coded decimal. Break that down for us.
So ideally, computers just like binary, pure
ones and zeros. But humans, we like. decimal.
We like our digits 0 through 9. BCD is a specific
way of encoding those human -friendly digits
using exactly four bits of binary code. Okay,
four bits. So that's 0, 0, 0, 0, 0, 0, 0, 1,
all the way up to weight. Four bits. That's 2
to the power of 4. That is 16 combinations. You
got it. Four bits can count from 0 up to 15.
That's 16 possible combinations. But for BCD,
which represents just a single decimal digit,
we only actually need 0 through 9. So what happens
to the binary codes for 10, 11, 12, 13, 14, and
15? They are the don't care terms. The sources
refer to them as pseudotetrates. Pseudotetrades.
That sounds like a dinosaur. Or maybe a very
specific weird geometric shape. Look out! A wild
pseudotetrade appeared. It does sound ancient.
But it simply means they are combinations of
four bits that in the specific language of BCD
are complete nonsense. They are totally unused.
So if I'm building a circuit for, say, an old
school alarm clock with those glowing red seven
segment displays. Perfect example. Everyone knows
those clocks. The displays have seven little
LED light bars arranged in a figure eight pattern.
To make the number one, you light up two bars
on the right. To make the number eight, you light
up all seven bars. Right. Now imagine you are
the engineer designing the circuit that controls
just the lower left vertical bar of that display,
just that one little line. Okay. You need to
tell that bar exactly when to turn on. It turns
on for the number zero, the number two, the number
six, the number eight. And it stays off for one,
three, four, five, seven, and nine. Correct.
But what about inputs 10 through 15? Well, the
clock is only ever supposed to show zero to nine.
If my bedside clock shows a 12 in a single digit
spot, something is very physically wrong with
my bedroom. So the chip should never, ever receive
a 12. Exactly. It's a can't happen term. So for
the inputs 10 through 15, which is binary, 1010
through 11111, the designer just says, I don't
care. If the chip did somehow receive a 12, it
literally does not matter if that lower left
bar lights up or not because the user is never
supposed to be in a situation to see it. And
by saying I don't care, the math for that lower
left bar gets easier. So much easier. The Boolean
function can be simplified significantly. Without
the don't cares, the logic equation might be
huge because you have to explicitly tell it to
be 0 off f for 10, 0 off for 11, 0 off for 12.
But if we treat those unused numbers as wildcards,
we can simplify the equation down to practically
nothing. You save actual transistors on the chip.
That is really cool. It's kind of like cleaning
your room by shoving everything into the closet,
but because nobody is ever allowed to open the
closet, the room is technically clean? In a way,
yes. You're optimizing the visible space by utilizing
the forbidden space. And there is actually another
super quirky example of this in vintage hardware.
Write -only registers. I saw this in the notes
and it confused me. A register is basically a
memory slot. So this is a slot that you can write
data to, but you literally cannot read data from.
That sounds useless. How do you know what's in
it? You don't. It is strictly one -way communication.
In older hardware... Reading data back out required
extra logic gates. It needed extra wires, extra
multiplexers. If the designer desperately needed
to save space or cost and the software didn't
strictly need to check what it just wrote, they
would just completely leave out the read circuitry.
So what happens if a programmer makes a mistake
and tries to read it anyway? Like, hey, computer,
what number's in that box? You get a don't care
result. You just get garbage data. So the computer
basically just shrugs and hands you whatever
random electricity happened to be nearby. Basically,
yeah. It might return all zeros, all ones, or
just pure static noise. It is a direct consequence
of don't care optimizations. The engineers actively
traded functionality, the ability to read for
hardware size. The output for read was explicitly
defined as a don't care. Okay, so we've got a
good handle on the hardware side, but I want
to transition to the software side, or at least
the simulation side of things, because the technical
docs mention this somewhat mysterious X value.
Ah, yes, the X factor. This is where things get
really interesting, and honestly, where the critical
distinction between simulation and physical reality
becomes so important. So in hardware description
languages, things like Verilog or VHDL, which
engineers use to code the hardware before it's
actually built in a factory, you have zero, you
have one, and you have... X. Right. In multivalued
logic for Verilog, it's literally just X. In
VHDL, it can be X for a forced unknown or sometimes
a W for a weak unknown. A weak unknown. That
sounds like me trying to remember where I parked
at the grocery store. But what does X actually
mean when you're running a simulation on a screen?
In a computer simulation of a chip, an X usually
represents a problem or a transition state. It
might mean you have a flip -flop, a little memory
storage unit that simply hasn't settled yet.
It's in flux. Okay. Or, and this is worse, it
could mean you have two different sources trying
to drive the exact same signal at the same time.
One wire is trying to say zero and the other
wire is trying to say one. So it's a conflict.
A massive conflict. The simulator basically throws
up its hands and says, X, I don't know who wins.
It's a very useful way for the simulation to
tell the engineer, hey, your logic is ambiguous
right here. And I think here is where it gets
really interesting for anyone trying to wrap
their head around this. You can't have an X in
real life, can you? If I took a physical voltmeter
to a physical wire inside my laptop motherboard,
the dial wouldn't say X. No, you cannot measure
an X. And this is a concept that actually trips
up a lot of students and even junior engineers.
An X value simply does not exist in synthesized
hardware in reality. Because physics is physics.
There is either a voltage on the wire or there
isn't. It's not magic. It's not Schrodinger's
cat. Exactly. The voltage is always either a
zero or a one, assuming it's within the logic
thresholds. The real problem is, because of that
conflict or that unsettled state we saw in the
simulation, we as the observers don't actually
know which one it is. It's just a ghost in the
machine. It truly is. The specific zero or one
is not determinable from the inputs. So in the
real physical world, the chip isn't outputting
a magical X state. It is outputting a 1 or a
0, but it might be flickering rapidly or it might
be stuck and we just can't predict it based on
the logic rules we wrote. That feels a little
unsettling, to be honest. We build these machines
to be these precise, perfectly logical beasts,
but deep down there are these pockets of we just
don't know. And that leads us directly to the
truly dangerous part of don't care terms. Because
sometimes what we think implies I don't care
actually turns into I really should have cared.
Right. Let's get into the danger zone. So we
established earlier that can't happen terms are
treated as don't cares. We optimize the chip,
assuming they will never, ever occur. But what
if a can't happen condition actually does happen?
This is the absolute nightmare scenario for reliability
engineers. And this is especially risky in circuits
that involve feedback. Things like state machines,
where the next step the machine takes depends
entirely on the previous step it just took. Okay,
so let's visualize this for the listener. Normally,
the machine bounces happily between state A,
state B, and state C. It is never supposed to
go to state D. So the engineers optimized state
D completely out of existence. They treated it
as a don't care. Correct. We didn't build a logic
path out of state D because we confidently assumed
we would never get into state D. But how does
the physical machine get to state D if it's supposed
to be mathematically impossible? The source material
lists a few major culprits. Power up is a really
big one. When you first turn a device on, for
just a split second, all the voltage levels across
the board are stabilizing. The flip -flops might
wake up in a completely random state. So you
turn on your smart toaster, and for one tiny
millisecond, it wakes up and thinks it is in
a state that doesn't logically exist. Exactly.
It wakes up in that forbidden state D. Another
cause could be that the power supply fluctuated.
And the other causes are even more exotic. Things
like electrical noise, extreme heat, or even
cosmic radiation. Cosmic radiation. I love that
actual space particles can fly through the atmosphere,
flip a single bit in my computer, and cause a
don't -care error. It sounds like pure science
fiction. It absolutely happens. In the industry,
it's called a soft error. A high -energy particle
hits a microscopic transistor and literally flips
a zero to a one. And here's the dire consequence.
If that physical flip pushes the machine into
a state it was never supposed to be in a forbidden
input, it might get entirely stuck. The source
text uses a very poetic phrase for this specific
lockup. It calls it a walled garden of states.
It is a beautiful image, isn't it? But a terrible,
terrible situation for a circuit. Imagine the
machine accidentally lands in one of those can't
-happen states. Because we treated it as a don't
care to save space, we didn't program a way out.
We didn't give it an instruction that says, if
you find yourself here, go back to start. We
just left the logic completely undefined. Yes.
So the machine might look at its internal logic
table, see that the next step from this impossible
state is just another impossible state. And it
just bounces around in this walled garden of
invalid states, unable to ever get back to normal
operation. That's a hardware lockup. That is
your computer freezing completely and needing
you to pull the plug for a hard reboot. It is
literally lost in a logic maze that has no physical
exit. That is terrifying for anything mission
critical. I mean, if my smart toaster freezes
in a walled garden, fine, I get bumped toast.
But what if it's a pacemaker or a communications
satellite or the anti -lock breaks on a car?
Exactly. That is where the stakes completely
change. So responsible designers have to ensure
that can't -happen states are truly impossible.
Or, if the safety is critical, they simply cannot
use those states for optimization. They have
to leave the efficiency on the table. So you
have to sacrifice the smaller chip size. You
have to leave the extra logic gate on the board
just in case. You have to. You design what the
documents call don't -care alarms. An alarm,
like a physical siren goes off. Essentially,
it's logic that detects if the machine enters
a forbidden state. It flags an emergency signal
to the main processor. Or, even more commonly,
you design the logic so that all those unused
states automatically transition back to a known,
safe, normal state. These are called transitory
states. Oh, like self -healing logic. Yes. It
assumes the impossible might actually happen
due to a cosmic ray or a power surge, and it
deliberately provides a path home. It explicitly
says, if you wake up in state D, go immediately
to state A. It costs more logic, it makes the
physical chip slightly bigger and more power
-hungry, but it prevents the walled garden scenario
entirely. So it really just comes down to, how
much do you trust the universe not to mess with
your voltages today? Pretty much. If you are
building a cheap plastic toy that sings when
you press a button, you optimize everything,
use all the don't cares, and you risk the lockup.
If you are building the navigation system for
an airplane, you build the path home. This has
been a really enlightening look at something
that sounds so dismissive on the surface, don't
care. But really, it is entirely about efficiency.
It is. It's about recognizing that not every
single input requires a specific, thoughtful
response. Strategically ignoring the impossible
allows the possible to run smoother, faster,
and cheaper. It's focusing purely on what matters.
Exactly. But as we learned with the walled gardens,
you have to be careful because if you ignore
the wrong thing or if the universe decides to
force an impossible input on you anyway, you
end up scrapped. It is the absolute classic engineering
tension. Optimization versus robustness. You
can strip everything down to the bare metal to
make it blazing fast, but you lose that critical
buffer against the unexpected. Before we go,
I want to leave you, the listener, with a thought
about that X value we discussed. You mentioned
that in reality, X doesn't exist. It is always
a zero or a one. We just don't know which one
it is. Right. The physics demands value. The
universe doesn't do undefined. I feel like there's
a real life philosophy buried in there. In the
simulation of our lives, our plans, our daily
worries, we have all these unknowns. These X
values where we feel like we are in conflict
or totally unsettled. We don't know the outcome.
But in the physical reality of the moment, everything
already has a value. We are where we are. The
outcome is already unfolding. We just might not
have the inputs to determine what it is yet.
That is... Surprisingly deep for a technical
discussion about logic gates. But it's true.
The value is always there, even if the simulator
in our heads can't quite see it yet. Just don't
let a cosmic ray knock you into a walled garden
today. Thanks for joining us for this deep dive.
Stay curious. We will see you downtime.