You know, when you think about the computers
running our world, you probably picture like
absolute precision, right? Ones and zeros. Every
single detail, meticulously calculated, mapped
out, and accounted for. Right. Yeah. It definitely
feels comforting to assume that underneath the
glass screens of our devices, there's this rigid
binary foundation, a place where nothing is left
to chance, you know, a perfectly ordered digital
universe. Yeah, exactly. But then you actually
peek under the hood of digital logic design and
suddenly that perfection starts to look a lot
more like, well, selective blindness. Selective
blindness. I like that. Right. Because they were
intentionally ignoring information just to save
power and space. And the craziest part? that
intentional ignorance is exactly what makes our
most advanced machines vulnerable to chaotic
real -world phenomena. Oh, absolutely. Things
like cosmic rays. Literally cosmic rays. So welcome
to the Deep Dive. Today we're exploring a dense,
highly technical concept in digital logic called
the Don't Care Term. It's a fundamental tension
in computer science really. Engineers are constantly
battling between elegant, optimized theory and
the messy, unpredictable physical reality we
actually live in. And that's our mission today.
We are going to break down how digital systems
rely on these blind spots to run faster and cheaper.
Because whether you are a programmer or just,
you know, someone reading this on a digital screen,
your devices are actively making don't care decisions
right now to save power. They absolutely are.
It's happening constantly. But we're also going
to look at what happens when the physical universe
forces those systems to care and to guide you
through this kind of counterintuitive logic.
I've got our expert here. Thanks, yeah. To really
grasp how these systems break down, we first
have to look at how they're built to ignore things
in the first place. Right. So at the very bottom
of computing, you have Boolean functions. That's
the underlying math of ones and zeros that makes
logic gates work. And within that math, engineers
use two distinct concepts. OK, what's the first
one? First is the don't care term. This is simply
an input sequence where the designer decides
the output of the function. Just doesn't matter.
It's just a giant shrug from the system. Pretty
much. And its sibling concept is the can't happen
term. This is an input that the system assumes
will literally never occur in the real world.
Never ever. Right. And when you dig into the
engineering history, you see decades of logic
designers wrestling with what to even call these
informational voids. Because don't care is pretty
informal. Right, exactly. So you'll see historical
aliases like redundancies, irrelevancies, vacuous
combinations, and my personal favorite, forbidden
combinations. Forbidden combinations. That sounds
like something out of an alchemy textbook, not,
you know, computer engineering. It really does.
OK, let's unpack this. Why would a designer whose
entire job is to build a precise, predictable
machine purposely introduce irrelevancies. Is
this like building a house with a blueprint,
but since you know nobody will ever walk on the
ceiling, you don't care about putting carpet
up there. That's a great way to think about it.
So by ignoring the ceiling, you just save money
and time on the overall build. Yes, exactly.
Just like the ceiling, leaving it undefined is
an intentional design choice to optimize the
whole structure. In logic design, it's all about
minimization. Minimization, meaning making things
smaller. Making them smaller, simpler, faster.
Every single logic gate you add to a circuit
requires physical space on a silicon chip. It
requires manufacturing materials. And crucially,
it requires electricity to run, which generates
heat. Right. And heat is the enemy of electronics.
Exactly. So by treating certain complex conditions
as don't cares, the designer doesn't have to
wire a specific physical path for them. They
just leave it blank. Well, they do better than
leave it blank. They strategically assign it
at one or zero based on whatever makes the rest
of the circuit simpler. Wait, how do they decide
that? Engineers use graphical methods like Karnavatch
maps to do this, or even algebraic methods like
the Quine -McCluskey algorithm. Let's stick to
the visual one. The map. Sure. Imagine a grid
on a piece of paper filled with ones and zeros
representing the desired outputs of a circuit.
To minimize the physical gates, engineers look
for groups of ones they can circle together.
Like finding a word in a word search puzzle.
Right. And the larger the circle they can draw,
the simpler the resulting algebraic equation
becomes, which means fewer physical wires. Now,
if you have a can't happen state, a square on
that grid that you know will never be triggered,
you can drop a don't care wild card into that
square. Just pretend it's whatever you need it
to be. Exactly. If treating that wild card as
So one allows you to draw a massive circle connecting
other real ones. You do it. You've just mathematically
squeezed the waste out of the design. Wow. OK,
but hasn't this always been the goal? It has.
But translating that theory into physical reality
took time. Back in 1958, a mathematician named
Seymour Ginsburg actually proved something really
frustrating for engineers. What did he prove?
He proved that minimizing the states of a machine
using these don't care conditions didn't necessarily
guarantee you were minimizing the actual physical
logic elements on the board. Wait, why not? If
you simplify the math, shouldn't the physical
machine automatically be simpler? Not necessarily.
Ginsburg showed that sometimes combining states
mathematically forces you to use more complex
physical wiring to write the signals between
those newly combined states. Oh, so the abstract
math doesn't perfectly match the physical silicon.
Right. And the real problem was, in 1958, they
didn't have the computational power to test the
millions of combinations required to perfectly
align the abstract state minimization with the
physical logic minimization. computationally
impractical for large systems at the time. Exactly.
They were mapping out the philosophy of intentional
ignorance long before they had the supercomputers
needed to fully exploit it. So to see how this
works in the real world today, let's look at
how computers handle basic human numbers. We
operate on a base -10 system, right? Zero through
nine. But computers operate in binary ones and
zeros. So engineers use something called binary
coded decimal or BCD to bridge the gap. And that
translation creates a fascinating mathematical
leftover. How so? Well, to represent the human
number 9, a computer needs a 4 -bit binary sequence,
specifically 1001. But a 4 -bit sequence can
actually hold up to 16 different values, from
0 all the way up to 15. Which means, if you are
only ever counting from 0 to 9, Those binary
values for 10, 11, 12, 13, 14, and 15 will absolutely
never be used. Right. This system assumes they
are impossible. And there's a specific engineering
term for these leftover impossible inputs, right?
Pseudotetraids. Yes, pseudotetraids. The term
itself highlights their phantom nature. They
were valid binary combinations that simply have
no meaning in a decimal context. And designers
exploit these pseudotetraids aggressively to
save money. I think the classic example of this
is the seven segment display. Oh, that's a perfect
example. Yeah, think of the blocky red digital
numbers on a bedside alarm clock or like a microwave.
Each number is made of up to seven little LED
bars. The logic circuit behind that display has
to constantly compute when to turn on say the
lower left bar. And this is where the minimization
we talked about earlier physically manifests.
To definitively tell that lower left bar what
to do, the circuit normally has to check all
four binary wires. Like it has to ask, are you
a zero? Are you a one? Are you an eight? Exactly.
But the designer knows the inputs for 10 through
15 will never happen. So they just don't bother
building the complex web of logic gates required
to check for those higher numbers. They just
drop those don't care wild cards onto their Karnaugh
map. Yes. They make an arbitrary mathematical
choice for those pseudo tetraids, allowing them
to strip the circuit down to a bare bones algebraic
equation. And what does that equation look like?
In the case of that lower or left bar, they reduce
the logic to simply checking two specific conditions,
usually written algebraically as A0B plus A0C.
They bypass checking the other wires entirely.
Wow. So that physically shrinks the microchip.
Significantly. They are literally erasing physical
wires from the manufacturing process just by
letting the math bleed over into the impossible
numbers. Exactly. And this isn't just for alarm
clocks. This exact same shortcut is used in complex
encoding schemes, too. Right, the source mentioned
schemes with names like Hertz, Chen Ho, and densely
packed decimal. Yes, all of them rely on this.
It also shows up in OLDL hardware with things
called write -only registers. Oh yeah, the write
-only registers. That was a direct consequence
of engineers trading off functionality just to
reduce the number of logic gates. But that brings
up a really specific mechanical question that
I had when reading the source. OK, what is it?
If you have a write -only register, and you attempt
to read it, the source says a don't care is read.
How does a computer actually read a don't care?
Does it just spit out garbage data to the user
because it traded functionality for Fewer gates?
What's fascinating here is that the system literally
shrugs. It shrugs. Yeah, it provides whatever
arbitrary output was cheapest to wire because
you were never supposed to ask it that question.
You were asking for data from a pathway that
was intentionally left blank to save a microsenon
manufacturing. So the physical circuitry just
like gives you whatever residual electricity
happens to be there. Basically, yes. OK. But
before a single wire is ever soldered or a chip
is printed, engineers have to test this stuff
on computers. How do they simulate a void of
information before the hardware is even built?
Well, in the realm of multi -valued logic systems,
they introduce the x -value. The x -value. Which
stands for don't know. Exactly. When engineers
write the code that will eventually be synthesized
into physical hardware, they use hardware description
languages. In Verilog, for example, these unknown
values are just denoted by the letter X. And
in another language called VHDL, things get even
more specific. They use X to mean a forced unknown,
but they also use the letter W key to represent
a weak unknown. That distinction is really important
for simulating real world physics. How so? Well,
a forced unknown The X happens in a simulation
if two sources are driving a signal simultaneously,
like they're basically fighting each other to
a draw. OK. And the weak unknown? A weak unknown,
the W -YO, is an undefined signal that isn't
being actively pushed by a strong power source,
so it can easily be overridden if a valid zero
or one comes along. It's the software saying,
I am confused. This pathway is currently undefined.
This can also happen if a tiny component called
a flip -flop hasn't reached a stable output yet,
right? Right. A flip -flop holds state. It's
the fundamental building block of computer memory.
Think of it like a tiny seesaw. It holds a one
when it's tilted up and a zero when it's down.
Okay, I'm picturing it. But during a transition,
that seesaw can balance perfectly in the middle
for a fraction of a microsecond. In the pure
math of a simulation software, the program can
look at that perfectly balanced seesaw and just
ladle it X. It can just sit there unresolved.
Here's where it gets really interesting. So in
the simulation software, X is a literal placeholder
for confused. But in the real world, the physical
wire has to hold a voltage. It is forced to pick
aside a 0 or a 1. Even if it's completely blind
to what it should be, it can't just hover in
an X state in physical hardware. This raises
an important question about the danger of assumptions.
When a simulation allows an X, a nice, safe,
undefined placeholder, but reality demands a
definitive 1 or 0, a dangerous gap opens up between
theory and physical execution. because the synthesized
physical hardware's actual value is indeterminable
from the inputs. Exactly. Which brings us to
the chaos of the physical universe. Since the
physical hardware must pick a 1 or a 0, what
actually happens when the real world forces the
system into one of those can't happen combinations?
The forbidden inputs the engineers assumed could
never happen. Things get truly precarious when
we look at circuits with feedback. These are
known as state machines. State machines. Right.
A state machine depends heavily on its own previous
outputs and external inputs. It has a memory
of the state it is currently in. It's like a
combination lock that remembers the last number
you spun to. Great analogy. And the environment
is constantly threatening that memory. The real
world can accidentally generate these forbidden
inputs. Like what kind of threats it could be
a voltage glitch during the circuits power -up
phase or Random interference from a heat spike.
It could be electrical noise or incredibly. It
can be cosmic radiation Wait cosmic radiation.
So an engineer perfectly designs a state machine
mathematically proves an input will never occur
and Then literally a particle from outer space
hits the computer and throws it into a forbidden
state. I know it sounds like sci -fi, but yes.
It's like building an impenetrable bank vault,
and then a meteor teleports inside of it. That
is exactly what it's like, and that's why nominally
can't -happen states are actually deeply dangerous,
because cosmic rays are high -energy particles
constantly raining down on Earth. And if one
hits the chip perfectly... It can inject a microscopic
charge. It physically flips a single bit from
a zero to a one. Which means the machine is suddenly
experiencing an input that the designer guaranteed
was impossible. Right. Suddenly, your combination
lock thinks it is set to a number that isn't
even painted on the dial. And because the logic
was minimized to save space, there's no pathway
instructing the lock on what to do. The results
of this are catastrophic, right? Completely.
In computer engineering, this is called a hardware
lockup or a soft error. Your device essentially
freezes into a coma. Because it enters what the
source called the walled garden of states. The
walled garden. Which usually sounds pleasant,
but here it is a scenario where the machine is
stuck between can't happen states with absolutely
no combination of inputs that can exit it back
into normal operation. Because they used don't
care logic to minimize the circuitry, they didn't
build an exit. The cosmic ray pushes the machine
into the walled garden and it is entirely trapped.
Just bouncing around inside the forbidden logic
loop forever? Exactly. Entirely unresponsive.
So if a random burst of heat or cosmic radiation
can permanently trap our technology in a walled
garden, how do engineers prevent our devices
from instantly becoming expensive paperweights?
Well, they have to design safety nets. There
are three main mitigation strategies designers
must employ for state machines. What's option
one? Option one is the brute force approach.
They must take extra physical steps to ensure
the forbidden combinations are truly made. Can't
happen. But doesn't that mean heavily shielding
the system? Wouldn't that completely ruin the
whole point of minimizing the circuit to save
money and space? It absolutely does. It's a massive
trade off, usually reserved for things like aerospace
or medical devices. Where failure is just not
an option. Right. So for everyday electronics,
option two is more common, making them transitory.
How does a transitory state work? The designer
explicitly wires the walled garden so that it
has a trap door. If the machine enters a forbidden
state, the very next tick of the clock will automatically
lead it back to a normal operational state. OK,
so it auto -fleshes the system. Exactly. The
user might experience a tiny glitch, but it won't
permanently lock up. That makes sense. But then
there is option three, which is creating a don't
care alarm to indicate an emergency state for
air detection. Yes, the don't care alarm. And
I have to call out this phrase from the source,
a don't care alarm. Isn't that the ultimate oxymoron?
How do you mean? Well, if you are taking the
time to design a specific emergency protocol
and an alarm for it, you explicitly do care.
If we connect this to the bigger picture. That
is the beautiful irony of digital design. Engineers
start out using don't cares to save time, space,
and money. They want to strip the system down
to its absolute most efficient form. Right. But
because the physical world is inherently chaotic,
they eventually have to spend time and money
building alarms just to protect the system from
its own optimized ignorance. Wow. They optimize
themselves into a corner and then have to build
a fire alarm inside that corner. That's the reality
of bridging abstract math with physical hardware.
It really reframes how you look at the devices
around us. We started this deep dive looking
at an obscure trick that don't care term. We
saw how things like Karnaugh maps allow for brilliant,
elegant minimization. Like stripping down the
seventh segment displays. Exactly. But we also
saw that the physical universe doesn't respect
our mathematical boundaries. The physical realities
of X values heat and cosmic radiation force engineers
to build safety nets like don't care alarms to
prevent permanent hardware lockups. It's a constant
invisible negotiation happening millions of times
a second inside your phone. Which leaves you
with something to chew on. We build our most
advanced systems to efficiently ignore the impossible.
But since the physical universe guarantees that
the impossible like a random cosmic ray will
eventually happen, our most optimized digital
systems inherently are most fragile. It's a great
question. Think about that the next time your
screen freezes for no apparent reason. Thank
you for joining us on this deep dive and keep
asking the big questions.