[Part 1] Unit 1.2 - Boolean Functions

Summary

This script delves into constructing Boolean functions from primitive operations, crucial for computer design. It explains transitioning from a truth table to a Boolean expression, using disjunctive normal form. The script highlights that any Boolean function can be represented using AND, OR, and NOT gates, but more remarkably, with just NAND gates. This insight underscores the power of basic logic gates in building complex computer systems.

Takeaways

• 😀 Boolean functions can be constructed from primitive operations using Boolean algebra, which includes AND, OR, and NOT gates.
• 📊 Representing Boolean functions with truth tables and expressions is fundamental, and one can transition from expressions to truth tables by evaluating inputs.
• 🔄 The process of deriving a Boolean expression from a truth table involves creating a disjunctive normal form formula, focusing on rows with a value of 1.
• 🛠️ In computer design, the challenge is to construct complex functions from primitive gates, which is analogous to deriving Boolean expressions from truth tables.
• 🌐 The disjunctive normal form is created by combining clauses that yield a value of 1 for specific input rows, using the OR operation.
• 🔧 Boolean expressions can be manipulated and optimized for efficiency, though finding the shortest or most efficient formula is an NP-hard problem.
• 🎯 Any Boolean function can be represented using only AND, OR, and NOT operations, showcasing the power of these basic logical constructs.
• 🚀 A remarkable theorem states that with just AND and NOT gates, one can construct any Boolean function, as OR can be synthesized from these.
• ❌ The NAND gate alone is sufficient to compute any Boolean function, as it can emulate both NOT and AND operations.
• 💡 The transition from abstract Boolean logic to actual computer gates is seamless, emphasizing the practical application of these logical principles in computer construction.

Q & A

What is the main focus of the last unit discussed in the transcript?

The main focus of the last unit was on Boolean functions, Boolean values, Boolean algebra, and Boolean formulas.

How can one represent a Boolean function?

A Boolean function can be represented through a Boolean expression or a truth table.

What is the process of constructing a Boolean function from a truth table?

The process involves creating a disjunctive normal form formula by focusing on rows with a value of 1, constructing expressions that yield 1 for those specific rows, and then combining these expressions using the OR operation.

Why is it important to convert a truth table into a Boolean expression?

Converting a truth table into a Boolean expression is important in computer design as it allows one to compose complex operations from primitive gates and operations.

What is a disjunctive normal form formula?

A disjunctive normal form formula is a standard way of constructing a Boolean function from a truth table by using OR operations to combine expressions that yield 1 for specific rows.

How can one manipulate a Boolean expression to make it more efficient?

One can manipulate a Boolean expression by combining clauses with common variables and fixed values to create shorter or more efficient expressions.

Why is finding the shortest Boolean expression for a given function considered an NP-hard problem?

Finding the shortest Boolean expression is NP-hard because there is no efficient algorithm to determine the shortest expression equivalent to a given one, and it is computationally complex.

What is the significance of the theorem stating that any Boolean function can be represented using only AND, OR, and NOT operations?

This theorem is significant because it demonstrates the universality of these basic operations in Boolean algebra, allowing for the construction of any logical function, which is foundational in computer design.

How can one prove that only AND and NOT gates are sufficient to compute any Boolean function?

One can prove this by showing that with AND and NOT gates, one can compute an OR operation, and since OR operations can compute any Boolean function, AND and NOT gates are sufficient.

What is the NAND function, and why is it significant in computer design?

The NAND function is a Boolean operation that outputs 0 only when both inputs are 1, and outputs 1 for all other cases. It is significant because it can be used to compute any Boolean function, allowing for the construction of complex logic with only NAND gates.

How does the shift from abstract logical operations to actual computer gates change the perspective in computer design?

This shift changes the perspective from thinking about operations in abstract terms to considering them as physical components within a computer that perform specific functionalities.

Outlines

00:00

🔧 Constructing Boolean Functions from Primitive Operations

This paragraph introduces the concept of constructing Boolean functions from more basic operations. It explains the importance of being able to derive a Boolean formula from a truth table, which is essential in computer design. The process involves creating a disjunctive normal form (DNF) formula by examining each row of the truth table that corresponds to an output of 1. For each such row, a Boolean function is constructed that outputs 1 only for that specific row. These individual functions are then combined using the OR operation to form a single Boolean function that matches the original truth table. The paragraph also touches on the possibility of simplifying the DNF formula by combining clauses and mentions that finding the shortest or most efficient formula is an NP-hard problem.

05:02

🔩 The Universality of AND, OR, and NOT Gates

This paragraph delves into the universality of Boolean operations, highlighting that any Boolean function can be represented using only AND, OR, and NOT operations. It contrasts this with integer functions, which cannot always be represented using basic arithmetic operations. The paragraph then presents a theorem stating that OR gates are not necessary, as any Boolean function can be constructed using just AND and NOT gates, demonstrated through De Morgan's laws. It further explores the possibility of reducing the number of required gates, concluding that NAND gates alone are sufficient to compute any Boolean function. The NAND operation is defined, and it is shown how NOT and AND operations can be performed using only NAND gates, leading to the conclusion that a computer's logic can be built from basic NAND operations.

Keywords

💡Boolean functions

Boolean functions are mathematical functions that take Boolean inputs (true or false) and produce a Boolean output. They are fundamental in digital electronics and computer science. In the video, Boolean functions are discussed in the context of constructing them from more primitive operations, which is crucial for designing computer hardware. The script mentions constructing Boolean functions from truth tables and primitive gates, highlighting their importance in computer design.

💡Boolean values

Boolean values represent the two binary states of a logical operation, which are typically denoted as 'true' and 'false'. These values are the building blocks of Boolean algebra and are essential for understanding Boolean functions. The script refers to Boolean values when discussing how to construct Boolean functions and when explaining the process of evaluating expressions to fill in truth tables.

💡Boolean algebra

Boolean algebra is a branch of algebra dealing with binary values, typically true and false, and operations such as AND, OR, and NOT. It is the basis for digital logic and is used to simplify and analyze Boolean functions. In the video, Boolean algebra is mentioned as a foundation for understanding how to construct Boolean functions from primitive operations and for designing logic gates in computers.

💡Boolean expression

A Boolean expression is a combination of variables, literals, and operators that evaluates to a Boolean value. It is a way to represent a Boolean function in a symbolic form. The video script discusses how to derive a Boolean expression from a truth table, which is a key step in the process of designing logic circuits in computers.

💡Truth table

A truth table is a mathematical table used in logic to compute the functional values of logical expressions on each of their functional arguments' possible values. It is a way to visually represent the outputs of a Boolean function for all possible combinations of inputs. The script describes the process of constructing a Boolean function from a truth table, emphasizing the importance of truth tables in computer logic design.

💡Disjunctive normal form

Disjunctive normal form (DNF) is a standard way of representing a Boolean function as a disjunction (OR) of conjunctions (ANDs) of literals. The video script explains how to construct a DNF formula for a Boolean function by focusing on rows with a value of 1 in the truth table and combining them with OR operations.

💡Primitive operations

Primitive operations refer to the basic logical operations such as AND, OR, and NOT that are used to construct more complex Boolean functions. The video emphasizes the importance of understanding how to construct Boolean functions from these primitive operations, as it is essential for designing computer logic gates.

💡De Morgan's laws

De Morgan's laws are a pair of rules in Boolean algebra that state that the negation of a disjunction is the conjunction of the negations, and the negation of a conjunction is the disjunction of the negations. In the video, De Morgan's laws are mentioned as a way to prove that OR gates can be constructed using only AND and NOT gates, which is a fundamental concept in reducing the complexity of logic circuits.

💡NAND gate

A NAND gate is a logic gate that outputs 1 only when both inputs are 1, otherwise, it outputs 0. The video script introduces the NAND function as a universal gate, meaning that any Boolean function can be constructed using only NAND gates. This concept is crucial for understanding the minimal components needed to build complex computer systems.

💡Computer logic design

Computer logic design involves creating the logical structure of a digital system, such as a computer, using Boolean functions and logic gates. The video script discusses the process of constructing Boolean functions from primitive operations and truth tables, which is a fundamental aspect of computer logic design. Understanding these concepts is essential for anyone involved in designing digital systems.

Highlights

Introduction to constructing Boolean functions from primitive operations.

Explanation of representing Boolean functions using Boolean expressions and truth tables.

The process of deriving a Boolean formula from a truth table description.

Importance of constructing Boolean functions in computer design.

Abstract treatment to understand the principles of constructing Boolean functions.

Constructing a disjunctive normal form formula from a truth table.

Focusing on rows with a value of 1 in the truth table to construct Boolean functions.

Creating an expression that gets a value of 1 only for specific rows.

Combining Boolean functions using the OR operation to create a single expression.

Manipulating the expression to find more efficient formats.

The challenge of finding the shortest or most efficient Boolean formula.

Theorem that any Boolean function can be represented using AND, OR, and NOT operations.

Proof that AND and NOT gates alone can construct any Boolean function.

Introduction of the NAND function and its significance in Boolean logic.

Theorem that NAND gates alone can compute every Boolean function.

Method to perform NOT and AND operations using only NAND gates.

Transition from abstract logical operations to practical computer gates.

The fundamental approach of building computers using basic NAND operations.

Transcripts

So, in the last unit we talked about what are Boolean functions, Boolean values,

Boolean algebra, Boolean formulas.

What we want to do now is actually talk about how we can construct

Boolean functions from more primitive operations.

So, we already saw two ways to represent the Boolean function,

a Boolean expression, and a truth table.

We also know how, already know how to go from the expression to the truth table.

You take the expression, evaluate it for each possible, possible value of the input

bits, and then you get the, and then you can fill the truth table.

What we want to do now is exactly the opposite.

You start with a description of a function,

let's say given as a truth table, and

our challenge is to come up with a formula that computes the same boolean function.

Why do we need to do that?

Well, that's exactly what we have to do when we come to design a computer.

We know what we want to do, we know what we want a certain unit to do, but

then we actually have to actually compose it from primitive gates,

from primitive operations.

So let's see how we can do it.

So again, we continue with our abstract treatment.

We're trying to see what is a basic logical way

we can actually construct such a Boolean function from primitive operations.

Later, once we actually talk about practically constructing them,

we will be more practical oriented.

We'll go step-by-step and so on.

At this point, we just want to make sure, what are the principles?

So let us start with a specification of a Boolean function given as a truth table.

How can we construct it?

Well, here's what I'm going to describe to you as a standard way of what's called

constructing a disjunctive normal form formula for it, and it goes like this.

We actually go row by row in the truth table.

We focus only on the rows that have a value of 1.

For example, the first row here has a value of 1.

Now, what we can do is,

we write an expression that basically gets a value of 1 only at this row.

For example, so, in particular, since here in this row, the values of x, y,

and z are 0, 0, and 0, if we look at the expression not x and

not y and not z, that is going to be a Boolean function,

the green Boolean function, that only gets a value 1 on this row.

So now we have one Boolean function.

We do the same thing, we construct another Boolean function,

another clause for each row that has the value of 1.

So for example, there's another second row with a value of 1.

This time, in here, this row, y equals to 1 while x and z equal to 0.

So the clause we write here is not x and y and not z.

Again, this is something that completely gets a value of 1 only on this row and

gets a value of 0 everywhere else.

We do that for every possible row that has a value of 1, Now the purple row.

Now we have a bunch of different functions that each one of them gets a value

of 1 only in its row and gets a value of 0 in all other rows.

But we desire a single function, a single expression,

that gets exactly the value 1 exactly on all of these rows and 0 on the other rows.

How do we do that?

Well, that's very simple.

We just or them together.

And now we get a single expression, a single Boolean function that gets value

1 exactly on the rows for which we built clauses for, and gets 0 everywhere else.

And now we've basically constructed our

function as a Boolean expression only using ands, nots, and ors.

Now, of course, once we have this expression,

we can start manipulating it in varied ways.

This is one way to write the function as an expression, but if you actually look at

it, you can see that we can start changing, start changing its format.

For example, if you look at the first two clauses, you can see that one of them

is not x and not y and not z, while the other is not x and y and not z.

Notice that the, what we have both possibilities for y and

exactly the same fixed value for x.

So instead of these two, two clauses, we can combine them into one clause

which does not ask about y, and only asks about not x and not z.

So we get an equivalent expression that's slightly shorter.

There are more manipulations that we can do.

Let's not go into them, but

we can actually write the same expression in many different ways.

Some of them will be shorter than others, some, some of them might be, might be

more efficient in terms of when we actually go implement them in a computer.

But the point is, logically, they're all completely equivalent.

Now, you might wonder, how do you actually find the shortest or

most efficient formula that's equivalent to the one we've just derived?

Well, that's not an easy problem in general.

It's not easy for humans, nor is there any algorithm that can do that efficiently.

In fact, this is an NP-hard problem

to actually find the shortest expression that's equivalent to a given one, or

even to verify if the expression that you're given is just a constant 0 or 1.

What's more interesting, what I really want to focus at this point,

is that we've really prove the really remarkable mathematical theorem

that any Boolean function, it doesn't really matter on how many variables and

what the boolean function is,

can be represented in an expression using only the operations AND, OR, and NOT.

Now, to notice how remarkable that is,

just think about, just think about functions on integers.

Integer functions that you've learned in elementary school.

Not every element, not every function and

integer numbers can be represented using let's say, addition or multiplication.

In fact, most functions cannot be represented just by arithmetic operations.

Yet, because of the finite world that we're living in, the Boolean algebra,

ever Boolean function can just be represented with ANDs, ORs, and NOTs.

And that is what exactly gives us the basic power to actually construct

computers only with these possible gates,

only with these possible operations, AND, OR, NOT.

But do we really need all of them?

Well, here's a better theorem, if you wish.

We don't really need OR gates.

Just with ANDs and NOTs, we can construct any Boolean function.

How can we prove that?

Well, we already know that if we have ORs, we can do everything.

We've just seen that.

And now the only thing that we need to show is that we can actually compute an OR

with AND and NOT gates.

But we know how to do that already, we remember, we recall De Morgan formula

that exactly give us a formula for OR that only uses NOT and AND gates.

So now we have a really more remarkable theorem, that we only need these two basic

operations to actually compute every, every Boolean function.

Can we go even less?

Can we give up, let's say, the AND gate?

Well, that doesn't make sense, because not only it takes one value to one value,

it doesn't even allow us to combine anything.

Can we give up the NOT gate?

Well, not really, because ANDs has this property

that if you only feed it zeroes it will always have zero as output.

And there are Boolean functions that when you feed 0 give you 1 as output, so AND

by itself wouldn't suffice.

But it turns out that there is yet another operation that by itself does

suffice to actually compute everything, so let me introduce the NAND function.

So the NAND function, here is a truth table.

It gives 0 only if both of its inputs are 1.

And every other possibility it gives 1.

Logically, x NAND y is defined to be the negation of x and y.

So what's so remarkable about this Boolean function?

Well, the nice thing is that we can prove the following theorem, that if you only

have NAND gates, you can already compute every Boolean function, you can already

represent every Boolean function as an expression using just these NAND gates.

How do we prove that?

Well, we know that if you can do NOT and if you can do AND, you can do everything.

So we just have to show how to do NOT with NAND gates and how to do AND

with NAND gates.

So here's how you do NOT.

If you just look what happened when you feed x to both inputs of the NAND gate,

you plug it into the truth table in the previous slide and

you can see that NOT x is really represented by x NAND x.

That's part one.

The second part we need to show, how do we do AND.

Well, x AND y turns out to be NOT of x NAND y.

But how do we have NOT, well, we've just seen that you can do NOT with NAND itself.

So now we've basically reduced the fact instead of using NOTs and

ANDs, we're just using NAND gates.

And we got an amazing theorem that just if you have a NAND gate,

you can compute every, everything.

And this is exactly what is going to be our approach when we actually go to build

a computer.

We will give you a basic, a primitive NAND gate operation, and

you will basically build the whole computer, all the complicated logic that

we'll actually ask you to build, using just from these basic NAND operations.

So at this point,

we've just finished our abstract point of view about Boolean logic.

And now we're doing a really interesting shift of perspective

from the abstract logical operations to the actual gates and,

the actual gates from which we build computers.

This switch really has no, there's no actual difference between the two,

two different things in terms of the kind of thing that we do.

But it's just in the way that we're thinking about them.

Until now, we ask you to think about everything as abstract Boolean logic.

From now, we're going to start asking you to think about everything

as actual little pieces in a computer that compute the functionality that we want.