reveal.js

## Gates, Truth Tables, and
# Boolean Algebra

---

CS 130 // 2021-11-01

## Administrivia
- Debrief on Exam 2
- 
 Readings for the next few weeks are on Blackboard (Appendix C of our textbook)

# Questions
## ...about anything?

# Course Themes

## Overarching Theme
- Learning how a high-level program is actually executed on your computer's processor

![Compiler-Assembler Diagram](/teaching/2021f/cs130/assets/images/COD/compiler-assembler.png)

## Trajectory of the Course
1. The C Programming Language
2. Assembly language programming
3. **Digital logic**
4. Processor architecture

## Basic CPU Operations
- Recall that most operations operate on one or more 32-bit registers
    + 
 `add $s0,` `$s1`, `$s2` takes two 32-bit numbers, adds them up, and produces a new 32-bit number
- 
 How can we build a machine that takes 0s and 1s as input and produces 0s and 1s as output?

# Digital Logic

NOTE: Discuss how hard electrical engineering is and why it is so useful to focus on the simplicity of digital logic

## Transistors
- CPU instructions are implemented with nanoscale switches called **transistors**

![Light bulb](https://makercise.com/wp-content/uploads/2015/11/vlcsnap-2015-11-17-20h50m57s882.png.jpg)

</div>
<div>

You can think of a transistor as a "switch" with states "on" and "off"

</div>
</div>

## Digital Logic
- Learning about electricity, capacitors, resistors, is beyond the scope of this course
- 
 For now, it is useful to abstract away the details of electrical engineering
- 
 We will think about circuits in terms of 0s and 1s going through the wires of a circuit

## Input/Output as Voltages
<div class="twocolumn">
<div>

- The 0s and 1s of our circuits will be "low" and "high" voltages across a wire, similar to a light bulb being "on" and "off"

</div>
<div>

![Bits as light bulbs](https://miro.medium.com/max/2400/1*Um-qKrB2QLxRU0C8CkCCeg.png)

</div>
</div>

## Logic Gates
- A **logic gate** is an elementary circuit that takes one or more bits as input and produces one or more bits as output
- 
 Many of these logic gates can be constructed with one or two transistors

## Name that Gate
![AND Gate](/teaching/2021f/cs130/assets/images/gates/and.png)

---

- 
 An **AND** gate

## Name that Gate
![OR Gate](/teaching/2021f/cs130/assets/images/gates/or.png)

---

- 
 An **OR** gate (inclusive OR)

## Name that Gate
![NOT Gate](/teaching/2021f/cs130/assets/images/gates/not.png)

---

- 
 A **NOT** gate

## Name that Gate
![NAND Gate](/teaching/2021f/cs130/assets/images/gates/nand.png)

---

- 
 A **NAND** gate

## Name that Gate
![NOR Gate](/teaching/2021f/cs130/assets/images/gates/nor.png)

---

- 
 A **NOR** gate

## Name that Gate
![XOR Gate](/teaching/2021f/cs130/assets/images/gates/xor.png)

---

- 
 An **XOR** gate (exclusive OR)

### What is the truth table for the following circuit?

---

![XOR Gate](/teaching/2021f/cs130/assets/images/gates/simple-circuit.png)

</div>
<div>

</div>
</div>

# Boolean Algebra

## Boolean Algebra
- We can represent logic gate computations as using **Boolean algebra**
    + 
 All variables `$A, B, C, \ldots$` are either 0 or 1
    + 
 `$A\cdot B$` means the **AND** of `$A$` and `$B$`
    + 
 `$A+B$` means the **OR** of `$A$` and `$B$`
    + 
 `$\overline{A}$` means the **NOT** of `$A$`

## Boolean Algebra Practice
- Let's convert the Boolean expression `$\overline{A}(B+C)$` into its equivalent logic gate representation

---

- 
 **EXERCISE**: Do the same with these:
    + `$(\overline{A}\cdot\overline{B})+(\overline{C}\cdot\overline{D})$`
    + `$(A+\overline{C})(\overline{D}+\overline{B})$`
    + `$A(B+\overline{A}+A)+B\overline{A}$`

## Universal Gates
- The **NAND** gate and the **NOR** gate are "universal" and can be used to construct every other gate

---

- 
 How can we construct a NOT using a NAND?
- 
 How can we construct an AND using only NANDs?
- 
 How can we construct an OR using only NANDs?
- 
 How can we construct an XOR using only NANDs?