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# Combinational Logic

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CS 130 // 2021-11-03

## Administrivia
- Noyce Scholarship
- 
 Quiz 4 on Monday (will be administered on Gradescope and be unlimited time)
    + Topics on elementary digital logic (basic gates, truth tables, Boolean algebra, etc.)
- 
 Released Assignment 4
    + Due next Wednesday

# Questions
## ...about anything?

# Digital Logic

## Computer Organization
- Computer processors perform their computations using **transistors**
    + 
 Transistors are essentially little **switches** that can be turned on and off

## Constructing Gates from Transistors
<iframe width="560" height="315" src="https://www.youtube.com/embed/sTu3LwpF6XI" frameborder="0" allow="accelerometer; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>

## Universal Gates
- The **NAND** gate and the **NOR** gate are "universal" and can be used to construct every other gate
- Using only **NAND** gates, construct circuits that are equivalent to NOT, AND, and OR gates

## Boolean Algebra
- We can represent logic gate computations as using **Boolean algebra**
- 
 $\overline{A}\cdot B+A\cdot C$ means the same thing as:
    + `((not A) and B) or (A and C)`

# Combinational Logic

## Combinational Logic
- **Combinational** logic circuits don't have memory
    + 
 Output depends on only the input
    + 
 Can be completely specified by a truth table
- 
 **Sequential** logic circuits do have memory
    + 
 Output depends on both the input and current state of memory

## Sum of Products Form
- All combinational circuits can be converted to a **sum of products** form that makes them especially easy to implement with gates

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- 
 Need to identify the 1s in the output
- 
 Construct a product term such as $\overline{A}\cdot\overline{B}\cdot C$ for each 1 in output
- 
 "Sum" them together

</div>
<div style="font-size: 60%">
<table>
  <tbody>
    <tr> <th>A</th> <th>B</th> <th>C</th> <th></th> <th>D</th> </tr>
    <tr> <th>0</td> <td>0</td> <td>0</td> <td></td> <td>0</td> </tr>
    <tr> <td>0</td> <td>0</td> <td>1</td> <td></td> <td>1</td> </tr>
    <tr> <td>0</td> <td>1</td> <td>0</td> <td></td> <td>1</td> </tr>
    <tr> <td>0</td> <td>1</td> <td>1</td> <td></td> <td>0</td> </tr>
    <tr> <td>1</td> <td>0</td> <td>0</td> <td></td> <td>1</td> </tr>
    <tr> <td>1</td> <td>0</td> <td>1</td> <td></td> <td>0</td> </tr>
    <tr> <td>1</td> <td>1</td> <td>0</td> <td></td> <td>0</td> </tr>
    <tr> <td>1</td> <td>1</td> <td>1</td> <td></td> <td>1</td> </tr>
  </tbody>
</table>
</div>
</div>

## Sum of Products Practice
- **EXERCISE**: Implement the following circuit
    + Inputs are X2, X1, X0. Outputs are Y1, Y0

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</div>
<div>

</div>
</div>

## Multiplexer
- Suppose we want to design a circuit that behaves like a **if statement**
- 
 Inputs are S, A, and B
- 
 What we want is the following:
    + If S is 1, then the output should be A
    + 
 If not, then the output should be B

## Multiplexor

<div>

- Its behavior is similar to:
    ```py
    def multiplexor(S, A, B):
        if S == 1:
            return A
        else:
            return B
    ```
- 
 Complete the truth table
- 
 Now let's implement the truth table as a circuit

</div>

</div>
</div>

## Decoder
- A **decoder** is another common circuit that has $n$ inputs and $2^n$ outputs
- 
 Only one output is a 1 at any given time (one for each possible combination of inputs)
- 
 If the input encodes the number 7, then the output bit for 7 is asserted and all others are zeros

## Decoder Truth Table

- Below is a 3-bit decoder truth table

![Decoder Truth Table](../../assets/images/COD/decoder.png)

# Assignment 3