# Assignment 1: Combinational Circuits

• Out on: February 10, 2017
• Due by: February 16, 2017 before 10:00 pm
• Collaboration: None
• Grading: Packaging 10%, Style 10%, Functionality 80%

## Overview

The first assignment is mostly about simple combinational circuits of the kind we discussed in lecture: gates, decoders, adders, etc. It’s possible for you to submit the assignment without drawing a single circuit, simply by writing out the correct formulas, discussing/showing how you derived them, and describing what their characteristics would be as circuits. If you want to include actual circuit diagrams, please either “draw” them as “ASCII art” or include a PDF file with clearly labeled diagrams; however, only the diagrams can be in PDF, the rest of the assignment must be in plain text!

## Problem 1: Universal Gates (30%)

In lecture we mentioned that NAND (and NOR) gates are universal in the sense that any combinational circuit can be built exclusively out of NAND (or NOR) gates without any other gates.

For this problem, first show how you can implement the standard AND, OR, and NOT gates using only NAND gates. Then show how to do the same using only NOR gates. Optionally draw those circuits.

Since the set of AND, OR, NOT is intuitively universal as well, once you’ve done this problem you should be very much convinced that NAND and NOR are indeed universal.

Note: Just to make sure, you’ll need to hand in six circuits for this problem; three will contain only NAND gates, and the circuits will implement AND, OR, and NOT; three will contain only NOR gates, and those circuits will implement AND, OR, and NOT. I hope this is clear?

## Problem 2: XOR and XNOR Gates (20%)

Below you’ll find the truth tables for XOR and XNOR gates. Show how you can implement XOR and XNOR gates in terms of just AND, OR, and NOT gates.

a  b | a XOR b     a  b | a XNOR b
------+---------   ------+----------
0  0 |    0        0  0 |    1
0  1 |    1        0  1 |    0
1  0 |    1        1  0 |    0
1  1 |    0        1  1 |    1

You should use the “design process” we talked about in lecture: From the truth tables, derive a formula for the circuit in either DNF or CNF (don’t mix normal forms!), and optionally draw the circuit. Do not simplify the circuits, leave them in CNF or DNF!

## Problem 3: Seven-Segment Display (50%)

Seven-segment displays can be found in a huge number of electronics applications, including on some fancy PC motherboards (where they usually indicate errors). Everybody knows what they look like, but just in case:

Each of the seven display elements can be switched on and off independently through one of seven pins. The pins are labeled A to G according to the following scheme (forget the decimal point, we won’t use it):

Here is how the hexadecimal digits from 0 to F are usually represented on a seven-segment display:

_       _   _       _   _   _   _   _   _       _       _   _
| |   |  _|  _| |_| |_  |_    | |_| |_| |_| |_  |    _| |_  |_
|_|   | |_   _|   |  _| |_|   | |_|  _| | | |_| |_  |_| |_  |

For this last problem, you have to design a circuit that takes a binary digit 0-7 encoded on three lines and displays it correctly on a seven-segment display by outputting 0/1 on the correct pins A-G of the seven-segment display. Your circuit takes the inputs X, Y, and Z on which the digits 0-7 are supplied as follows:

Digit | X Y Z
-------+-------
0   | 0 0 0
1   | 0 0 1
2   | 0 1 0
3   | 0 1 1
4   | 1 0 0
5   | 1 0 1
6   | 1 1 0
7   | 1 1 1

The outputs of your circuit are A-G connected to the seven-segment display as shown above. Good luck!

You should once again use the “design process” we talked about in lecture: From the truth tables, derive formulas for the circuit(s) in DNF or CNF (it’s okay to mix normal forms but be sure to explain why you picked one over the other), and optionally draw the circuit. Do not simplify the circuits, leave them in CNF or DNF!

Note: If you’re really into circuits, design the circuit for 4-bit numbers as well. But this is in addition to the one for 3-bit numbers!

## Hints

• Be careful when including a PDF file, it’s easy to make the archive too large by accident, causing the submission system to time out.
• You must include derivations even for things that seem “obvious” to you. If we cannot tell whether you actually figured something out or simply copy/pasted from another website, we’ll assume the latter.

## Deliverables

Please follow the submission instructions as detailed on Piazza. Make sure that your tarball contains no derived files whatsoever (i.e. no executable files), but allows building all required derived files. Finally, make sure to include your name and email address in every file you turn in (well, in every file for which it makes sense to do so anyway)!

## Grading

For reference, here is a short explanation of the grading criteria; some of the criteria don’t apply to all problems, and not all of the criteria are used on all assignments.

Packaging refers to the proper organization of the stuff you hand in, following both the guidelines for Deliverables above as well as the general submission instructions for assignments on Piazza.

Style refers to programming style, including things like consistent indentation, appropriate identifier names, useful comments, suitable documentation, etc. or to the clarity and readability of your solution for a written problem. Simple, clean, readable code is what you should be aiming for. Make sure you follow the style guide posted on Piazza!

Functionality refers to your programs being able to do what they should according to the specification given above; if the specification is ambiguous, ask for clarification! (It also refers to you simply doing the required work correctly, which may not be programming alone.)