Pitch Black
36 Pages
English
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Pitch Black

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Learn all about the services we offer
36 Pages
English

Description

Pitch Black by Clive Duncan _____________________________________________________ Worksheet by Kristina Leitner & Andrew Milne-Skinner
  • only fuels othello‟s rage
  • othello
  • desdemona
  • desdemona‟s handkerchief
  • iago‟s wife
  • othello‟s mind
  • iago
  • interpret jago‟s answer
  • play

Subjects

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Published by
Reads 19
Language English

Exrait

From now on:
• Combinatorial Circuits:
– Study analyses and design of circuits made up of gates without
memory
– Design components needed for a simple “Computer”
• Sequential Circtuits:
– Study basic memory elements
– Study Ckts that can be designed using memory elements and gates
• Design a computer (and understand design of computers in general)
CS231 Boolean Algebra 1Combinatorial Circuits
• Analysis of Circuits:
– Given a circuit with gates (but no feedback : i.e. no memory), figure
out what it does
– Express it as “boolean expressions”
• Methods for simplifying circuits
– Boolean algebra
– Karnaugh Maps (K-maps)
CS231 Boolean Algebra 2Functions
• Computers take inputs and produce outputs, just like functions in math!
• Mathematical functions can be expressed in two ways:
An expression is A function table is
finite but not unique unique but infinite
x y f(x,y)
f(x,y) = 2x + y
= x + x + y
00 0
= 2(x + y/2)
…… …
= ...
22 6
…… …
23 41 87
…… …
• We can represent logical functions in two analogous ways too:
– A finite, but non-unique Boolean expression.
– A truth table, which will turn out to be unique and finite.
CS231 Boolean Algebra 3Basic Boolean operations
• There are three basic operations for logical values.
NOT
AND (product) OR (sum) of (complement)
Operation:
of two inputs two inputs on one input
Expression:
xy, or x •yx + y x’
Truth table:
x y xy x y x+y x x’
00 0 01
00 0
01 0 01 1 10
10 1
10 0
11 1 11 1
CS231 Boolean Algebra 4Boolean expressions
• We can use these basic operations to form more complex expressions:
f(x,y,z) = (x + y’)z + x’
• Some terminology and notation:
– f is the name of the function.
– (x,y,z) are the input variables, each representing 1 or 0. Listing the
inputs is optional, but sometimes helpful.
– A literal is any occurrence of an input variable or its complement.
The function above has four literals: x, y’, z, and x’.
• Precedences are important, but not too difficult.
– NOT has the highest precedence, followed by AND, and then OR.
– Fully parenthesized, the function above would be kind of messy:
f(x,y,z) = (((x +(y’))z) + x’)
CS231 Boolean Algebra 5Truth tables
• A truth table shows all possible inputs and outputs of a function.
• Remember that each input variable represents either 1 or 0.
– Because there are only a finite number of values (1 and 0), truth
tables themselves are finite.
n
– A function with n variables has 2 possible combinations of inputs.
• Inputs are listed in binary order—in this example, from 000 to 111.
f(x,y,z) = (x + y’)z + x’
x y z f(x,y,z)
00 0 1
00 1 1
f(0,0,0) = (0 + 1)0 + 1 = 1
01 0 1
f(0,0,1) = (0 + 1)1 + 1 = 1
01 1 1
f(0,1,0) = (0 + 0)0 + 1 = 1
10 0 0
f(0,1,1) = (0 + 0)1 + 1 = 1
f(1,0,0) = (1 + 1)0 + 0 = 0 101 1
f(1,0,1) = (1 + 1)1 + 0 = 1
11 0 0
f(1,1,0) = (1 + 0)0 + 0 = 0
11 1 1
f(1,1,1) = (1 + 0)1 + 0 = 1
CS231 Boolean Algebra 6Primitive logic gates
• Each of our basic operations can be implemented in hardware using a
primitive logic gate.
– Symbols for each of the logic gates are shown below.
– These gates output the product, sum or complement of their inputs.
NOT
AND (product) OR (sum) of (complement)
Operation:
of two inputs two inputs on one input
Expression:
xy, or x
•yx + y x’
Logic gate:
CS231 Boolean Algebra 7Expressions and circuits
• Any Boolean expression can be converted into a circuit by combining
basic gates in a relatively straightforward way.
• The diagram below shows the inputs and outputs of each gate.
• The precedences are explicit in a circuit. Clearly, we have to make sure
that the hardware does operations in the right order!
(x + y’)z + x’
CS231 Boolean Algebra 8Circuit analysis
• Circuit analysis involves figuring out what some circuit does.
– Every circuit computes some function, which can be described with
Boolean expressions or truth tables.
– So, the goal is to find an expression or truth table for the circuit.
• The first thing to do is figure out what the inputs and outputs of the
overall circuit are.
– This step is often overlooked!
– The example circuit here has three inputs x, y, z and one output f.
CS231 Boolean Algebra 9Write algebraic expressions...
• Next, write expressions for the outputs of each individual gate, based
on that gate’s inputs.
– Start from the inputs and work towards the outputs.
– It might help to do some algebraic simplification along the way.
• Here is the example again.
– We did a little simplification for the top AND gate.
– You can see the circuit computes f(x,y,z) = xz + y’z + x’yz’
CS231 Boolean Algebra 10