Models of Computation Boolean logic Logic circuits January 08, - - PowerPoint PPT Presentation

CPSC 121 Models of Computation

Boolean logic Logic circuits

January 08, 2020 Patrice Belleville / Geoffrey Tien 1

Announcements

• Pre-lecture quiz #2 due Tuesday Jan.14, 19:00

– Read Epp 4e/5e: Chapter 2.2

• HW1 out soon, due Tuesday, Jan.21, 19:00
• Pre-lecture quiz #3 due Sunday, Jan.19, 19:00

– Read Epp 4e/5e: Chapter 2.5 – http://www.students.cs.ubc.ca/~cs-121/current/handouts /signed-binary- decimal-conversions.html – http://en.wikipedia.org/wiki/Binary_number

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Logical operators

• AND

∧ (between two variables or expressions)

• OR

∨ (between two variables or expressions)

• NOT

∼ (in front of a variable or expression)

• Examples:

– 𝑍 = 𝐵 ∧ 𝐶 is read as "𝑍 equals 𝐵 AND 𝐶" – 𝑔 = 𝑦 ∨ 𝑧 is read as "𝑔 equals 𝑦 OR 𝑧" – 𝑎 = ~𝐵 is read as "𝑎 equals NOT 𝐵"

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Truth tables

• Shows the relationship between the values of the input

variables, and the value of the result of the operation

– compute the result, based on all possible combinations of the inputs

• Example, inputs 𝑌, 𝑍, result 𝑎:

– let 𝑌 be "I'm nervous" – let 𝑍 be "My socks are too loose" – let 𝑎 be "Having a bad day at school", 𝑎 = 𝑌 ∧ 𝑍

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AND 𝑌 𝑍 𝑎 F F F F T F T F F T T T OR 𝐵 𝐶 𝐷 F F F F T T T F T T T T NOT 𝐺 𝐻 F T T F 𝑎 = 𝑌 ∧ 𝑍 𝐷 = 𝐵 ∨ 𝐶 𝐻 = ~𝐺

Larger truth tables

• As we introduce more variables, truth tables become larger to

accommodate all combinations of input values

– arrange the input combinations in a consistent, predictable order

• Quick way:

– entries in rightmost input column alternate as "F T F T F T F T..." – entries in second column from right alternate as "F F T T F F T T ..." – entries in third column from right repeat as "F F F F T T T T ..." – fourth from right? – ... – What's the pattern for a system with 𝑙 variables?

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Determining truth values for compound statements

• Should I go to class? Let:

– 𝑏 = need it to graduate – 𝑐 = class is boring – 𝑑 = I'm stuck in traffic – 𝑒 = I slept in

• 𝑕 = 𝑏 ∨ ~𝑐 ∧ ~(𝑑 ∨ 𝑒)

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via intermediate values

𝑏 𝑐 𝑑 𝑒 ? ? ? ? 𝑕 F F F F F F F T F F T F F F T T F T F F F T F T F T T F F T T T T F F F T F F T T F T F T F T T T T F F T T F T T T T F T T T T

Digital logic circuits

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Data representation

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• Information in the real world can take many values, e.g.

– Temperature, weight, pressure (continuous, numeric) – Price, quantity (discrete, numeric) – Names, descriptions (discrete, alphabetic)

• Data representation in digital systems uses voltage

– Voltage is also continuous – Control the voltage to two discrete levels

• Two discrete voltage levels allow a binary alphabet

– {high, low}, {true, false}, {1, 0}

high low

Logic gates

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• Logic gates are electronic circuits that operate on one or more

input signals to produce an output signal(s).

– signal: a connection that can transmit one bit of information at a time

• Electric signals exist throughout a digital system as high or

low voltages (we will call 1 or 0).

• Gates are electronic circuits which produce the high and low

voltage equivalents of logic-1 (T) and logic-0 (F) output signals according to their specified truth tables

AND OR NOT, aka inverter

From circuits to propositions

• These gates correspond to the Boolean operations introduced

last class

– Can build circuits to evaluate logical propositions – e.g. "should I go to class?" 𝑕 = 𝑏 ∨ ~𝑐 ∧ ~(𝑑 ∨ 𝑒)

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...and back

AND OR NOT, aka inverter

• Another useful (non-Boolean) function: XOR (exclusive-OR)

XOR 𝑌 𝑍 𝑎 F F F F T T T F T T T F 𝑎 = 𝑌 ⊕ 𝑍

From circuits to propositions

• Propositions can be used to model circuits

– Each variable corresponds to an input – The value of the proposition (for a given combination of input values) is the output of the circuit

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• Example: the segments in a 7-segment display

– Each integer from 0 to 9 is represented by a specific combination of input values – We can write a proposition for each segment, which combinations of input values will activate the segment

From circuits to propositions

• Consider the following circuit:

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Exercise

• Write a proposition corresponding to this circuit
• What does this circuit compute?

– Complete a truth table for the proposition you obtained above – Study the truth table for patterns or characteristics, to express the circuit behaviour

The "bubble" means the value is inverted before entering the AND gate

From circuits to propositions

• What is the simplest logical expression that corresponds to the

following circuit?

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Exercise

Note the inversions on the outputs of the OR and AND gates These are called NOR and NAND gates and are logically equivalent to inverting the OR/AND functions. Also note the 3-input AND function.

A (sort of) real problem

• Consider the problem of designing a light that changes state

whenever any of the switches that control it is flipped. Ideally your solution would work with any number of switches!

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Light switches

?

Light switches

• How do we approach this?

– We first need to make sure we understand what we are designing – Then we use propositional logic to model the circuit's desired output

• It's helpful to start with very simple versions of the problem

– First try 1 switch – Then try 2 switches – Then try 3 switches – Then see if we can generalize to 𝑜 switches

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Light switches

• Making sure we understand what we are designing
• Which of these would be usable (most useful) as the output of
• ur circuit?

a) the switch is flipped b) the switch is on c) the light is on d) the light changed state

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Output

Light switches

• Making sure we understand what we are designing
• Which of these would be usable (most useful) as the input of
• ur circuit?

a) the switch is flipped b) the switch is on c) the light is on d) the light changed state

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Input

Light switches

• Make sure we understand the problem
• For one switch, is the light on or off when the switch is "on"?

a) Always on b) Always off c) Depends, but a correct solution should always do the same thing d) Depends, and a correct solution might do different things at different times

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Requirements

Light switches

• Which circuit(s) is/are correct solution(s)?

a) b) c) d) two of (a), (b), (c) e) all three of (a), (b), (c)

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One switch

Readings for this lesson

• Pre-class quiz #2 due Tuesday Jan.14, 19:00

– Read Epp 4e/5e: Chapter 2.2

• Homework #1 will be available soon!

– due Tuesday, Jan.21, 19:00

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