Snick snack CPSC 121: Models of Computation 2012 Summer Term 2 Introduction & Motivation Steve Wolfman, based on notes by Patrice Belleville and others

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CPSC 121: Models of Computation2012 Summer Term 2

Introduction & Motivation

Steve Wolfman, based on notes by Patrice Belleville and others

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This work is licensed under a Creative Commons Attribution 3.0 Unported License.

Learning Goals: In-Class

By the end of the unit, you should be able to:– Give an example of how we can apply formal

reasoning and computers to a simple, real-world task.

– Give an example of how a computational solution to a simple task might go wrong.

– Describe the two “big stories” of CS121: reasoning about computation and building computers.

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Outline

• Introductions

• Introductions Exercise

• The CS121 Story

• Course Administration

• Next Lecture Notes

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Introductions

Steven Wolfman <[email protected]>

ICICS 239; office hours listed on the websiteI also have an open door policy:If my door is open, come in and talk!Also, I will usually be available after class.And, you can make appointments with me.Additionally, you can use TA office hours. (Specific times are TBA on the web soon!)

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Outline

• Introductions


• The CS121 Story



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More Introductions

Introduce yourselves… how?

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Introduce Yourselves in Groups of 4-ish

FIND ~3 people around you, preferably people you’ve never met. Form a group.

Have everyone in the group introduce themselves to everyone else in the group. (Count the number of intros this takes.)

Tell everyone why you’re here, your favorite career that you’ll never have, and one unusual thing about you.

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Problem: How Many Introductions?

Problem: How many introductions does it take for everyone in a group to meet everyone else in a group?

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Concept Q: Intros for 4

How many introductions does a group of 4 people take?

a.3

b.4

c.6

d.12

e.None of these

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Problem: How Many Introductions?

Problem: How many introductions does it take for everyone in a group to meet everyone else in a group?

To solve this problem, we need to model it more formally.10

How Many Introductions?

Model: One “introduction” is one person introducing themselves to another person. (Two people need two introductions to introduce themselves to each other.)

A group has “introduced themselves” when every group member has introduced themselves to every other member. (No self-introductions!)

Hi, I’m Grace H.

Hi Grace, I’m Alan T.Intro #1

Intro #211


Problem: How many introductions does it take for a group of n people to introduce themselves?

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For 2 people?

For 3 people?

For 4 people?

For 5 people?

…

For n people?

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Our examples.Should 0 and 1 be examples?


For 100 people?

For 8675309 people?

For 1526097757 people?

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Program for Introductions

int introductions(int n)

{

return n * (n - 1);

}

(define (introductions n)

(* n (- n 1)))

Do you believe it?15

(in Java)

(in Racket)

Program for Introductions: Testing

Java version with 100: 9900









Java version with 1526097757: -645820308

Um.. Do you believe it?Does this fit with your “model” of computation?

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Racket version with 100: 9900



Do you believe it?Will Racket always get the right answer?

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Alternate Introductions Program

;; Model in math, translate to Racket.


(* n (- n 1)))

;; Model as “I know what happens

;; a) in a group of 0 people, and

;; b) when someone new enters a group.”

;; Translate to Racket.


(if (= n 0)

0

(+ (introductions (- n 1))) ; the smaller group

(* 2 (- n 1))) ; the extra intros20

Are both correct?

Outline

• Introductions


• The CS121 Story



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Questions that CPSC121 Answers

How can we prove that our formula for the number of introductions is correct? (predicate logic proof for n*(n-1) version, induction for “recursive” version)

What went wrong in our Java implementation but right in Racket? (number representation)

How does the computer (e.g., Racket) decide if the characters of your program represent a name, a number, or something else? How does it figure out if you have mismatched " " or ( )? (DFAs)

How do we model and reason about computational systems at various levels of abstraction? (propositional and predicate logic, proof, sets, functions, DFAs, relations, ...)

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CPSC 121: The Big Stories

Theory

How do we model computational systems (programs/computers) in order to design and analyze them?

Grand goal: Reason about what is and isn’t possible to compute.

Hardware

How do we build devices that can compute out of real materials (“sand and rocks and water”)?

Grand goal: break

down a full computer

into simple “gates”.

Bonus end goal: Develop tools to communicate computational ideas clearly and precisely. 23

Our Working Computer

24The whole thing (mostly wires connecting boxes).

Our Working Computer(zoomed in on one “box”)

25Just the “ALU” (Arithmetic/Logic Unit). You’ll see a pared-down version in lab in a couple of weeks.

CPSC 121: The (By?)Products

Theory

Products:• Propositional logic• Predicate logic• Sets and functions• Proof techniques (especially

induction!)

• Finite Automata/Regular Expressions

• Universal machines• Uncomputability

Hardware

Products:• Gates• Combinational circuits• Binary data

representations• Sequential circuits• A full computer

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What is CPSC 121 Good For?

With CPSC121’s help, you will be able to:• model important problems so that they are easier

to discuss, reason about, solve, and test.• learn new modeling formalisms more easily.• communicate clearly and unambiguously with

other CS experts on complex topics.• characterize algorithms (CS problem solutions),

such as proving their correctness or comparing their performance.

• critically read proofs: justifying why each step is correct and judging what the proof means.

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Outline

• Introductions


• The CS121 Story



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Course Administration

Explore the CPSC 121 website:

http://www.ugrad.cs.ubc.ca/~cs121/

You are required to be familiar with the course website. Ignorance of information on the website may harm you.

At minimum: read everything in the “course information” area, skimming the interactive schedule and the detailed learning goals.

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Additional Administrative Notes

The first quiz is “any marks gives full marks”. So, if you get more than 0%, we’ll count it as 100%.

Labs, tutorials, and office hours all start this week!

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Outline

• Introductions


• The CS121 Story


• Summer Bonus: True, False, and Gates; Why Start Here?


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Logic for Reasoning about Truth: Where Should We Start?

I will suppose that ... some malicious demon of the utmost power and cunning has employed all his energies in order to deceive me. I shall think that the sky, the air, the earth, colours, shapes, sounds and all external things are merely the delusions of dreams which he has devised to ensnare my judgement. I shall consider myself as not having hands or eyes, or flesh, or blood or senses, but as falsely believing that I have all these things.

- René Descartes

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Logic as Model for Physical Computations

Input a Input b

a b

~aa

5V

0V

http://alumni.media.mit.edu/~paulo/courses/howmake/mlfabfinalproject.htm

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a b

“OR” operator and gate

a b

output

Physical System

“Truth Table” model

a b a b

T T T

T F T

F T T

F F F

We think of “flowing water” as true and “no water” as false, and the physical world becomes an effective representation for our ideas!

Propositional logic model: a b means “a OR b”

Circuit diagram model:the “OR” gate

Outline

• Introductions


• The CS121 Story



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Learning Goals: In-Class

By the end of the unit, you should be able to:– Give an example of how we can apply formal

reasoning and computers to a simple, real-world task.

– Give an example of how a computational solution to a simple task might go wrong.

– Describe the two “big stories” of CS121: reasoning about computation and building computers.

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Next Lecture Learning Goals: Pre-Class

By the start of class, you should be able to:– Translate back and forth between simple

natural language statements and propositional logic.

– Evaluate the truth of propositional logic statements using truth tables.

– Translate back and forth between propositional logic statements and circuits that assess the truth of those statements.

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Next Lecture Prerequisites

Read Sections 1.1 and 1.4/2.1 and 2.4

Read propositional logic supplement: http://www.ugrad.cs.ubc.ca/~cs121/current/Handouts/prop-circuit-xlate.html

You should have completed the open-book, untimed quiz on Vista that’s due by 9PM the day before class.

(You are responsible for ensuring that you have submitted the quiz by 9PM!)

38Readings: 3rd ed in black/4th ed in red

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Some Things to Try...

(on your own if you have time, not required)

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What Works is NOT Always Obvious

Let’s try out what we’ll call the “Radix” Algorithm. General form:

1. Put all items together in a list in order

2. Make an empty list for each possible value of each “digit” in the items (or letter in a word, etc.).

3. For each digit of the items in the main list, starting with the rightmost digit:

a) For each item in the main list, in order: Put it at the back of the list indicated by the current digit in that item

b) For each digit list, in order: Put the list at the back of the main list.

What does this do?How does it work?

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What Doesn’t Work isNot Always Obvious (1 of 2)

Class Main { public static void main(String[] args) { // Let's add up 4 quarters. System.out.println("4 quarters gives us:"); System.out.println(0.25 + 0.25 + 0.25 + 0.25);

// Let's do something a hundred times. int i = 100; do { // Make i one smaller. i--; } while (i > 0);

System.out.println("Done!"); System.out.println("i ended up with the value: " + i); System.out.println("It went down by: " + (100 - i)); }}

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What Doesn’t Work isNot Always Obvious (2 of 2)

Class Main { public static void main(String[] args) { // Let's add up 10 dimes. System.out.println("10 dimes gives us:"); System.out.println(0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1 + 0.1); // Let's try do something a hundred times.. // but accidentally go forever int i = 100; do { // Make i one LARGER. Oops! i++; } while (i > 0);

System.out.println("Done!"); System.out.println("i ended up with the value: " + i); System.out.println("It went down by: " + (100 - i)); }}

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Even Bigger Story:“Clear Thought”

Computer Science is the science of “clear thought”, but not like philosophy (or religion, or poetry, or...).

CS is the science of thoughts so clear in their expression and intent that we can realize them: execute them or test their truth in the world.

CPSC121 provides and applies the fundamentals you need to model the world with “clear thought”.

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Documents

Snick snack CPSC 121: Models of Computation 2012 Summer Term 2 Introduction & Motivation Steve Wolfman, based on notes by Patrice Belleville and others