sept 4, 2003© vadim bulitko - cmput 272, fall 2003, uofa1 cmput 272 formal systems & logic in...

Sept 4, 2003 © Vadim Bulitko - CMPUT 272, Fall 2003, UofA 1

CMPUT 272CMPUT 272Formal Systems & Logic in Formal Systems & Logic in

CSCS

CMPUT 272CMPUT 272Formal Systems & Logic in Formal Systems & Logic in

CSCS

I. E. LeonardUniversity of Alberta

http://www.cs.ualberta.ca/~isaac/cmput272/f03


TodayTodayTodayToday

Organizational details

Purpose of the course

Material coverage

Section 1.1 of the text


Internet ResourcesInternet ResourcesInternet ResourcesInternet Resources

Course webpage: http://ugweb.cs.ualberta.ca/~c272/

Class webpage:http://www.cs.ualberta.ca/~isaac/

cmput272/f03


LecturesLecturesLecturesLectures

Instructor: I. E. Leonard

Time: Tues Thurs 9:30 – 10:50

Place: CSC B 10

Text: S.Epp. “Discrete Math. &

App.”

Attendance: strongly encouraged,

limited slide coverage,

read the text before class


TextbookTextbookTextbookTextbook

Attending the lectures is not a substitute for reading the text

Lectures merely highlight some examples

Read the text and do as many exercises as humanly possible


SlidesSlidesSlidesSlides

To refresh a lecture (e.g., while preparing for an exam) consult the slides and the text!!!

The slides are on the web (posted after each lecture)

NO substitute for the textbook!!!


Office HoursOffice HoursOffice HoursOffice Hours

Instructor: I. E. Leonard

Time: See Schedule on

Webpage

Place: Athabasca 323

Attendance: encouraged whenever

you have questions


SeminarsSeminarsSeminarsSeminars

Instructors: I. E. Leonard

Time: Monday 3:00 – 3:50

Place: CAB 265

Purpose: - interactive environment- answers to your

questions/problems

- more practice in problem solving

Attendance: beneficial to you

Start: Monday, Sept 8th


LabsLabsLabsLabs

Instructors: TAs: See the Schedule

for times and places

Purpose: 1-on-1 interaction with

TA’s

Start: Week of Sept 8th


Marking SchemeMarking SchemeMarking SchemeMarking Scheme

6 assignments 24%

2 Midterms: 36%

1 Final Exam: 40%


In-Class ParticipationIn-Class ParticipationIn-Class ParticipationIn-Class Participation

As far as is possible the class will be an

interactive problem solving session


AssignmentsAssignmentsAssignmentsAssignments

Each of the 6 assignments will count : 4%

Each will consist of problems taken from the text

All problems will be marked

Solutions will be published on the web

There will be about 2 weeks per assignment


AssignmentsAssignmentsAssignmentsAssignments

No late assignments will be acceptedCan consult classmates, TA, instructor, etc.The first assignment is out TODAYAvailable on the course webpageDue : Thursday, Sept 25th All assignments are due in class (at the beginning of class) on the due date


ExamsExamsExamsExams

Midterm examination I:In-class : 50 minutesThursday, October 9th

Midterm examination II:In-class : 50 minutesThursday, November 6th

Final exam:Tuesday, December 9th

Both comprehensive!Both open book!


Teaching Assistants Teaching Assistants (TA)(TA)

Teaching Assistants Teaching Assistants (TA)(TA)

TAs:Are busy studentshave other things to do in life…may not necessarily be thrilled about marking

It is in YOUR best interest:To make your assignment/exam as clear as possible:

Write legibly or typeExplain all assumptions, notation, etc.Credit ALL collaboration, reference materials

To ensure the entire package gets to TA (staple, write your name, Unix ID, etc.) See course outline!


Collaboration - Collaboration - AssignmentsAssignments

Collaboration - Collaboration - AssignmentsAssignments

YES! You are encouraged to discuss problems in the assignments and solution ideas

Nevertheless, each assignment handed in must be done by the person submitting it

No labor division:“You do half of the problems and I’ll do half…”


AcknowledgementsAcknowledgementsAcknowledgementsAcknowledgementsAll discussions, hints, TA’s help, office hours, internet material, etc. must be properly acknowledged

Examples:

"I discussed problem 3.43 with my classmates P. Brosnan and R. Crowe. On problem 3.49 I received an office-hour consultation from my instructor Y.F. Chow. Additionally I used sources [1] and [2] for problem 3.78.

[1] A.Jolie. "Fast Numeric Methods for Curvature Approximation", Journal of Geeky Gamers, volume 36, issue C, June 2001.

[2] F.Oz. "On Using the Force as a Theorem Proving Technique", Jedi Archives, volume 666, number 34, May 2002."


Collaboration - ExamsCollaboration - ExamsCollaboration - ExamsCollaboration - Exams

Absolutely NO collaboration on:MidtermsFinal exam

Counter-measures include:Tight surveillance (proctor)ID checkingUnannounced other means…


ProsecutionProsecutionProsecutionProsecutionAny detected case of academic dishonesty will be automatically passed to the office of the Dean after the student is confronted

The violator(s) will go there for a friendly chat with the associate dean

Recent cases:Failure in the course3 months – 1 year student status suspensionMay jeopardize one’s student visa statusPermanently goes on the student’s record


Some ArithmeticSome ArithmeticSome ArithmeticSome Arithmetic

Suppose you are running late and the due date is near…

… and suppose you copy it from a friend:We will most likely detect itYou will lose 100% of the final markYour friend will be prosecuted as wellPlus: suspension, possible deportation, etc.

The choice is always yours…


Questions?Questions?Questions?Questions?


Why This Course?Why This Course?Why This Course?Why This Course?

Relation to real life:

Algorithm correctness ~ programming, reverse-

engineering, debugging

Propositional logic ~ hardware (including VLSI)

design

Sets/relations ~ databases (Oracle, MS Access, etc.)

Predicate logic ~ Artificial Intelligence, compilers

Proofs ~ Artificial Intelligence, VLSI, compilers,

theoretical physics/chemistry


Why This Course?Why This Course?Why This Course?Why This Course?


Code CorrectnessCode CorrectnessCode CorrectnessCode Correctness

Millions of programmers code away daily…How do we know if their code works?


ImportanceImportanceImportanceImportanceUSS Yorktown, a guided-missile cruiser --- the first to be outfitted with Smart Ship technology09/97: suffered a widespread system failure off the coast of Virginia. After a crew member mistakenly entered a zero into the data field of an application, the computer system proceeded to divide another quantity by that zero. The operation caused a buffer overflow, in which data leak from a temporary storage space in memory, and the error eventually brought down the ship's propulsion system. The result: the Yorktown was dead in the water for more than two hours.


More Software Bugs…More Software Bugs…More Software Bugs…More Software Bugs…On June 4, 1996, the maiden flight of the European Ariane 5 launcher crashed about 40 seconds after takeoff. Media reports indicated that the amount lost was half a billion dollars -- uninsured.

The exception was due to a floating-point error: a conversion from a 64-bit integer to a 16-bit signed integer, which should only have been applied to a number less than 2^15, was erroneously applied to a greater number, representing the "horizontal bias" of the flight.

There was no explicit exception handler to catch the exception, so it followed the usual fate of uncaught exceptions and crashed the entire software, hence the on-board computers, hence the mission.


Where is CMPUT 272 ?Where is CMPUT 272 ?Where is CMPUT 272 ?Where is CMPUT 272 ?

How do we find such bugs in software?

TracingDebug statementsTest casesMany software testers working in parallel…

All of that had been employed in the previous casesYet the disasters occurred…


C272 : Program C272 : Program CorrectnessCorrectness

C272 : Program C272 : Program CorrectnessCorrectness

Logic : provides a means to prove correctness of software

Sometimes it can be semi-automated

Can also verify a provided correctness proof


Argument #1Argument #1Argument #1Argument #1

All men are mortalSocrates is a man

Therefore, Socrates is mortal


Argument #2Argument #2Argument #2Argument #2

Nothing is better than a 9

A sandwich is better than nothing

Therefore, a sandwich is better than a 9


ValidityValidityValidityValidity

An argument is valid if and only if given that its premises hold its conclusion also holds

So…Socrates argument: Valid or Invalid?Sandwich argument: Valid or Invalid?


How can we tell ?How can we tell ?How can we tell ?How can we tell ?

Common sense?Voting?Authority?What is valid argument anyway?Who cares?

???


CMPUT 272CMPUT 272CMPUT 272CMPUT 272

Logic : a formal way to assess the validity of an argument

Can prove theorems in a semi-automatic fashion

Can verify a given proof that an argument is valid


Material CoverageMaterial CoverageMaterial CoverageMaterial Coverage

Chapter 1: propositional calculusChapter 2: 1st order predicate calculusChapter 3: number theoryChapter 4: mathematical inductionChapter 5: set theoryChapter 6: counting / probabilitiesChapter 7: functionsChapter 10: relations (not the kind)


What is the name of this What is the name of this man?man?

What is the name of this What is the name of this man?man?


Arguments in PuzzlesArguments in PuzzlesArguments in PuzzlesArguments in Puzzles

The Island of Knights and Knaves

Never lie

Always lie


Example #1Example #1Example #1Example #1

You meet two people: A, BA says:

I am a Knave or B is a Knight.

Who is A?

Who is B?


SolutionSolutionSolutionSolutionThe original statement can be written as:S = X or YX = “A is a Knave”Y = “B is a Knight”Suppose A is a KnaveThen S must be false since A said itThen both X and Y are falseIf X is false then A is not a KnaveContradiction : A cannot be a Knave and not a Knave !So A must be a KnightSo S is true and X is not trueThus, to keep S true Y must be trueSo B is a Knight too


How about…How about…How about…How about…

You meet just one guy : A

A says:“I’m a Knave!”

Who is A?


Features Of An Features Of An ArgumentArgument

Features Of An Features Of An ArgumentArgument

arguments involve things or objectsthings have propertiesarguments consist of statements.statements may be composedan argument starts with assumptions which create a context.each step yields another statement which is true, within its context.arguments may contain sub-argumentsit is absurd for a statement to be both true and false.


FormalizationFormalizationFormalizationFormalization

Why formalize?

to remove ambiguity

to represent facts on a computer and use it for proving, proof-checking, etc.

to detect unsound reasoning in arguments


Graphically…Graphically…Graphically…Graphically…


LogicLogicLogicLogic

Mathematical logic is a tool for dealing with formal reasoningIn a nutshell:

Logic does:Assess if an argument is valid/invalid

Logic does not directly:Assess the truth of atomic statements


DifferencesDifferencesDifferencesDifferences

Logic can deduce that:Edmonton is in Canada

given these facts:Edmonton is in AlbertaAlberta is a part of Canada

and the definitions of:‘to be a part of’‘to be in’

Logic knows nothing of whether these facts actually hold in real life!


Propositional Calculus Propositional Calculus (Ch (Ch 1.)1.)

Propositional Calculus Propositional Calculus (Ch (Ch 1.)1.)

Simplest kind of mathematical logic

Dealing with:

Propositions: X,P,Q,…each can be true or falseExamples: P=“I’m a knave”

Q=“He is a knight”

Connectives: &, v, , , ~, …connect propositions: X v Y


ConnectivesConnectivesConnectivesConnectives

Different notation is in useWe will use the common math notation:

~ notV or (non-exclusive!)

& , and

implies (if … then …) if and only if for all exists

See the reverse of the text’s front cover


FormulaeFormulaeFormulaeFormulae

A statement/proposition: true or false

Atomic: P, Q, X, Y, …

Unit Formula: P, ~P, (formula), …

Conjunctive: P & Q, P & ~Q, …

Disjunctive: P v Q, P v (P & X),…

Conditional: P Q

Biconditional: P Q


Determining Truth of A Determining Truth of A FormulaFormula

Determining Truth of A Determining Truth of A FormulaFormula

Atomic formulae: given

Compound formulae: via meaning ofthe connectives

Suppose: P is trueQ is false

How about: (P v Q)

Truth tables


Truth TablesTruth TablesTruth TablesTruth Tables

Suppose: P is falseQ is falseX is true

How about:P & Q & XP v Q & XP & Q v X


PrecedencePrecedencePrecedencePrecedence

~ highest&v, lowest

Avoid confusion - use ‘(‘ and ‘)’:P & Q v X(P & Q) v X


ParenthesizingParenthesizingParenthesizingParenthesizing

Parenthesize & build truth tablesSimilar to arithmetic:

3*5+7 = (3*5)+7 but NOT 3*(5+7)A&B v C = (A&B) v C but NOT A&(B v C)

So start with sub-formulae with highest-precedence connectives and work your way out

Let’s do the knave & knight problem in T T


T T for K&KT T for K&KT T for K&KT T for K&K

S = X or YX = “A is a Knave”Y = “B is a Knight”

A B S X Y X v YAbsurd------------------------------------------------------------------------------Knave Knave false true false true yesKnave Knight false true true true yesKnight Knave true false false false yesKnight Knight true false true true no

sept 4, 2003© vadim bulitko - cmput 272, fall 2003, uofa1 cmput 272 formal systems & logic in...

Documents

interactive problem

beginning of class

class participationas

problem solvingattendance

textall problems

tas help

office hoursinstructor

thfinal exam