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Weixiong ZhangWashington University in St. Louis

http://www.cse.wustl.edu/~zhang/teaching/cse240/Spring10/index.html

CSE 240Logic and Discrete Math

Lecture notes Introduction and Propositional

Logic

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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

This is the “calculus” course for the computer science

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Code Correctness

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

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ImportanceUSS 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.

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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.

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Where is CSE 240?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…

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CSE240 : Program Correctness

Logic : means to prove correctness of software

Sometimes can be semi-automated

Can also verify a provided correctness proof

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Argument #1

All men are mortalSocrates is a man

Therefore, Socrates is mortal

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Argument #2

Nothing is better than GodA sandwich is better than nothing

Therefore, a sandwich is better than God

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Argument #3

An island

A barber cannot cut his/her own hair

The barber cuts everyone’s hair on the island

Jon is a barber

So, Jon will cut his own hair

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Validity

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?Barber argument: Valid or Invalid?

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How can we tell ?

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

???

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CSE 240

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

Can prove theorems in a semi-automatic fashion – construct a proof

Can verify given proofs that an argument is valid

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Material CoverageChapter 1: propositional calculusChapter 2: 1st order predicate calculusChapter 3: number theory and proof techniquesChapter 4: mathematical inductionChapter 5: set theoryChapter 6: counting / probabilitiesChapter 7: functionsChapter 10: relationsChapter 11: graphs and trees

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Questions?

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Arguments in Puzzles

The Island of Knights and Knaves

Never lie

Always lie

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Example #1

You meet two people: A, BA says:

I am a Knave or B is a Knight.

Who is A?

Who is B?

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A possible argument (solution)The 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 Knave(Contradiction : A can be a Knave and cannot a Knave !)So A must be a KnightSo S is true (why?) and X is not trueThus, to keep S true Y must be trueSo B is a Knight too

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How about…

You meet just one guy : A

A says:“I’m a Knave!”

Who is A?

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Features Of An Argumentarguments involve things or objectsthings have propertiesarguments consist of statementsstatements 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

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Formalization

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

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Graphically…

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Logic

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

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DifferencesLogic can deduce that:

St Louis is in the USAgiven these facts:

St Louis is in MissouriMissouri is a part of the USA

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

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

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Questions?

Lecture 04 © Vadim Bulitko : CMPUT 272, Winter 2004, UofA 26

Announcement

It is Ok for an assignment problem to refer to the material that is:

Not covered in class by the due dateNot covered in class at all

Why?

Because the lectures only highlight the material in the textbookThey are not a substitute for reading the text

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Propositional Calculus (Ch 1.)

Simplest kind of math 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

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Connectives

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

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Formulae

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

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Determining Truth of A Formula

Atomic formulae: givenCompound formulae: via meaning of

the connectivesSuppose: P is true

Q is falseHow about: (P v Q)

Truth tables

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Truth Tables

Suppose: P is falseQ is falseX is true

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

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Precedence

~ highest&v, lowest

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

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Parenthesizing

Parenthesize & build truth tablesSimilar to arithmetics:

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 TT

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TT for K&KS = X or YX = “A is a Knave”Y = “B is a Knight”

A B S X Y X v Y Absurd------------------------------------------------------------------------------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

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Questions?

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Conditionals

“If I go to Save-on-Foods tomorrow I will buy oranges there”

S = (go to SOF) (buy oranges)

When is S true?When I went to SOF and bought orangesWhen I didn’t go there at all!

When is S false?When I went there but didn’t buy oranges

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Truth Table

N independent atomic formulae (variables) 2N rowsN=20: 220 ~ 1 millionConsider A B:

A B A BF F TF T TT F FT T T

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More terminology

A B

A is called:AssumptionPremiseAntecedent

B is calledConclusion

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Bi-Conditionals“Marion will take 240 if and only if Norma does so”

S = (Marion takes 240) (Norma takes 240)

When is S true?When Marion takes is and Norma takes itWhen Marion doesn’t take it and neither does Norma

When is S false?When one of them takes it but not the other

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Truth Tables

A B A BF F TF T FT F FT T T

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, and Daily LifeSuppose:

“Buy Ferrari” = B“Ferrari is on sale” = S

I will buy a Ferrari if it is on sale:S B

I will buy a Ferrari if and only if it is on sale:

S BI will buy a Ferrari only if it is on sale:

~S ~BB S

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Sufficient & Necessary Conditions

Suppose the Ferrari salesperson wants to figure out when you are ready to buy one of their cars…

B = “ready to buy”X = some condition of you

If they find X such that: X BThen they have a sufficient conditionExample: X = “got a WashU scholarship”

If they find X such that: B XThen they have a necessary conditionExample: X = “it is winterized”

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CriteriaIf they find X such that: B X

Then they have a criterion

What would be an example?How about a conjunction of the sufficient and necessary conditions?

X = “got WashU scholarship and it is winterized”

Doesn’t work: you may get a Ferrari Christmas gift certificate from your grandma and you will just go and pick one up (regardless of whether you got the scholarship)

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SummarySuppose X is a statement

A is a sufficient condition for X iff we can prove that:“A implies X”

A is necessary condition for X iff we can prove that:“X implies A”

A is a criterion for X iff we can prove that:“X holds if and only if A holds”

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Contradictions & Tautologies

Contradiction:A statement that is always false regardless of the values of its variables

Examples: A & ~A

Tautology:A statement that is always true regardless of the values of its variables

Examples: A v ~A

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Contingencies

What is I have a formula that is sometimes true and sometimes false?

It is called a contingency

Example:A & B

If A and B are independent variables

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Interpretation

In propositional logic interpretation (instantiation) is a mapping from symbols in your formulae to {true, false}

Example:Formula: A v BInterpretation: A = true, B = false

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Formula Classification

Tautology : all interpretations satisfy the formula

Contradiction : all interpretations falsify the formula

Contingency : some interpretations satisfy and some falsify the formula

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Models / Counter-models

An interpretation is called a model for formula F iff:

It satisfies F (i.e., makes F=true)

An interpretation is called a counter-model for formula F iff:

It falsifies F (i.e., makes F=false)

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Logic EquivalencePropositions/statements/formulae A and B are logically equivalent when:

A holds if and only if B holdsNotation: A ≡ BExamples:

A v A is equivalent to:A

A v ~A is equivalent to:true

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Equivalence & Tautology

Suppose A and B are logically equivalentConsider proposition (A B)What can we say about it?It is a tautology!Why?A B A BF F TT T T

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More Equivalences

Alex is not unemployedAlex is employed

P ≡ ~(~P) : Double-negationIt is not true that she is single and she is cute

She is not single or she is not cute

~(A & B) ≡ ~A v ~B : De Morgan’s lawIt is not true that she is single or she is cute

She is not single and she is not cute

~(A v B) ≡ ~A & ~B : De Morgan’s law

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Boolean AlgebraPage 14 presents Theorem 1.1.1 with 21 equivalencesHowever, only the first 10 are neededThe rest can be derived from them

Example: let’s prove that P v P ≡ PP v C ≡ P (4)P v (P & ~P) ≡ P (5)(P v P) & (P v ~P) ≡ P (3)(P v P) & T ≡ P (5)P v P ≡ P (4)

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Challenge

Derive the rest (e.g., #8) from the first 5 equivalences…

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Proving Equivalences

Suppose P and Q can take on T and Fonly Then all equivalences can be proven by definition using truth tablesP Q ~(P v Q) ~P & ~QF F T F T T TF T F T T F FT F F T F F TT T F T F F F

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Another Example

Prove P Q ≡ ~P v QTruth tables…

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Questions?

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Uses of Equivalences

SimplificationSuppose someone gives you

~P v (AB) v ~(C v D H) v P v XY

and asks you to compute it for all possible input valuesYou can either immediately draw a truth table with 28 = 256 rows

Or you can simplify it first

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Simplification

~P v (AB) v ~(C v D H) v P v XY~P v P v (AB) v ~(C v D H) v XYT v (AB) v ~(C v D H) v XYT YY

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Test Drive

Let’s take our equivalence tool box for a spin…What can we tell about

A B

and its contraposition:~B ~A

?They are equivalent !

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Proof?

Truth-tables

EquivalencesA B~A v BB v ~A~B ~A

Isn’t this fun?

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Test Drive

Let’s try another one!What can we tell about

A B

and its inverse:~A ~B

?Oh-oh…They are not equivalent!Counter-model: A is false and B is true…

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Summary

ImplicationA B

is equivalent to:~B ~A : its contraposition

But is not equivalent to:~A ~B : its inverse

Or to:B A : its converse

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Another Lesson Learned

Proving equivalencesvia truth-tablesvia other equivalences

Proving non-equivalence:Finding an instantiation that makes one formula hold and the other doesn’t

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Bi-conditionals

How can we express:A B

With &, v, ~ ?It is simply:

(A & B) v (~A & ~B)

How can we express it with , & ?Why, but of course it is just:

(A B) & (B A)

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Another Spin…

Ok, let’s try this fun ride:(P Q) & P

In English it would sound:(If P is true then Q is true) and (P is true)

What does it tell us?Naturally Q is true!

Let’s prove it…

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Proving…

(P Q) & P(~P v Q) & P~P & P v Q & PF v Q & PQ & PBut we want just Q not Q & P!Is Q & P ≡ Q ?No -- counter-model: Q=T and P=F …

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What is the matter?

Not all arguments have the form of a chain of equivalencesExample:

PQ = “If Socrates is human then Socrates is mortal”P=“Socrates is human”A conjunction of these two is NOT equivalent to Q= “Socrates is mortal”Why? Name a dog Socrates. It is mortal (Q holds). But it is NOT human (P does not hold)…

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Entailment

A collection of statements P1,…,Pn (premises) entails statement Q (conclusion) if and only if:Whenever all premises hold the conclusion holds

Example:Premises:

P1 = “If Socrates is human then Socrates is mortal”P2 = “Socrates is human”

Conclusion:Q = “Socrates is mortal”

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Valid/Invalid Arguments

Suppose someone makes an argument:P1,..,PN therefore Q

The argument is called valid iff:P1,…,PN logically entail Q

That is:Q must hold when all Pi hold

Otherwise the argument is called invalid

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Relation to Tautologies

We already know that formulae A and B are equivalent iff (A B) is a tautology (i.e., holds for any interpretation)

How about entailment?

Formula A entails formula B iff (A B) is a tautologyIn general: premises P1,…,PN entail Q iff (P1 & … & PN Q) is a tautology

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Questions?

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Today

Refresher: Chapter 1.2

Chapter 1.3 : Arguments

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Interpretation

In propositional logic interpretation is a mapping from variables in your formulae to {true, false}

Example:Formula: A v BInterpretation 1: A = true, B = falseInterpretation 2: A = false, B = false

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Interpretations

How many interpretations do the following formulae allow?A B

4

(AB & A) B4

Why not 8 or 16?

The number of interpretations is 2N where Nis the number of independent variables

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Questions?

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ConditionsSuppose we care about statement X

X = “this assignment is copied”We want to evaluate X (true/false?)

Suppose we know A such that AXA is a sufficient conditionA=“the cheater is caught in the act”

Suppose we know B such that XBB is a necessary conditionB=“there was an original assignment to copy from”

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Criteria

Suppose we know C such that CXC is a criterionC=“someone has copied this assignment”

Graphically:

X

A

B

C

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Wanted : Criteria

Medical testsSoftware/hardware correctnessFraud/cheatingFinancial marketPsychology (e.g., in sales)Science : mathematics, physics, chemistry, etc.Logic :

If C is a criterion for X then C ≡ X !

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PracticeIt is frequently non-trivial to derive a criterion for a real-life property X

Then we have to settle for:Sufficient conditions:

“If this quality test passes then the product is fine”Necessary conditions:

“If the patient breaks a leg they will be in pain”

Statistical validity : the condition works most of the timeIn logic : the condition works all the time!

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Derivation of Criteria

Logic/Mathematics/Theoretical sciences:Equivalent transformationsProofs by contradiction

Empirical sciences:Statistical testsFunction approximation

Artificial Intelligence:Machine learning

These methods are

notguaranteed to produce true

criteria…

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Questions?

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Logic EquivalencePropositions/statements/formulae A and B are logically equivalent when:

A holds if and only if B holdsNotation: A ≡ BExamples:

A v A is equivalent to:A

A v ~A is equivalent to:true

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Challenge

Theorem 1.1.1 : Boolean AlgebraDerive the rest (e.g., #8) from the first 5 equivalences…

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Use of Equivalences

Deriving equivalent formulae!Of course, but why do we care?

Simplification of formula

Simplification of code

Simplification of hardware (e.g., circuits)

Derivation of criteria!

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Limitations

Not all statements are equivalent!Of course not, but what else is there?

Some formulae are stronger than others

They imply or entail other formula but not the other way around…

Equivalences cannot directly help us proving such entailments…

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EntailmentA collection of statements P1,…,Pn (premises) entails statement Q (conclusion) if and only if:

Whenever all premises hold the conclusion holds

For every interpretation I that makes all Pj hold, I also makes Q hold

Example:Premises:

P1 = “If Socrates is human then Socrates is mortal”P2 = “Socrates is human”

Conclusion:Q = “Socrates is mortal”

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Valid/Invalid Arguments

Suppose someone makes an argument:P1,..,PN therefore Q

The argument is called valid iff:P1,…,PN logically entail Q

That is:Q must hold if all Pi hold

Otherwise the argument is called invalid

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Example

Sample argument:P1 = “If Socrates is human then Socrates is mortal”P2 = “Socrates is human”Therefore:Q = “Socrates is mortal”

Valid / invalid?

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Entailment

Then is the following argument valid?P1

P2

entailsQ

Yes?Very well, but what if my interpretation I sets P1 and P2 to true but Q to false?

Then by definition Q is not entailed by P1 and P2

So do P1,P2 entail Q or do they not?

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What Happened…We considered P1, P2, and Q under a particular(common sense) interpretation:

P1 = “If Socrates is human then Socrates is mortal” trueP2 = “Socrates is human” trueQ = “Socrates is mortal” true

Thus, they were merely logical constants to us:P1=trueP2=trueQ=true

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Generality…Thus our argument was:

TrueTrueentailsTrue

Well, this is not very useful because it doesn’t tell us anything about validity of other arguments. For example:

P1=“If J.B. broke his leg then J.B. is in pain”P2=“J.B. broke his leg”entailsQ=“J.B. is in pain”

Is this argument valid?

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Extracting the Essence

How do we know it is valid?Because regardless of who J.B. is and what happened to him/her, we somehow know that:

If P1 and P2 holdThen Q will hold

But how do we know that?How can we extract the essence of the “dead Socrates” and “J.B. in pain” arguments?

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General Structure!Recall both arguments:

P1 “If Socrates is human then Socrates is mortal” “If J.B. broke his leg then J.B. is in pain”P2 “Socrates is human” “J.B. broke his leg”entailsQ “Socrates is mortal” “J.B. is in pain”

Note that P1, P2, and Q are related!Both arguments share the same structure:

P1 If X then YP2 XentailsQ Y

Then for any interpretation I as long as I satisfies P1and P2, interpretation I must satisfy Q

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Modus PonensThe “generalized” argument

P1 = X YP2 = XentailsQ = Y

…is much more useful!Why?

Because it captures the essence of both arguments and can be used for infinitely many more

“method of affirming” (Lat.)

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Valid Arguments (Revisited)Suppose someone makes an argument:

P1,..,PN therefore QThe argument is called valid iff:

P1,…,PN logically entail QThat is:

For any interpretation I that satisfies all Pj, interpretation I must necessarily satisfy Q

Usually: Pj and Q are somehow related formulae and P1 & … & PN can be true or false depending on the interpretation I

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Questions?

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How Do We:Tell between a valid argument and an invalid argument:

People are mortal. Socrates is a man. Socrates is mortal.Ducks fly. F-16 flies. F-16 is a duck.

Prove that something logically follows from something else:

1: Everybody likes Buddha2: Everybody likes someone

Prove that something is logically equivalent to something else:

1: Everybody likes cream and sugar2: Everybody likes cream and everybody likes sugar

Prove that there is a contradiction?

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Propositional Logic

Method #1:Go through all possible interpretations and check the definition of valid argument

Method #2:Use derivation rules to get from the premises to the conclusion in a logically sound way

“derive the conclusions from premises”

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Method #1

Section 1.3 in the text proves many arguments/inference rules using truth tables

Suppose the argument is:P1,…,PN therefore Q

Create a truth table for formulaF=(P1 & … & PN Q)

Check if F is a tautology

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But Why? Recall:

Formula A entails formula Biff (A B) is a tautology

In general:premises P1,…,PN entail Qiffformula F=(P1 & … & PN Q)is a tautology

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Example #1

P QP entails Q

valid/invalid?Modus ponens

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Example #2

P QQ entails P

valid/invalid?

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Example #3

P Q~Q entails ~P

valid/invalid?Modus tollens

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Example #4

P Qentails ~Q ~P

valid/invalid?

In fact, we proved last time that: (P Q) ≡ (~Q ~P)

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Example #5

P v Q~P & ~Qentails P & Q

valid/invalid?

Any argument with a contradiction in its premises is valid by default…

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Example #6

P v Q v R~R entails P v Q

valid/invalid?

(example 1.3.2 in the book, p. 31)

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Example #7

P v Q v R~R entails Q

valid/invalid?

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Pros & Cons

Method #1:Pro: straight-forward, not much creativity machines can do

Con: the number of interpretations grows exponentially with the number of variables cannot do for many variables

Con: in predicate and some other logics even a small formula may have an infinitenumber of interpretations

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Questions?

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Method #2 : Derivations

To prove that an argument is valid:

Begin with the premisesThen which high level strategy?

Equivalence, entailment, (contradiction)

Use valid/sound inference rules

Arrive at the conclusion

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Inference Rules

But what are these “inference rules”?They are simply…

…valid arguments! Table 1.1.1

Example:X & Y X & Y Z & WthereforeZ & W by modus ponens

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Example #1

(X&Y Z&W) & KX&YthereforeZ&WHow?(X&Y Z&W) & KX&Y Z&W by conjunctive simplificationX&YZ&W by modus ponens

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Derivations

The chain of inference rules that starts with the premises and ends with the conclusion…is called a derivation:

The conclusion is derived from the premises

Such a derivation makes a proof of argument’s validity

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Example #1

(X&Y Z&W) & KX&YthereforeZ&WHow?(X&Y Z&W) & KX&Y Z&W by conjunctive simplificationX&YZ&W by modus ponens

derivation

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Pros & Cons

Method #2:Pro: often can get a dramatic speed-up over truth tables.

Con: requires creativity and intuition harder to do by machines some times

Con: semi-decidable : there is no algorithm that can prove any first-order predicate logic argument to be valid or invalid

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Questions?

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Fallacies

An error in derivation leading to an invalid argumentVague formulations of premises/conclusionMissing stepsUsing non-sound inference rules, e.g.:

Converse errorInverse error

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Converse Error

If John is smart then John makes a lot of moneyJohn makes a lot of moneyTherefore:John is smart

Tries to use this non-sound “inference rule”:AB, BThus: A

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Inverse Error

If John is smart then John makes a lot of moneyJohn is not smartTherefore:John doesn’t make a lot of money

Tries to use this non-sound “inference rule”:AB, ~AThus: ~B

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Questions?

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Truth of facts vs. Validity of Arguments

The premises are assumed to be true ONLY in the context of the argument

The following argument is valid:If John Lennon was a rock star then he was a womanJohn Lennon was a rock starThus:John Lennon was a woman

But the 1st premise doesn’t hold under the common sense interpretation

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Inference Rules

Table 1.3.1 on page 40 – study them!!

If practice with the rules then will be more fluent using them

If are more fluent using them then will be more likely to get a better mark on exams

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Summary

Equivalence:A ≡ BA holds iff B holdsA is a criterion for BB is a criterion for AA entails BB entails AA and B are “equivalently strong”Formula F=(AB) is a tautology

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Summary

Entailment:A entails BB follows from AA∴B is a valid argumentA is a sufficient condition for BB is a necessary condition for AIf A holds then B holdsA may be “stronger than” BFormula F=(AB) is a tautology

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The Big Picture

Logic is being used to verify validity of arguments

An argument is valid iff its conclusion logically follows from the premises

Derivations are used to prove validity

Inference rules are used as part of derivations

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Questions?

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Summary propositional logic

Interpretation – T/FTruth tableLogical equivalences theorem 1.1.1- page 14Order of operationsRules of inference (reading assignment)Necessary and sufficient conditions