CPSC 121: Models of Computation 2009 Winter Term 1

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CPSC 121: Models of Computation2009 Winter Term 1

Proof (First Visit)

Steve Wolfman, based on notes by Patrice Belleville, Meghan Allen and others

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Outline

• Prereqs, Learning Goals, and Quiz Notes

• Prelude: What Is Proof?

• Problems and Discussion– “Prove Your Own Adventure”– Why rules of inference? (advantages + tradeoffs)

– Onnagata, Explore and Critique

• Next Lecture Notes

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Lecture Prerequisites

Read Section 1.3.

Solve problems like Exercise Set 1.3, #1, 3, 4, 6-32, 36-44. Of these, we’re especially concerned about problems like 12-13 and 39-44. Many of these problems go beyond the pre-class learning goals into the in-class goals, but they’re the tightest fit in the text.

Complete the open-book, untimed quiz on WebCT that was due before class.

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Learning Goals: Pre-Class

By the start of class, you should be able to:– Use truth tables to establish or refute the

validity of a rule of inference.– Given a rule of inference and propositional

logic statements that correspond to the rule’s premises, apply the rule to infer a new statement implied by the original statements.

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Learning Goals: In-Class

By the end of this unit, you should be able to:– Explore the consequences of a set of

propositional logic statements by application of equivalence and inference rules, especially in order to massage statements into a desired form.

– Critique a propositional logic proof, identifying flaws in its reasoning and application.

– Devise and attempt multiple different, appropriate strategies for proving a propositional logic statement follows from a list of premises.

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Quiz 4 Notes (1 of 2)

Applying a rule:

a b, b c a c

p (q r), q s ?

Validity of a rule:

p ~p ?

6Second set of quiz notes will come much later!

Outline






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What is Proof?

A rigorous formal argument that unequivocally demonstrates the truth of a proposition, given the truth of the

proof’s premises.

Adapted from MathWorld: http://mathworld.wolfram.com/Proof.html8

Problem: Meaning of Proof

Let’s say you prove the following:

Premise 1

Premise 2

⁞

Premise n

Conclusion

What does this mean?

a.Premises 1 to n are trueb.Conclusion is truec.Premises 1 to n can be trued.Conclusion can be truee.None of the above

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Tasting Powerful Proof: Some Things We Might Prove

• We can build a “three-way switch” system with any number of switches.

• We can build a combinational circuit matching any truth table.

• We can build any digital logic circuit using nothing but NAND gates.

• We can sort a list by breaking it in half, and then sorting and merging the halves.

• We can find the GCD of two numbers by finding the GCD of the 2nd and the remainder when dividing the 1st by the 2nd.

• There’s (sort of) no fair way to run elections.• There are problems no program can solve.

Meanwhile...10

What Is a Propositional Logic Proof?

An argument in which (1) each line is a propositional logic statement, (2) each

statement is a premise or follows unequivocally by a previously established rule of inference from the truth of previous statements, and (3)

the last statement is the conclusion.

A very constrained form of proof, but a good starting point.Interesting proofs will usually come in less structured

packages than propositional logic proofs.11

Outline






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Prop Logic Proof Problem

To prove:

~(q r)(u q) s

~s ~p___

~p

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“Prove Your Own Adventure”

To prove:

~(q r)(u q) s

~s ~p___

~p

Which step is the easiest to fill in?

1. ~(q r) Premise2. (u q) s Premise3. ~s ~p Premise[STEP A: near the start] [STEP B: in the middle]

[STEP C: near the end][STEP D: last step]

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D: Last Step

To prove:

~(q r)(u q) s

~s ~p___

~p

1. ~(q r) Premise2. (u q) s Premise3. ~s ~p Premise

...~q ~r De Morgan’s (1)~q Specialization (?)

...((u q) s) Bicond (2) (s (u q))

...~s~p Modus ponens (3,?)

How do we know to put ~p at the end?

a.~p is the proof’s conclusionb.~p is the end of the last premisec.every proof ends with ~pd.None of these but some other reasone.None of these because we don’t know

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C: Near the End

To prove:

~(q r)(u q) s

~s ~p___

~p



...((u q) s) Bicond (2) (s (u q))


How do we know to put the blue line/justification at the end?

a.~s ~p is the last premiseb.~s ~p is the only premise that mentions ~sc.~s ~p is the only premise that mentions pd.None of these but some other reasone.None of these because we don’t know

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A: Near the Start

To prove:

~(q r)(u q) s

~s ~p___

~p



...((u q) s) Bicond (2) (s (u q))


How do we know to put the blue lines/justifications in?

a.~(q r) is the first premiseb.~(q r) is a useless premisec.We can’t work directly with a premise with a negation “on the outside”d.Neither the conclusion nor another premise mentions re.None of these

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B: In the Middle

To prove:

~(q r)(u q) s

~s ~p___

~p



...((u q) s) Bicond (2) (s (u q))


How do we know to put the blue lines/justifications in?

a.(u q) s is the only premise leftb.(u q) s is the only premise that mentions uc.(u q) s is the only premise that mentions s without a negationd.We have no rule to get directly from one side of a biconditional to the othere.None of these

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Prop Logic Proof Strategies

• Work backwards from the end• Play with alternate forms of premises• Identify and eliminate irrelevant information• Identify and focus on critical information• Alter statements’ forms so they’re easier to

work with• “Step back” from the problem frequently to

think about assumptions you might have wrong or other approaches you could take

And, if you don’t know that what you’re trying to prove follows...switch from proving to disproving and back now and then.

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Continuing From There

To prove:

~(q r)(u q) s

~s ~p___

~p

1. ~(q r) Premise2. (u q) s Premise3. ~s ~p Premise4. ~q ~r De Morgan’s (1)5. ~q Specialization (4)6. ((u q) s) Bicond (2) (s (u q))7. ?????? Specialization (6)


Which direction of goes in step 7?

a.(u q) s because the simple part is on the rightb.(u q) s because the other direction can’t establish ~sc.s (u q) because the simple part is on the leftd.s (u q) because the other direction can’t establish ~se.None of these

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Finishing Up (1 of 3)

To prove:

~(q r)(u q) s

~s ~p___

~p

1. ~(q r) Premise2. (u q) s Premise3. ~s ~p Premise4. ~q ~r De Morgan’s (1)5. ~q Specialization (4)6. ((u q) s) Bicond (2) (s (u q))7. s (u q) Specialization (6) 8. ???? ????9. ~(u q) ????10. ~s Modus tollens (7, 9)11. ~p Modus ponens (3,10)

We know we needed ~(u q) on line 9 because that’s what we created line 7 for!

Now, how do we get ~(u q)?

Working forward is tricky. Let’s work backward. What is ~(u q) equivalent to? 21


To prove:

~(q r)(u q) s

~s ~p___

~p

1. ~(q r) Premise2. (u q) s Premise3. ~s ~p Premise4. ~q ~r De Morgan’s (1)5. ~q Specialization (4)6. ((u q) s) Bicond (2) (s (u q))7. s (u q) Specialization (6) 8. ~u ~q ????9. ~(u q) De Morgan’s (8)10. ~s Modus tollens (7, 9)11. ~p Modus ponens (3,10)

All that’s left is to get to ~u ~q. How do we do it?

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To prove:

~(q r)(u q) s

~s ~p___

~p

1. ~(q r) Premise2. (u q) s Premise3. ~s ~p Premise4. ~q ~r De Morgan’s (1)5. ~q Specialization (4)6. ((u q) s) Bicond (2) (s (u q))7. s (u q) Specialization (6) 8. ~u ~q Generalization (5)9. ~(u q) De Morgan’s (8)10. ~s Modus tollens (7, 9)11. ~p Modus ponens (3,10)

As usual in our slides, we made no mistakes and reached no dead ends. That’s not the way things really go on difficult proofs!

Mistakes and dead ends are part of the discovery process! So, step back now and then and reconsider your assumptions and approach! 23

Outline






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Limitations of Truth Tables

Why not just use truth tables to prove propositional logic theorems?

a.No reason; truth tables are enough.b.Truth tables scale poorly to large problems.c.Rules of inference and equivalence rules can

prove theorems that cannot be proven with truth tables.

d.Truth tables require insight to use, while rules of inference can be applied mechanically.

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Limitations of Logical Equivalences

Why not use logical equivalences to prove that the conclusions follow from the premises?

a.No reason; logical equivalences are enough.b.Logical equivalences scale poorly to large

problems.c.Rules of inference and truth tables can prove

theorems that cannot be proven with logical equivalences.

d.Logical equivalences require insight to use, while rules of inference can be applied mechanically.

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Outline




– Onnagata: Explore and Critique


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Problem: Onnagata

Problem: Critique the following argument.

Premise 1: If women are too close to femininity to portray women then men must be too close to masculinity to play men, and vice versa.

Premise 2: And yet, if the onnagata are correct, women are too close to femininity to portray women and yet men are not too close to masculinity to play men.

Conclusion: Therefore, the onnagata are incorrect, and women are not too close to femininity to portray women.

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Quiz 4 Notes (2 of 2)

Approaches:

• Use our model!

• Prove with a truth table

• Trace the argument

• Build a new argument and see where it leads

• Assume the opposite of the conclusion and see what happens

• Question the premises

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Contradictory Premises?

Do premises #1 and #2 contradict each other (i.e., is premise1 AND premise2 logically equivalent to F)?

a. Yes

b. No

c. Not enough information to tell.

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Defining the Problem

Which definitions should we use?

a. w = women, m = men, f = femininity, m = masculinity, o = onnagata, c = correct

b. w = women are too close to femininity, m = men are too close to masculinity, pw = women portray women, pm = men portray men, o = onnagata are correct

c. w = women are too close to femininity to portray women, m = men are too close to masculinity to portray men, o = onnagata are correct

d. None of these, but another set of definitions works well.

e. None of these, and this problem cannot be modeled well with propositional logic.

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Translating the Statements

Which of these is not an accurate translation of one of the statements?

a.w m

b.(w m) (m w)

c.o w ~m

d.~o ~w

e.All of these are accurate translations.

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Problem: Now, Explore!

Critique the argument by either:

(1) Proving it correct (and commenting on how good the propositional logic model’s fit to the context is).

How do we prove prop logic statements?

(2) Showing that it is an invalid argument.

How do we show an argument is invalid? (Hint: think back to the quiz!)

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Outline






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Next Lecture Learning Goals: Pre-Class

By the start of class, you should be able to:– Evaluate the truth of predicates applied to

particular values.– Show predicate logic statements are true by

enumerating examples (i.e., all examples in the domain for a universal or one for an existential).

– Show predicate logic statements are false by enumerating counterexamples (i.e., one counterexample for universals or all in the domain for existentials).

– Translate between statements in formal predicate logic notation and equivalent statements in closely matching informal language (i.e., informal statements with clear and explicitly stated quantifiers).

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Next Lecture Prerequisites

Review Chapter 1 and be able to solve any Chapter 1 exercise.

Read Sections 2.1 and 2.3 (skipping the “Negation” sections in 2.3 on pages 102-104)

Solve problems like Exercise Set 2.1 #1-24 and Set 2.3 #1-12, part (a) of 14-19, 21-22, 30-31, part (a) of 32-38, 39, parts (a) and (b) of 45-52, and 53-56.

You should have completed the open-book, untimed quiz on Vista that was due before this class.

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More problems to solve...

(on your own or if we have time)

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Problem: Who put the cat in the piano?

Hercule Poirot has been asked by Lord Martin to find out who closed the lid of his piano after dumping the cat inside. Poirot interrogates two of the servants, Akilna and Eiluj. One and only one of them put the cat in the piano. Plus, one always lies and one never lies.

Akilna says:– Eiluj did it.– Urquhart paid her $50 to help him study.

Eiluj says:– I did not put the cat in the piano.– Urquhart gave me less than $60 to help him study.

Problem: Whodunit?

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Problem: Automating Proof

Given:p q

p ~q r(r ~p) s ~p

~r

Problem: What’s everything you can prove?

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Problem: Canonical Form

A common form for propositional logic expressions, called “disjunctive normal form” or “sum of products form”, looks like this:

(a ~b d) (~c) (~a ~d) (b c d e) ...In other words, each clause is built up of simple

propositions or their negations, ANDed together, and all the clauses are ORed together.

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Problem: Canonical Form

Problem: Prove that any propositional logic statement can be expressed in disjunctive normal form.

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Mystery #1

Theorem:

p qq (r s)~r (~t u)p t u

Is this argument valid or invalid?Is whatever u means true?

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Mystery #2

Theorem:

p

p rp (q ~r)~q ~s s

Is this argument valid or invalid?Is whatever s means true?

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Mystery #3

Theorem:

q

p mq (r m)m q p

Is this argument valid or invalid?Is whatever p means true?

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Practice Problem (for you!)

Prove (with truth tables) that hypothetical syllogism is a valid rule of inference:

p qq r p r

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Prove (with truth tables) whether this is a valid rule of inference:

q

p q p

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Are the following arguments valid?

This apple is green.If an apple is green, it is sour. This apple is sour.

Sam is not barking.If Sam is barking, then Sam is a dog. Sam is not a dog.

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Are the following arguments valid?

This shirt is comfortable.If a shirt is comfortable, it’s chartreuse. This shirt is chartreuse.

It’s not cold.If it’s January, it’s cold. It’s not January.

Is valid (as a term) the same as true or correct (as English ideas)?48

More Practice

Meghan is rich.

If Meghan is rich, she will pay your tuition.

Meghan will pay your tuition.

Is this argument valid?Should you bother sending in a check for your

tuition, or is Meghan going to do it?49

Problem: Equivalent Java Programs

Problem: How many valid Java programs are there that do exactly the same thing?

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Resources: Statements

From the Java language specification, a standard statement is one that can be:

http://java.sun.com/docs/books/jls/third_edition/html/statements.html#14.551

Resources: Statements

From the Java language specification, a standard statement is one that can be:


What’s a “Block”?

Back to the Java Language Specification:


What’s a “Block”?

A block is a sequence of statements, local class declarations and local variable declaration statements within braces.

…

A block is executed by executing each of the local variable declaration statements and other statements in order from first to last (left to right).

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What’s an “EmptyStatement”

Back to the Java Language Specification:


Problem: Validity of Arguments

Problem: If an argument is valid, does that mean its conclusion is true? If an argument is invalid, does that mean its conclusion is false?

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Problem: Proofs and Contradiction

Problem: Imagine I assume premises x, y, and z and prove F. What can I conclude (besides “false is true if x, y, and z are true”)?

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Proof CritiqueTheorem: √2 is irrational

Proof: Assume √2 is rational, then...

There’s some integers p and q such that √2 = p/q, and p and q share no factors.

2 = (p/q)2 = p2/q2 and p2 = 2q2

p2 is divisible by 2; so p is divisible by 2.

There’s some integer k such that p = 2k.

q2 = p2/2 = (2k)2/2 = 2k2; so q2 and q are divisible by 2.

p and q do share the factor 2, a contradiction!

√2 is irrational. QED

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Problem: Comparing Deduction and Equivalence Rules

Problem: How are logical equivalence rules and deduction rules similar and different, in form, function, and the means by which we establish their truth?

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Problem: Evens and Integers

Problem: Which are there more of, (a) positive even integers, (b) positive integers, or (c) neither?

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Documents

CPSC 121: Models of Computation 2009 Winter Term 1