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CPSC 121: Models of Computation2008/9 Winter Term 2

Proof (First Visit)

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

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.

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.

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.

Note: in this learning goal, we are not asking you to memorize the inference rules or their names! However,

as you become familiar with them, you’ll find that your exploration is easier and more effective.

Quiz 4 Notes (1 of 2)

Correctness problems:

Mostly, problems with “none of the above”:• a a b

p ?

• a b, b c a c p (q r), q s ?

Quiz 4 Notes (2 of 2)

Onnagata problems:• If the onnagata are

incorrect, is the argument valid?

• Do the two premises contradict each other?

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

What is Proof?

A rigorous mathematical argument which unequivocally demonstrates

the truth of a given proposition, given the truth of the proof’s premises.

MathWorld definition: http://mathworld.wolfram.com/Proof.htmlMy addition in italics.

Problem: Evens and Integers

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

Tasting Powerful Proof: Some Things We Might Prove

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

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

Meanwhile...

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.

Problem: Prop Logic Proof

Problem: prove that...

~(q r)(u q) s

p s_____

~p

Concept Q: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.

Concept Q: 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.

Problem: Onagata

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.

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?

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).

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)

Problem: Automating Proof

Given:p q

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

~r

Problem: What’s everything you can prove?

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.

Problem: Canonical Form

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

Mystery #1

Theorem:

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

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

Mystery #2

Theorem:

p

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

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

Mystery #3

Theorem:

q

p mq (r m)m q p

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

Practice Problem (for you!)

Prove that hypothetical syllogism is a valid rule of inference:

p qq r p r

Practice Problem (for you!)

Prove whether this is a valid rule of inference:

q

p q p

Practice Problem (for you!)

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.

Practice Problem (for you!)

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)?

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?

Problem: Equivalent Java Programs

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

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#14.5

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#14.5

What’s a “Block”?

Back to the Java Language Specification:

http://java.sun.com/docs/books/jls/third_edition/html/statements.html#14.2

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).

What’s an “EmptyStatement”

Back to the Java Language Specification:

http://java.sun.com/docs/books/jls/third_edition/html/statements.html#14.6

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?

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”)?

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

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?