CPSC 121: Models of Computation 2012 Summer Term 2

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CPSC 121: Models of Computation2012 Summer Term 2

Proof Techniques(Part A)

Steve Wolfman, based on notes by Patrice Belleville and others

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Outline

• Learning Goals, Quiz Notes, and Big Picture• Problems and Discussion: Generally Faster?

– Breaking Down Big Proofs– Witness Proofs, also known as

Proofs of Existence– Without loss of generality (WLOG), also known as

Generalizing from the Generic Particular– Antecedent Assumption– Proving Inequality (and equivalences/equality)– Breaking Down Big Proofs, Revisited

• Coming Soon: The Rest2

Learning Goals: “Pre-Class”

Be able for each proof strategy below to:– Identify the form of statement the strategy can prove.– Sketch the structure of a proof that uses the strategy.

Strategies: constructive/non-constructive proofs of existence ("witness"), disproof by counterexample, exhaustive proof, generalizing from the generic particular ("WLOG"), direct proof ("antecedent assumption"), proof by contradiction, and proof by cases.

Alternate names are listed for some techniques.3

Learning Goals: In-Class

By the end of this unit, you should be able to:– Devise and attempt multiple different, appropriate

proof strategies—including all those listed in the “pre-class” learning goals plus use of logical equivalences, rules of inference, universal modus ponens/tollens, and predicate logic premises—for a given theorem.

– For theorems requiring only simple insights beyond strategic choices or for which the insight is given/hinted, additionally prove the theorem.

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Where We Are inThe Big Stories

Theory

How do we model computational systems?

Now: With our powerful modelling language (pred logic), we can begin to express interesting questions (like whether one algorithm is faster than another “in general”).

Hardware

How do we build devices to compute?

Now: We’ve been mostly on the theoretical side for

a while, and we’ll stay there for a littlewhile longer. Never fear, though, we’ll

return!

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Outline






Our “GenerallyFaster”

GenerallyFaster(a1, a2) = i Z+, n Z+, n i Faster(a1, a2, n).

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Alg A

Alg Bproblem size

time

Our Algorithms

(a) Ask each student for the list of their MUG-mates’ classes, and check for each class whether it is CPSC 121. If the answer is ever yes, include the student in my count.

(b) For each student s1 in the class, ask the student for each other student s2 in the class whether s2 is a MUG-mate. If the answer is ever yes, include s1 in my count.

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Alg A

Alg Bproblem size

time

Our Algorithms At Particular Sizes

(a) For 10 students: 10 minutes For 100 students: 100 minutes

For 400 students: 400 minutes

(b) For 10 students: ~10*10 seconds For 100 students: ~100*100 seconds

For 400 students: ~400*400 seconds

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Alg A

Alg Bproblem size

time

Proving “GenerallyFaster”

GenerallyFaster(a1, a2) = i Z+, n Z+, n i Faster(a1, a2, n).

Can we prove algA is generally faster than algB?

GenerallyFaster(algA, algB) i Z+, n Z+, n i Faster(algA, algB, n). i Z+, n Z+, n i 60n < n2.

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Alg A

Alg Bproblem size

time

(The last line is what we really mean in this case.)


Theorem: i Z+, n Z+, n i 60n < n2.

Which of these is the best overall description of this statement?

a.It’s a big “AND”.

b.It’s a big “OR”.

c.It’s a conditional.

d.It’s an inequality.

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Alg A

Alg Bproblem size

time



We can always pick out the “outermost” operator:i Z+, P(i), where…

P(i) = n Z+, n i 60n < n2

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Alg A

Alg Bproblem size

time




P(i) = n Z+, Q(i,n),

Q(i,n) = n i 60n < n2

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Alg A

Alg Bproblem size

time




P(i) = n Z+, Q(i,n),

Q(i,n) = R(i,n) S(n),R(i,n)= n i,S(n) = 60n < n2

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Alg A

Alg Bproblem size

time




P(i) = n Z+, Q(i,n),

Q(i,n) = R(i,n) S(n),R(i,n)= n i,S(n) = 60n < n2

So to get started, we can think about how to prove an existential…

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Alg A

Alg Bproblem size

time

Outline






Proof of Existence or “witness proofs”

Pattern to prove x D, R(x).

Prove R(x) for any one x in D. Pick the one that makes your job easiest!

The x you use for your proof is the “witness” to the existential… it “testifies” that your existential is true.

(We’re proving one of the disjuncts of a big “OR”.)

17proving

Witness Proof Example: A Touch of Brevity

Theorem: There’s a valid Racket program shorter than this (45-character) Java program:

class A{public static void main(String[]a){}}

Problem: prove the theorem.

18Where “valid” means “runnable using the java/racket commands with no flags”.

Proving “GenerallyFaster” Our Strategy So Far


We pick i = ??.

Then, we prove: n Z+, n i 60n < n2.

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Alg A

Alg Bproblem size

time

LEAVE this blank until you know what to pick.Take notes as you learn more about i.

Form of Our “TODO Item”

Partial Theorem: n Z+, n i 60n < n2.






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Alg A

Alg Bproblem size

time



We pick i = ??.

Then, we prove: n Z+, n i 60n < n2.

That’s the same as:

Q(i,n) = n i 60n < n2.

n Z+, Q(i,n).

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Alg A

Alg Bproblem size

time

So, how do we prove a universal?

Outline






Generalizing from the Generic Particular /Without Loss of Generality (WLOG)


Pick some arbitrary x, but assume nothing about which x it is except that it’s drawn from D

Then prove R(x).

That is: pick x “without loss of generality”!

23proving

Why Does This Work?


Pick some arbitrary x, but assume nothing about which x it is except that it’s drawn from D. Then prove R(x).

This is a big “AND”. To prove it, we must prove each “conjunct”.

Can we generate each individual proof from this one generic proof?

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WLOG Example: My Machine Speaks Racket

Theorem: Any valid Racket program can be represented in my computer’s machine language.


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WLOG Example: My Machine Speaks Racket

Theorem: Any valid Racket program can be represented in my computer’s machine language.

Proof: Without loss of generality, consider a valid Racket program p.

Since p is valid, my Racket interpreter (DrRacket) can interpret it on my computer. However, all commands that my computer runs are expressed in its machine language.

Therefore, p can be expressed (as the combination of the compiled interpreter and the input program) in my computer’s machine language.

QED

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We pick i = ??.

Without loss of generality, let n be an arbitrary positive integer.

Then, we prove: n i 60n < n2.

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Alg A

Alg Bproblem size

time

Form of Our “TODO Item”

Partial Theorem: n i 60n < n2.






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Alg A

Alg Bproblem size

time



We pick i = ??.


Then, we prove: n i 60n < n2.

With appropriate helpers, that’s just:R(i,n) S(n)

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Alg A

Alg Bproblem size

time

So, how do we prove a conditional?

Outline






A New Proof Strategy“Antecedent Assumption”

To prove p q:

Assume p.

Prove q.

You have then shown that q follows from p, that is, that p q, and you’re done.

But this is a prop logic technique?Can we use those for pred logic?

31proving

Why Does This Work?

To prove p q:

Assume p.

Prove q.

p q is “really” an OR like ~p q.

If our assumption is wrong, is the OR true?

If our assumption is right, is the OR true?

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Partly Worked Problem: Universality of NOR Gates

Theorem: If a circuit can be built from NOT gates and two-input AND, OR and XOR gates, then it can be built from NOR gates alone.


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Opening steps:

(1) Without loss of generality, consider an arbitrary circuit.

(2) [Assume the antecedent.] Assume the circuit can be built from NOT gates and two-input AND, OR and XOR gates.

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Insight: We can “rewrite” each of the gates in this circuit as a NOR gate. How?

AND OR XORNOT

Once you’ve shown this rewriting, you’ve proven the theorem.35


Which of these NOR gate configurations is equivalent to ~p?

e. None of these36

p

pT

pF

pq

a.

b.

c.

d.


Insight: Now that we can build NOT, can we rewrite the rest in terms of NOR and NOT?

AND OR XOR

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We pick i = ??.


Assume n i.

Then, we prove: 60n < n2.

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Alg A

Alg Bproblem size

time

So, how do we prove an inequality?

Outline






“Rules” for Inequalities

Proving an inequality is a lot like proving equivalence.

First, do your scratch work (often solving for a variable).

Then, rewrite formally:• Start from one side.• Work step-by-step to the other.• Never move “opposite” to your inequality (so, to

prove “<”, never make the quantity smaller).• Strict inequalities (< and >): have

at least one strict inequality step.

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We pick i = ??.


Assume n i.Then, we prove: 60n < n2.

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Alg A

Alg Bproblem size

time

Scratch work: We need to pick an i so that 60n < n2.

Scratch Work

Partial Theorem: 60n < n2.

We need to pick an i so that 60n < n2.

Let’s try solving for n in our scratch work!

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Alg A

Alg Bproblem size

time

Polished Work

Partial Theorem: 60n < n2.

With i = ____:

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Alg A

Alg Bproblem size

time

Outline






Finishing the Proof


We pick i = 61.


Assume n i.Observe that:60n < 61n

= i*n n*n (since n i)= n2

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Alg A

Alg Bproblem size

time QED!

Notation note…

Remember that this:

60n < 61n= i*n n*n= n2

Actually means this:

60n < 61n61n = i*ni*n n*nn*n = n2

Since 60n is less than 61n, and 61n is equal to i*n, 60n is less than i*n.

And, since i*n is less than or equal to n*n, 60n is less than n*n.

And so on…

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Alg A

Alg Bproblem size

time

How Did We Build the Proof?


We pick i = 61.


Assume n i.Observe that:60n < 61n

= i*n n*n (since n i)= n2

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Alg A

Alg Bproblem size

time

QED!

Strategies So Far

x D, P(x). with WLOG

x D, P(x). with a witness

p q by assuming the LHS

p q by proving each part

p q by proving either part

48Those last two are prop logic strategies,

and we can still use the rest of those as well!

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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More Practice: Always a Bigger Number

Prove that for any integer, there’s a larger integer.

Note: our proofs will frequently be purely in words now.BUT, translate the theorem into predicate logic

so you can structure your proof!This is x Z, y Z, y > x.

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More Practice: Always a Bigger Number


Which strategy or strategies should we use?

a.Witness proof alone

b.WLOG with a witness proof inside

c.Without loss of generality, twice.

d.Witness proof, twice.

e.None of these51

x Z, y Z, y > x

Worked Problem: Always a Bigger Number


Proof: Without loss of generality, let the first number x be an integer. Let the second number y be x + 1. Then, y = x + 1 > x. QED

The proof uses WLOG then witness.

52And… the predicate logic version makes that order obvious!

WLOG outside for x Z, witness inside for y Z.

x Z, y Z, y > x.

Outline






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Extra Slides

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Why Does This Work?

To prove p q:

Assume p.

Prove q.

By the way, does this look like a propositional logic proof? When is such a proof valid? When p q is a tautology!

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CPSC 121: Models of Computation 2012 Summer Term 2

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