CPSC 121: Models of Computation 2013W2

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CPSC 121: Models of Computation2013W2

Describing the World with Predicate Logic

Steve Wolfman, based on notes by Patrice Belleville and others

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Outline

• Prereqs, Learning Goals, and Quiz Notes

• Prelude: Scope and Predicate Definition

• Problems and Discussion– Liszt Etudes– Sorted Lists– Comparing Algorithms

• Next Lecture Notes

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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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Learning Goals: In-Class

By the end of this unit, you should be able to:– Build statements about the relationships

between properties of various objects—which may be real-world like “every candidate got votes from at least two people in every province” or computing related like “on the ith repetition of this algorithm, the variable min contains the smallest element in the list between element 0 and element i”)—using predicate logic.

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Where We Are inThe Big Stories

Theory

How do we model computational systems?

Now: Finally getting a tool that we can use to model the complex systems we really want to reason about!

Hardware

How do we build devices to compute?

Now: Still taking a bit of a vacation in lecture!

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Outline

• Prereqs, Learning Goals, and Quiz Notes• Prelude: Motivation, Scope & Defining Predicates

• Problems and Discussion– Liszt Etudes– Sorted Lists– Comparing Algorithms

• Next Lecture Notes

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Problem: Translate to Propositional Logic

• Steve is married to Rachel.

• Steve is married to Rachel and Rachel is married to Steve.

• Steve is not married to anyone other than Rachel.

10NO predicates allowed, just things like p, q, r, etc.

Limitations of Propositional Logic as a Model

Which of the following can propositional logic model effectively?

a. Relationships among factory production lines like “wheel assembly and frame welding both connect to the undercarriage line”.

b. Defining what it means for a number to be prime.c. Generalizing from examples to abstract patterns

like “everyone takes off their shoes at airport security”.

d. Prop logic can model all of these effectively.e. Prop logic cannot model any of these effectively.

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What Does Predicate Logic Model?

• Relationships among real-world objects

• Generalizations about patterns

• Infinite (or even just big) domains

12Problems where the properties of different concepts,

ideas, parts, or entities depend on each other.

But... Would You Ever Really Use Pred Logic?• Data Structures Example: “...every key is less than

or equal to all of its children’s keys...”• AI example: “...let h' be a ‘heuristic’ function

evaluating game states and h be the true value of the state. For all nodes n, h'(n) h(n)...”

• Java example: “...there is no path via references from any variable in scope to any memory location available for garbage collection...”

• Economics/elections example: “...for any distinct pair of candidates c1 and c2, if all voters prefer c1 to c2, then society must rank c1 above c2...”

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Quantifier Syntax

A quantifier in CPSC 121 is always of the form:

_____ _____ ______, _______________ or variable set subexpression that can

mention the variable

Together, this forms a single subexpression.

For example, x N, (x > 3) (x + 4 < 12).

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Quantifier Syntax

You cannot treat any part of this (like x Z or x) as a subexpression. It just isn’t.

So, no saying: (x Z) p.

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Quantifier Scope

A quantifier applies to everything to its right until a closing parenthesis stops it.

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Quantifier Scope

A quantifier applies to everything to its right until a closing parenthesis stops it.

x D, (y E, Q(x,y) z F, R(y,z)) P(x).

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one subexpression (all under the universal)

Quantifier Scope

A quantifier applies to everything to its right until a closing parenthesis stops it.

x D, (y E, Q(x,y) z F, R(y,z)) P(x).

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a smaller subexpression

Quantifier Scope

A quantifier applies to everything to its right until a closing parenthesis stops it.

x D, (y E, Q(x,y) z F, R(y,z)) P(x).

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another smaller subexpression (all under the existential)

Quantifier Scope

A quantifier applies to everything to its right until a closing parenthesis stops it.

x D, (y E, Q(x,y) z F, R(y,z)) P(x).

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another smaller subexpression

A Bit of Syntax: Quantifier Scope

Which of the following placements of parentheses yields the same meaning as: x Z, y Z, x < y Even(y).

a.()x Z, y Z, x < y Even(y).

b.(x) Z, y Z, x < y Even(y).

c.(x Z), y Z, x < y Even(y).

d.(x Z, y Z, x < y) Even(y).

e.(x Z, y Z, x < y Even(y)).21

Reminder: Z is the set of integers.

Aside: Where Can You “Cut” a Propositional Logic Statement?

This doesn’t make sense: (x w) ( y).

Why not?

Every use of looks just like every other use. They all look like p q.

The left side must make sense as an equivalent to p: p x w here.

The right side must make sense as an equivalent to q. Does q y make sense?

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Aside: Where Can You “Cut” a Predicate Logic Statement?

This doesn’t make sense: x Z Tasty(y).

Why not?

Every use of looks just like every other use. They all look like P(x, y, …) Q(x, y, …).

The RIGHT side must make sense as an equivalent to Q(…). Q(y) Tasty(y) here.

The left side must make sense as an equivalent to P(…): Does P(x) x Z make sense?

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Aside: Where Can You “Cut” a Predicate Logic Statement?

So does this make sense: x Z, Costs(y, x) Tasty(y)?

Does it look like P(x, y, …) Q(x, y, …)?

P(y) x Z, Costs(y, x)?

Q(y) Tasty(y)?

No! The is inside the universal quantifier. It applies to Costs(y, x) and Tasty(y).

P(x, y) Costs(y, x).

Q(y) Tasty(y).

x Z, P(x, y) Q(y).24

Aside: Where Can You “Cut” a Predicate Logic Statement?

How about: (x Z, Costs(y, x)) (w F, Salty(w))?

P(y) x Z, Costs(y, x).

q w F, Salty(w).

Note: q alone is essentially a predicate Q() with no arguments.

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A Bit of Syntax: Negation Scope

Which of the following placements of parentheses yields the same meaning as: ~x Z+, y Z+, x < y Even(y).

a.(~)x Z+, y Z+, x < y Even(y).

b.(~(x)) Z+, y Z+, x < y Even(y).

c.(~(x Z+)), y Z+, x < y Even(y).

d.(~(x Z+, y Z+, x < y)) Even(y).

e.(~(x Z+, y Z+, x < y Even(y))).26

A Bit of Semantics: Unbound Variables

What is the truth value of the following formula? x Z, x*x = y.

(Warning: a bit of a trick question.)

a.True, because (for example) 5*5=25.b.True, because every y=(sqrt y)*(sqrt y)c. False, because of counterexamples like no

integer multiplied by itself equals 3.d. It depends on y, but given a value for y, we

could calculate a truth value.e.None of the above.

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5

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Defining a Predicate Using Expressions with “Unbound” Variables

A pred. logic formula with only bound variables is a proposition, something that is either true or false:

x Z, x*x = 25. true

x Z, x*x = 3. false

x Z, x*x = robson. false (malformed?)

y Z, x Z, x*x = y. false

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5

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Defining a Predicate Using Expressions with “Unbound” Variables

A pred. logic formula with unbound variables is itself a predicate, something whose truth depends on its unbound variables’ values:

PerfectSquare(y): x Z, x*x = y, where y Z

PerfectSquare(25). true

PerfectSquare(3). false

PerfectSquare(robson). malformed

y Z, PerfectSquare(y). false

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5

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Notice that this is much like defining a function in programming:(1) The meaning of the predicate/function depends on the argument given.(2) You can use the predicate/function over and over again.

Unbound Variables Check

Which variable(s) are unbound?

(In other words, which variables does this formula’s truth depend on?)

i N, (i > n) ~v N, Elt(a, i, v).

a.i and v

b.a and n

c.n and v

d.i and n

e.None of these is correct.30

(DO NOT worry about what it means!)

Outline

• Prereqs, Learning Goals, and Quiz Notes

• Prelude: Scope and Predicate Definition

• Problems and Discussion– Liszt Etudes

• Intermezzo on Idioms

– Sorted Lists– Comparing Algorithms

• Next Lecture Notes

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Modeling ListOfInteger

Let Elt(a, i, v) be a predicate indicating that list a at index i has the integer value v, where indexes must be 0.

2 4 5 7 6 10

0 1 2 3 4 5

Elt(mylist, 3, 7) is true.Elt(mylist, 2, 0) and Elt(mylist, 7, 2) are false.

mylist

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ListOfInteger

;; ListOfInteger is one of:

;; - empty

;; - (cons Integer ListOfInteger)

Also:

(where index 0 gets us the first element)

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List Element Warmup

Elt(a, i, v) list a has value v at index i.

Which of the following should describe a valid list (assume all other Elt(x, y, z) are false)?

a.Elt(list, 0, 7), Elt(list, 1, 4), Elt(list, 2, 3)

b.Elt(list, 0, 7), Elt(list, 1, 4), Elt(list, 3, 3)

c.Elt(list, -1, 7), Elt(list, 0, 4), Elt(list, 1, 3)

d.Elt(list, 0, 7), Elt(list, 1, 7), Elt(list, 1, 3)

e.Nothing. (~i N, v Z, Elt(list, i, v).)

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List Element Exercise

Define a predicate Contains(a, v) meaning “list a contains the value v”.

Contains(a, v):a.a A, i N, v Z, Elt(a,i,v)b. a A, i N, v Z, Elt(a,i,v)c. i N, Elt(a,i,v)d. i N, Elt(a,i,v)e. None of these

35Hint: As with HtDP, design some tests. An empty

list, a list with one element, a list with two, etc.

List Length Properties

We’ll agree on all of the following properties for a list:

• Every list has a length.

• No list has two different lengths.

• No list has elements at any index as large or larger than (“beyond”) its length.

• A list has exactly one element at every index up to (not including) its length.

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Outline

• Prereqs, Learning Goals, and Quiz Notes

• Prelude: Scope and Predicate Definition

• Problems and Discussion– Liszt Etudes

• Intermezzo on Idioms

– Sorted Lists– Comparing Algorithms

• Next Lecture Notes

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Idioms: “a form of expression natural to a language”

To define lists, we need to say things like: •“No list has property P”, •“Some list has…”, •“Every list has…”, •“Some non-empty lists have…”, •“All non-empty lists have…”•“At most one element in a list has…”•“Exactly one element in a list has…”

These are common predicate logic “idioms”. Let’s learn them.

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Idioms: “None, some, not every, all”

Let A be the set of animals.

Honey(x) means “x likes honey”

Bear(x) means “x is a bear”

None of the animals likes honey: ~x A, Honey(x).

Some of the animals like honey: x A, Honey(x).

Not every animal likes honey: ~x A, Honey(x).

All animals like honey: x A, Honey(x).

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Idioms: “None, some, not every, all” (put another way)

Let A be the set of animals.

Honey(x) means “x likes honey”

Bear(x) means “x is a bear”

Every animals dislikes honey: x A, ~Honey(x).

Not every animal dislikes honey: ~x A, ~Honey(x).

Some animal dislikes honey: x A, ~Honey(x).

No animal dislikes honey: ~x A, ~Honey(x).

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Idioms: “Restricting the Domain of an Existential”

Let A be the set of animals.

Honey(x) means “x likes honey”

Bear(x) means “x is a bear”

Some animal likes honey:

x A, Honey(x).

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Idioms: “Restricting the Domain of an Existential”

Let A be the set of animals.

Honey(x) means “x likes honey”

Bear(x) means “x is a bear”

Some bear likes honey. We’d like to say:

x SetOfBears, Honey(x).

But we don’t have the set of bears.

How do we “restrict” the set of animals to just bears for an existential? You did it on the quiz!

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Idioms: “Restricting the Domain of an Existential”

Let A be the set of animals.

Honey(x) means “x likes honey”

Bear(x) means “x is a bear”

Some bear likes honey…Some animal is a bear and also likes honey…There is an animal that is a bear and likes honey.

x A, Bear(x) Honey(x).

43This far, you did on the quiz!

Idioms: “Restricting the Domain of an Existential”

Let A be the set of animals.

Honey(x) means “x likes honey”

Bear(x) means “x is a bear”

But wait.. by the commutative law:

x A, Bear(x) Honey(x)

x A, Honey(x) Bear(x)

So that also means “Some honey-liker is a bear.”

Yes, “some bear likes honey” is the same as “some honey-liker is a bear.”

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Idioms: “Restricting the Domain of an Existential”

More generally:

Given set D and predicate P(x), you want to say “Some <thing in D for which P is true> has the property Q.”

You say: x D, P(x) Q(x).

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Some <thing in D for which P is true>…

Idioms: “Restricting the Domain of a Universal”

Let A be the set of animals.

Honey(x) means “x likes honey”

Bear(x) means “x is a bear”

Does x A, Bear(x) Honey(x) mean “Every bear likes honey”?

(Try this test case: Steve is a wolf who dislikes honey; George is a bear who likes honey. Try to pick the best answer.)

a.Yes.

b.No, because then no bear could dislike honey.

c.No, because then all wolves would have to like honey.

d.No, because then there could be no wolves.

e.There’s not enough information to tell.

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Idioms: “Restricting the Domain of a Universal”

Let A be the set of animals.

Honey(x) means “x likes honey”

Bear(x) means “x is a bear”

“Every bear likes honey.”

So, we must let wolves (and other non-bears) off the hook:“Every animal likes honey (or isn’t a bear so we don’t care).”

x A, Honey(x) ~Bear(x).

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Every animal likes honey… or isn’t a bear (so we don’t care).

Idioms: “Restricting the Domain of a Universal”

Let’s play with logical equivalences…

“Every bear likes honey.”

x A, Honey(x) ~Bear(x)

x A, ~Bear(x) Honey(x) [by COM]

x A, Bear(x) Honey(x) [by IMP]

For every animal: if it’s a bear, it must like honey. (If it’s not a bear, we don’t care whether it likes honey.)

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Idioms: “Restricting the Domain of a Universal”

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More generally:

Given set D and predicate P(x), you want to say “All <things in D for which P is true> have the property Q.”

You say: x D, P(x) Q(x).

All <things in D for which P is true>…

Idioms Questions

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Let P be the set of Racket Programs.

Let Buggy(x) be true iff the program x contains errors.

Let Terminate(x) be true iff the program x ever terminates (finishes running).

Which of these means:“Every non-terminating program is buggy.”

a.p P, ~Terminate(p) Buggy(p).

b.p P, ~Terminate(p) Buggy(p).

c.p P, ~Terminate(p) Buggy(p).

d.p P, ~Terminate(p) Buggy(p).

e.None of these.

Idioms Questions

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Let P be the set of Racket Programs.

Let Buggy(x) be true iff the program x contains errors.

Let Terminate(x) be true iff the program x ever terminates (finishes running).

Which of these means:“Some buggy program does not terminate.”

a.p P, Buggy(p) ~Terminate(p).

b.p P, Buggy(p) ~Terminate(p).

c.p P, Buggy(p) ~Terminate(p).

d.p P, Buggy(p) ~Terminate(p).

e.None of these.

Idioms: “At Least One”, “At Least Two”

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“At least one animal likes honey.” means the same as “Some animal likes honey.”

x A, Honey(x).

“At least two animals like honey.” means “Some animal and some other animal like honey.” or“Some animal and some animal that is not the same animal like honey.” Just “restricting the domain”!

Idioms: “At Least One”, “At Least Two”

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“At least one animal likes honey.” means the same as “Some animal likes honey.”

x A, Honey(x).

“At least two animals like honey.” means “Some animal and some other animal like honey.” or“Some animal and some animal that is not the same animal like honey.”

x A, Honey(x) y A, x y Honey(y) x A, y A, x y Honey(x) Honey(y).

Just “restricting the domain”!

(We can move Honey(x) inside the that doesn’t mention x,but beware of negations, (bi)conditionals, and XORs!)

Idioms: “At Most One”

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“No things in A…” ~x A, …

“At least one thing in A…” x A, …

“At least two things in A…” x A, y A, x y … …

“At most one thing in A…” means the same as “It is not the case that at least two things in A…”

~x A, y A, x y … …

The same predicate applied to x and y.

The same predicate applied to x and y.

Idioms: “Exactly One”

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“Exactly one thing in A…” means both “at least one thing in A” and “at most one thing in A…”

“Exactly one animal likes honey.”

“Some animal likes honey and no animal that is not that animal likes honey.”

So, either:

z A, Honey(z) ~x A, y A, x y Honey(x) Honey(y).

Or just:

x A, Honey(x) ~y A, x y Honey(y).

Idiom Summary

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“None…”/“No x…” ~x D, …

“At least one…”/“Some…”/“A (particular) x…”

x D, …

“Every…”/“All…”/“Any x…”/“A (arbitrary) x…”

x D, …

“Some P-ish x…” (restricting the domain)

x D, P(x) …

“Every P-ish x…” (restricting the domain)

x D, P(x) …

“At least two…” x D, yD, xy … …

“At most one…” ~x D, y D, xy … …

“Exactly one…” x D, … ~y D, xy …

And yes, sometimes “An x has property P” means “Some” and sometimes it means “All”

In each case with two “…” below, should be the same predicate applied to x and y.

Outline

• Prereqs, Learning Goals, and Quiz Notes

• Prelude: Scope and Predicate Definition

• Problems and Discussion– Liszt Etudes

• Intermezzo on Idioms (DONE)

– Sorted Lists– Comparing Algorithms

• Next Lecture Notes

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List Length Properties

Reminder, we want to say:• Every list has a length.• No list has two different lengths.• No list has elements at any index as large

or larger than (“beyond”) its length.• A list has exactly one element at every

index up to (not including) its length.

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List Length Warmup

Elt(a, i, v) a has value v at index i.

Let Length(a, n) mean list a is n items long.

What does a A, n N, Length(a, n) mean?

(Hint: first figure out what n N, Length(mylist, n) means; then, switch from my list to a A,…)

a. Every list is n items long.

b. There are many lists that are n items long.

c. Every list has a length.

d. No list has more than one length.

e. None of these.

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We will assume henceforth that this statement is true.

List Element Exercise

Elt(a, i, v) a has value v at index i. Let A be the set of all lists.Which means: “a list has at most one element at any index”.a. a A, i N, v Z, Elt(a,i,v).

b. a A, i N, ~v1 Z, v2 Z, Elt(a,i,v1) Elt(a,i,v2).

c. a A, i N, ~v1 Z, v2 Z, v1 v2 Elt(a,i,v1) Elt(a,i,v2).

d. a A, i N, v1 Z, v2 Z, (Elt(a,i,v1) Elt(a,i,v2)) v1 = v2.

e. All of these We will assume henceforth that a list cannot have more than one element at any index.

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Better “No More Than One”?

Use your knowledge of how to break down problems into separate functions!

If “a list has at most one element at any index” is too hard, write “list a has at least two elements at index i”.

That’s a new predicate! You can break down complex predicates (or statements) in terms of simpler predicates, just as you break down complex functions in terms of simpler ones.

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Better “No More Than One”?

If “a list has at most one element at any index” is too hard, write “the list a has at least two elements at index i”.

Let’s make the predicate:

MultipleElts(a,i) v1 Z, v2 Z, Elt(a,i,v1) Elt(a,i,v2)

That’s not quite right. What’s missing?

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Better “No More Than One”?

We wrote “the list a has at least two elements at index i”:

MultipleElts(a,i) v1 Z, v2 Z, v1 v2 Elt(a,i,v1) Elt(a,i,v2).

Now, how do we write “a list has at most one element at any index”? In other words: “a list does not have multiple elements at any index.”

Easy: a A, i N, ~MultipleElts(a,i).

63Go back and check that this is the same as what we got in the clicker question!

Assumptions/Definitions for Lists

• A list cannot have more than one element at any indexa A, i N, ~v1 Z, v2 Z,

v1 v2 Elt(a,i,v1) Elt(a,i,v2).

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Assumptions/Definitions for Lists

• A list cannot have more than one element at any indexa A, i N, ~v1 Z, v2 Z,

v1 v2 Elt(a,i,v1) Elt(a,i,v2).

• Every list has a length.a A, n N, Length(a, n).

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No More than One Length

Which of the following means that no list has more than one length?a. a A, n1 Z, n2 Z,

n1 n2 (~Length(a,n1) ~Length(a,n2)).

b. a A, ~n1 N, n2 N, Length(a,n1) Length(a,n2) n1 n2.

c. a A, ~n1 N, n2 N, n1 n2 (Length(a,n1) Length(a,n2)).

d. a A, n N, Length(a,n1) ~Length(a,n2).

e. All of these

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We will assume henceforth that this statement is true.

Defining Length(a, n)

Consider: Length(a,n) i N, (v N, Elt(a, i, v)) (i < n).

In English: There’s an element at i if and only if i is a valid index (less than the list’s length).

Why does this guarantee that no list has more than one length?a. Because it states that no list has more than one length.b. Because a list with two lengths would now both have and not

have a value at some entry.c. Because i now must be less than the length of the list.d. None of these, but it is guaranteed.e. None of these, and it’s not guaranteed.

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We will assume henceforth that this statement is true.

Assumptions/Definitions for Lists

• A list cannot have more than one element at any index.a A, i N, ~v1 Z, v2 Z,

v1 v2 Elt(a,i,v1) Elt(a,i,v2).

• Every list has a length.a A, n N, Length(a, n).

• A list “has a length n” exactly when there’s an element at i if and only if i is less than n. Length(a,n) i N, (v Z, Elt(a, i, v)) (i < n).– So: No list has more than one length, and

– there’s an element at every index up to the list’s length, and

– There’s no element at any index greater than or equal to length.68

Outline

• Prereqs, Learning Goals, and Quiz Notes

• Prelude: Scope and Predicate Definition

• Problems and Discussion– Liszt Etudes– Sorted Lists– Comparing Algorithms

• Next Lecture Notes

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Problem: Sorted Lists

Problem: Give a definition for the predicate Sorted(a) in terms of Elt(a, i, v).

Assume lists cannot have more than one element at an index, every list has a length, and lists are “filled” to their length.

?

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Problem: Sorted Lists

Problem: Give a definition for the predicate Sorted(a) in terms of Elt(a, i, v).

Which of the following is a problem with this definition?Sorted(a) i N, Elt(a,i,v1) < Elt(a,i+1,v2).

a. a isn’t quantified.

b. v1 and v2 aren’t quantified.

c. We can’t use < on Elt (or any other predicate)

d. a and b

e. b and c

?

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Problem: Sorted Lists

Problem: Give a definition for the predicate Sorted(a) in terms of Elt(a, i, v).

What’s wrong with the following definition?Sorted(a) i N, v1 Z, v2 Z,

(Elt(a,i,v1) Elt(a,i+1,v2)) v1 < v2.

a. It’s missing quantifiers.

b. It’s too restrictive (e.g., for equal values).

c. It doesn’t handle the “boundary case” when i = length-1.

d. a and b

e. b and c

?

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Intuition ♥ Formality

“...when we become comfortable with formal manipulations, we can use them to check our intuition, and then we can use our intuition to

check our formal manipulations.”

- Epp (3rd ed), p. 106-107

We’ll often use predicate logic informally in the future, but the ability to express and reason about ideas formally keeps us honest and helps us discover points we may overlook otherwise.

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Outline

• Prereqs, Learning Goals, and Quiz Notes

• Prelude: Scope and Predicate Definition

• Problems and Discussion– Liszt Etudes– Sorted Lists– Comparing Algorithms

• Next Lecture Notes

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Efficiency of Algorithms

Let’s say each student is in a “MUG” (1st year orientation group). For each of their MUG-mates, each student has a list of all of their classes.

Assume each MUG has 13 students and each student is taking 5 classes.

I want to determine how many students in my class have a MUG-mate in my class.

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Which algorithm is generally faster?

(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.

(c) Neither.76

Concrete Examples: 10 students

Say checking if a class on a list is CPSC 121 takes 1 second and checking if a classmate is in your MUG takes 1 second.

Algorithm (a) takes ~10*12*5 seconds = 10 minutes.

Algorithm (b) takes ~10*10 seconds < 2 minutes.

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Concrete Examples: 100 students

Say checking if a class on a list is CPSC 121 takes 1 second and checking if a classmate is in your MUG takes 1 second.

Algorithm (a) takes ~100*12*5 seconds = 100 minutes.

Algorithm (b) takes ~100*100 seconds 167 minutes.

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Concrete Examples: 400 students

Say checking if a class on a list is CPSC 121 takes 1 second and checking if a classmate is in your MUG takes 1 second.

Algorithm (a) takes ~400*12*5 seconds 7 hours.

Algorithm (b) takes ~400*400 seconds 44 hours.

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Which algorithm is generally faster?

(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.

(c) Neither.80

Comparing at One Input Size

Let the predicate Faster(a1, a2, n) mean algorithm a1 is faster than algorithm a2 on a problem of size n, where n is a positive integer.

Alg A

Alg Bproblem size

time

81We’ll assume the Faster predicate is given to us.

How Faster Works (1 of 3)

Faster(a1,a2,n): a1 is faster than a2 at size n.Which of the following means “no algorithm is

ever faster than itself”?a.n Z+, a A, ~Faster(a,a,n).b. n Z+, a1 A, a2 A,

Faster(a1,a2,n) ~Faster(a2,a1,n).

c. n Z+, a1 A, a2 A, a3 A, (Faster(a1,a2,n) Faster(a2,a3,n))

Faster(a1,a3,n).

d.None of these.82

Alg A

Alg Bproblem size

time

We will assume this statement is true.

How Faster Works (2 of 3)

Faster(a1,a2,n): a1 is faster than a2 at size n.

Which of the following means “two algorithms cannot be faster than each other”?

a. n Z+, a1 A, a2 A, Faster(a1,a2,n) ~Faster(a2,a1,n).

b. n Z+, a1 A, a2 A, Faster(a1,a2,n) ~Faster(a2,a1,n).

c. n Z+, a1 A, a2 A, Faster(a1,a2,n) Faster(a2,a1,n).

d. b and c.e. All of these.

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Alg A

Alg Bproblem size

time

We will assume this statement is true.

How Faster Works (3 of 3)

Faster(a1,a2,n): a1 is faster than a2 at size n.

What does the following statement mean?n Z+, a1 A, a2 A, a3 A,

(Faster(a1,a2,n) Faster(a2,a3,n)) Faster(a1,a3,n).

a. Three algorithms cannot be faster than each other.b. Three algorithms are the same “speed”.c. Of any three algorithms, one is the fastest.d. An algorithm cannot be faster than itself.e. None of these.

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Alg A

Alg Bproblem size

time

We will assume this statement is true.

How Faster Works (4 of 3, bonus!)

Faster(a1,a2,n): a1 is faster than a2 at size n.

We assume Faster is:• “Anti-reflexive”: No algorithm is faster than itself.• “Anti-symmetric”: No two algorithms are faster

than each other.• “Transitive”: If one algorithm is faster than a

second, which is faster than a third, then the first algorithm is faster than the third. (We can “chain” fasters together, analagously to the transitivity rule for propositional logic.)

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Alg A

Alg Bproblem size

time

General Efficiency of Algorithms

Faster(a1,a2,n): a1 is faster than a2 at size n.

Problem: Create a definition of GenerallyFaster(a1, a2) in terms of Faster(a1, a2, n) that you can live with.

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Alg A

Alg Bproblem size

time

Desirable Properties of Generally Faster

Which of these properties should Generally Faster share with Faster?

a.Anti-reflexivity

b.Anti-symmetry

c.Transitivity

d.Should share all of these.

e.Should share none of these.

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Alg A

Alg Bproblem size

time

English-Language Generally Faster Definitions

Which one do you want? [Your definitions here.]

a.

b.

c.

d. None of these.

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Alg A

Alg Bproblem size

time

FormalGenerally Faster Definitions

Which one do you want? [Your definitions here.]

a.

b.

c.

d. None of these.

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Alg A

Alg Bproblem size

time

Which algorithm is generally faster?

(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.

(c) Neither.90

Alg A

Alg Bproblem size

time

Outline

• Prereqs, Learning Goals, and Quiz Notes

• Prelude: Scope and Predicate Definition

• Problems and Discussion– Liszt Etudes– Sorted Lists– Comparing Algorithms

• Next Lecture Notes

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Learning Goals: In-Class

By the start of class, you should be able to:– Build statements about the relationships

between properties of various objects—which may be real-world like “every candidate got votes from at least two people in every province” or computing related like “on the ith repetition of this algorithm, the variable min contains the smallest element in the list between element 0 and element i”) using predicate logic.

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Next Lecture Learning Goals: Pre-Class

By the start of class, you should be able to:– Determine the negation of a quantified statement in logical

notation as well as English. (But, feel free to entirely or partially translate English to logic in the process!)

– Given a quantified statement and an equivalence rule, apply the rule to create an equivalent statement (particularly the De Morgan’s and contrapositive rules).

– Prove and disprove quantified statements using the “challenge” method (Epp, 3rd ed 98-99, Epp, 4th ed 118-119).

– Apply universal instantiation, universal modus ponens, and universal modus tollens to predicate logic statements that correspond to the rules’ premises to infer statements implied by the premises.

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Next Lecture Prerequisites

Reread Sections 3.1 and 3.3 (including the negation part that we skipped previously).

Read Sections 3.2 and 3.4.(See course website for other texts’ sections.)(You needn’t learn the “diagram” technique, but

it may make more sense than other explanations!)

Complete the open-book, untimed quiz that’s due before the next class.

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snick

snack

More problems to solve...

(on your own or if we have time)

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Problem: Java Collections

Problem: Translate the following text from the Java 1.6.0 API page for the Collection interface into predicate logic.[T]he specification for the contains(Object o) method says: "returns true if and only if this collection contains at least one element e such that (o==null ? e==null : o.equals(e))."

c ? a : b acts essentially like a multiplexer.If c is true, it evaluates to a; otherwise, it evaluates to b.

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Problem: Java Collections

Problem: The API goes on to say:This specification should not be construed to imply that invoking Collection.contains with a non-null argument o will cause o.equals(e) to be invoked for any element e.

Explain whether and how this is consistent with your definition.

97

More Quantifier Examples

Someone is in charge.

Everyone except the person in charge reports to someone else.

98

More Quantifier Examples

n is a prime number.

Note: we use x|y as a predicate meaning x divides y (i.e., x “goes into” y with no remainder).99

More Quantifier Examples

n is a prime number.

Let’s define a new predicate P(x) in terms of this “clause”. Then, let’s express…

There’s some prime number larger than 10.

There’s some prime number larger than every natural number.

100

Yet More Examples

Eating food causes Alice to grow or shrink.

Solution:

x F, E(x) g s.

F = set of all foods

E(x): Alice eats x

g: Alice grows

s: Alice shrinks

101

Yet More Examples

Alice shrank when she ate some food.

Solution:

x F, E(x) s.

F = set of all foods

E(x): Alice eats x

g: Alice grows

s: Alice shrinks

102

Yet More Examples

All lions are fierce.

Solution:

x F, L(x) F(x).

F(x): x is a fierce creature

L(x): x is a lion

C(x): x drinks coffee

Domain for all is the set of all creatures.

103

Yet More Examples

Some lions do not drink coffee.

F(x): x is a fierce creature

L(x): x is a lion

C(x): x drinks coffee

Domain for all is the set of all creatures.

104

Yet More Examples

All fierce creatures are not lions.

F(x): x is a fierce creature

L(x): x is a lion

C(x): x drinks coffee

Domain for all is the set of all creatures.

Is that English sentence ambiguous?105

Yet More Examples

Is x, K(x, y) a proposition?

Why or why not?

F(x): x is a fierce creatureL(x): x is a lionC(x): x drinks coffee

K(x, y): x has been in y’s kitchen

Domain for all is the set of all creatures (with kitchens?).

106

Yet More Examples

Every creature has been in its own kitchen.

Some creature has not been in its own kitchen.

F(x): x is a fierce creatureL(x): x is a lionC(x): x drinks coffee

K(x, y): x has been in y’s kitchen

Domain for all is the set of all creatures (with kitchens?).

107

Yet More Examples

There is a creature that has been in every creature’s kitchen.

Every creature’s kitchen has had some creature in it.

F(x): x is a fierce creatureL(x): x is a lionC(x): x drinks coffee

K(x, y): x has been in y’s kitchen

Domain for all is the set of all creatures (with kitchens?).

Are these the same?108

Yet More Examples

Every creature has been in every other creature’s kitchen.

Every creature’s kitchen has had every creature in it.

F(x): x is a fierce creatureL(x): x is a lionC(x): x drinks coffee

K(x, y): x has been in y’s kitchen

Domain for all is the set of all creatures (with kitchens?).

Are these the same?What if we removed the word “other”?

109

Problem: Voting Database

Consider a database that tracks the votes in an election. In the database, the predicate Tally(d, c, n) means that district d reported that candidate c received n votes, where n is an integer 0.

Problem: Define a predicate GotVote(c) in terms of Tally whose truth set is the set of all candidates who received at least one vote.

110

Problem: Voting Database

Problem: Define a predicate whose truth set is the set of all candidates who won at least one district.

Why work so hard on defining predicates?This is essentially how we query databases.111

Problem: Voting Database

Let’s assume that every candidate has exactly one vote total for every district. That is, there’s no missing and no duplicate data.

Problem: Write a logical statement that describes this constraint.

Predicates are clumsy for expressing this common idea.Later, we’ll use functions to do a better job.

112

Problem: Voting Database

Let Winner(c) indicate that candidate c is the winner of the election.

Problem: Write a logical statement that means that the winner of the election must have received at least one vote.

113

Problem: Voting Database

Let D be the set of all districts and C be the set of all candidates.

Problem: Determine what the following statement means, whether it is necessarily true (no matter what the actual vote tallies are), and justify your stance:

c1C, dD, c2C, Winner(c1) c1c2 n1Z, n2Z, Tally(d,n1,c1) Tally(d,n2,c2) n1>n2

114