Discrete Mathematics - University of Colorado...

transcript

Discrete Mathematics

Jeremy Siek

Spring 2010

Jeremy Siek Discrete Mathematics 1 / 118

Outline of Lecture 1

1. Course Information

2. Overview of Discrete Mathematics

Course Information

I Class web page:http://ecee.colorado.edu/~siek/ecen3703/spring10

I Textbooks:I Discrete Mathematics and its Applications, 6th Edition, by Rosen. (At

the CU bookstore.)I A Tutorial Introduction to Structured Isar Proofs, by Nipkow.

(Available online.)I Isabelle/HOL – A Proof Assistant for Higher-Order Logic, by Nipkow,

Paulson, and Wenzel. (Available online.)I How to Prove It: A Structured Approach, by Daniel J. Velleman.

I Grading:Quizzes 30%Midterm exam 30%Final exam 40%

Course Information: Homework

I There are weekly homework assignments.I The quizzes and exams are based on the homework.I Every students gets a personal tutor named Isabelle. Isabelle is a

logic language, a programming language, and a most importantly,a proof checker.http://www.cl.cam.ac.uk/research/hvg/Isabelle/

I You know your proofs are correct when you convince Isabelle.

Overview of Discrete Mathematics

Discrete

Mathematics

I What is Math anyways?

I Is it the study of numbers?I Mathematics is actually much more broad.

DefinitionMathematics is the study of any truth regarding well-definedconcepts.

Numbers are just one kind of well-defined concept.

Mathematics

I What is Math anyways?I Is it the study of numbers?

I Mathematics is actually much more broad.

Mathematics

I What is Math anyways?I Is it the study of numbers?I Mathematics is actually much more broad.

Mathematics

I What is Math anyways?I Is it the study of numbers?I Mathematics is actually much more broad.

Discrete

DefinitionSomething is discrete if is it composed of distinct, separable parts. (Incontrast to continuous.)

Discrete Continuousintegers real numbersgraphs rational numbers

state machines differential equationsdigital computer radiosquantum physics Newtonian physics

Discrete Mathematics

DefinitionDiscrete Mathematics is the study of any truth regarding discreteentities.

I That’s pretty broad. So what is it really?I Discrete math is the foundation for the rigorous understanding of

computer systems.

A Discrete Problem: Sudoku

I What are the rules of Sudoku?

I Spend the next few minutes filling in this board.I Write down the rules of Sudoku on a sheet of paper.I Pass your paper to the person on your right. Are the rules that

you’ve been passed correct? If not, give an example.

I What are the rules of Sudoku?I Spend the next few minutes filling in this board.

I Write down the rules of Sudoku on a sheet of paper.I Pass your paper to the person on your right. Are the rules that

I What are the rules of Sudoku?I Spend the next few minutes filling in this board.I Write down the rules of Sudoku on a sheet of paper.

I Pass your paper to the person on your right. Are the rules thatyou’ve been passed correct? If not, give an example.

I What are the rules of Sudoku?I Spend the next few minutes filling in this board.I Write down the rules of Sudoku on a sheet of paper.I Pass your paper to the person on your right. Are the rules that

Abstracting Sudoku

I What aspects of the game of Sudoku don’t really matter?I What could you change such that an expert Sudoku player would

immediately be an expert of the modified game?I What aspects of the game really matter?

Sudoku Solver

I Write down a pseudo-code algorithm for solving Soduku.I What data structures did you use?I What kind of algorithm did you use?I Does your algorithm always solve the puzzle?I How long does your algorithm take to finish in the worst case?

Why Study Discrete Mathematics?

I It’s the basic language used to discuss computer systems. Youneed to learn the language if you want to converse with othercomputer professionals.

I It’s a toolbox full of the problem-solving techniques that you willuse over and over in your career.

I But best of all, studying discrete math will enhance your mind,turning it into a high-precision machine!

Uses of Discrete Math are Everywhere

I Circuit designI Computer architectureI Computer networksI Operating systemsI Programming: algorithms and data structuresI Programming languagesI Computer security, encryptionI Error correcting codesI Graphics algorithms, game enginesI . . .

Themes in Discrete Math

Mathematical Reasoning: read, understand, and create precisearguments.

Discrete Structures: model discrete systems and study theirproperties.

Algorithmic Thinking: create algorithms, verify that they work,analyze their time and space requirements.

Combinatorial Analysis: counting (not always as easy as it sounds!)

Advice

I Read in advance.I Do the homework.I Form a study group.I Form an intense love/hate relationship with Isabelle.

1. Propositional Logic

2. Syntax and Meaning of Propositional Logic

I Logic defines the ground rules for establishing truths.I Mathematical logic spells out these rules in complete detail,

defining what constitutes a formal proof.I Learning mathematical logic is a good way to learn logic because

it puts you on a firm foundation.I Writing formal proofs in mathematical logic is a lot like computer

programming. The rules of the game are clearly defined.

Propositional Logic

I Propositional logic is a language that abstracts away from contentand focuses on the logical connectives.

I Uppercase letters like P and Q are meta-variables that areplaceholders for propositions.

I The following rules define what is a proposition.

I A propositional variable (lowercase letters p, q, r) is aproposition. These variables model true/false statements.

I The negation of a proposition P, written ¬ P, is a proposition.I The conjunction (and) of two propositions, written P ∧ Q, is a

proposition.I The disjunction (or) of two propositions, written P ∨ Q, is a

proposition.I The conditional statement (implies), written P −→ Q, is a

proposition.I The Boolean values True and False are propositions.

Propositional Logic

I The following rules define what is a proposition.I A propositional variable (lowercase letters p, q, r) is a

proposition. These variables model true/false statements.

I The negation of a proposition P, written ¬ P, is a proposition.I The conjunction (and) of two propositions, written P ∧ Q, is a

Propositional Logic

proposition. These variables model true/false statements.I The negation of a proposition P, written ¬ P, is a proposition.

I The conjunction (and) of two propositions, written P ∧ Q, is aproposition.

I The disjunction (or) of two propositions, written P ∨ Q, is aproposition.

I The conditional statement (implies), written P −→ Q, is aproposition.

I The Boolean values True and False are propositions.

Propositional Logic

proposition. These variables model true/false statements.I The negation of a proposition P, written ¬ P, is a proposition.I The conjunction (and) of two propositions, written P ∧ Q, is a

proposition.

I The disjunction (or) of two propositions, written P ∨ Q, is aproposition.

Propositional Logic

proposition.

Propositional Logic

proposition.

Propositional Logic

I Different authors include different logical connectives in theirdefinitions of Propositional Logic. However, these differences arenot important.

I In each case, the missing connectives can be defined in terms ofthe connectives that are present.

I For example, I left out exclusive or, P ⊕ Q, but

P ⊕ Q = (P ∧ ¬ Q) ∨ ¬ P ∧ Q

Propositional Logic

I How expressive is Propositional Logic?I Can you write down the rules for Sudoku in Propositional Logic?

I It’s rather difficult if not impossible to express the rules of Sudokuin Propositional Logic.

I But Propositional Logic is a good first step towards more powerfullogics.

Propositional Logic

I How expressive is Propositional Logic?I Can you write down the rules for Sudoku in Propositional Logic?I It’s rather difficult if not impossible to express the rules of Sudoku

in Propositional Logic.I But Propositional Logic is a good first step towards more powerful

logics.

Meaning of Propositions

I A truth assignment maps propositional variables to True or False.The following is an example:

A ≡ {p 7→ True, q 7→ False, r 7→ True}A(p) = True A(q) = False A(r) = True

I The meaning of a proposition is a function from truthassignments to True or False. We use the notation JP K for themeaning of proposition P .

JpK(A) = A(p)

J¬P K(A) =

{True if JP K(A) = False

False otherwise

Meaning of Propositions, cont’d

JP ∧QK(A) =

{True if JP K(A) = True, JQK(A) = True

False otherwise

JP ∨QK(A) =

{False if JP K(A) = False, JQK(A) = False

True otherwise

JP −→ QK(A) =

{False if JP K(A) = True, JQK(A) = False

True otherwise

Example Propositions

Suppose A = {p 7→ True, q 7→ False}.

I JpK(A) = True

I JqK(A) = False

I Jp ∧ pK(A) = True

I Jp ∧ qK(A) = False

I Jp ∨ qK(A) = True

I Jp −→ pK(A) = True

I Jq −→ pK(A) = True

I Jp −→ qK(A) = False

I J(p ∨ q) −→ qK(A) = False

I JpK(A) = True

I JqK(A) = False

I JpK(A) = True

I JqK(A) = False

I JpK(A) = True

I JqK(A) = False

I JpK(A) = True

I JqK(A) = False

I JpK(A) = True

I JqK(A) = False

I JpK(A) = True

I JqK(A) = False

I JpK(A) = True

I JqK(A) = False

I JpK(A) = True

I JqK(A) = False

I JpK(A) = True

I JqK(A) = False

Tautologies

DefinitionA tautology is a proposition that is true in any truth assignment.

Examples:

I p −→ p

I q ∨ ¬q

I (p ∧ q) −→ (p ∨ q)

There are two ways to show that a proposition is a tautology:

1. Check the meaning of the proposition for every possible truthassignment. This is called model checking.

2. Contruct a proof that the proposition is a tautology.

Model Checking

I One way to simplify the checking is to only consider truthassignments that include the variables that matter. For example,to check p −→ p, we only need to consider two truth assignments.

1. A1 = {p 7→ True}, Jp −→ pK(A1) = True

2. A2 = {p 7→ False}Jp −→ pK(A2) = True

I However, in real systems there are many variables, and thenumber of possible truth assignments grows quickly: it is 2n for nvariables.

I There are many researchers dedicated to discovering algorithmsthat speed up model checking.

Stuff to Rememeber

Propositional Logic:

I The kinds of propositions.I The meaning of propositions.I How to check that a proposition is a tautology.

1. Proofs and Isabelle

2. Proof Strategy, Forward and Backwards Reasoning

3. Making Mistakes

Theorems and Proofs

I In the context of propositional logic, a theorem is just a tautology.I In this course, we’ll be writing theorems and their proofs in the

Isabelle/Isar proof language.I Here’s the syntax for a theorem in Isabelle/Isar.

theorem "P"proof -

step 1step 2...step n

qedI Each step applies an inference rule to establish the truth of some

proposition.

Inference Rules

I When applying inference rules, use the keyword have to establishintermediate truths and use the keyword show to conclude thesurrounding theorem or sub-proof.

I Most inference rules can be categorized as either an introductionor elimination rule.

I Introduction rules are for creating bigger propositions.I Elimination rules are for using propositions.I We write “Li proves P ” if there is a preceeding step or assumption

in the proof that is labeled Li and whose proposition is P .

Introduction Rules

And If Li proves P and Lj proves Q, then write

from Li Lj have Lk: "P ∧ Q" ..

Or (1) If Li proves P , then write

from Li have Lk: "P ∨ Q" ..

Or (2) If Li proves Q, then write

Implies

have Lk: "P −→ Q"proof

assume Li: "P"...· · · show "Q" · · ·

Introduction Rules, cont’d

Not have Lk: "¬ P"proof

assume Li: "P"...· · · show "False" · · ·

Hint: The Appendix of our text Isabelle/HOL – A Proof Assistant forHigher-Order Logic lists the logical connectives, such as −→ and ¬, andfor each of them gives two ways to input them as ASCI text. If youuse Emacs (or XEmacs) to edit your Isabelle files, then the x-symbolpackage can be used to display the logic connectives in their traditionalform.

Using Assumptions

I Sometimes the thing you need to prove is already an assumption.In this case your job is really easy!

I If Li proves P , write

from Li have "P" .

Example Proof

theorem "p −→ p"

proof -

show "p −→ p"

proofassume 1: "p"

from 1 show "p" .qed

Instead of proof -, you can apply the introduction ruleright away.

theorem "p −→ p"

proofassume 1: "p"

from 1 show "p" .qed

Exercise

theorem "p −→ (p ∧ p)"

Solution

proofassume 1: "p"

from 1 1 show "p ∧ p" ..qed

Elimination Rules

And (1) If Li proves P ∧Q, then write

from Li have Lk: "P" ..

And (2) If Li proves P ∧Q, then write

from Li have Lk: "Q" ..

Or If Li proves P ∨Q, then write

note Li

moreover { assume Lj: "P"...· · · have "R" · · ·} moreover { assume Lm: "Q"...· · · have "R" · · ·} ultimately have Lk: "R" ..

Elimination Rules, cont’d

Implies If Li proves P −→ Q and Lj proves P , then write

from Li Lj have Lk: "Q" ..

(This rule is known as modus ponens.)

Not If Li proves ¬P and Lj proves P , then write

from Li Lj have Lk: "Q" ..

False If Li proves False, then write

from Li have Lk: "P" ..

Example Proof

theorem "(p ∧ q) −→ (p ∨ q)"

proofassume 1: "p ∧ q"

from 1 have 2: "p" ..from 2 show "p ∨ q" ..

Another Proof

theorem "(p ∨ q) ∧ (p −→ r) ∧ (q −→ r) −→ r"

proofassume 1: "(p ∨ q) ∧ (p −→ r) ∧ (q −→ r)"

from 1 have 2: "p ∨ q" ..from 1 have 3: "(p −→ r) ∧ (q −→ r)" ..from 3 have 4: "p −→ r" ..from 3 have 5: "q −→ r" ..note 2

moreover { assume 6: "p"

from 4 6 have "r" ..} moreover { assume 7: "q"

from 5 7 have "r" ..} ultimately show "r" ..

Exercise

theorem "(p −→ q) ∧ (q −→ r) −→ (p −→ r)"

Solution

theorem "(p −→ q) ∧ (q −→ r) −→ (p −→ r)"

proofassume 1: "(p −→ q) ∧ (q −→ r)"

from 1 have 2: "p −→ q" ..from 1 have 3: "q −→ r" ..show "p −→ r"

proofassume 4: "p"

from 2 4 have 5: "q" ..from 3 5 show "r" ..

qedqed

Forward and Backwards Reasoning

And-Intro (forward) If Li proves P and Lj proves Q, then write

from Li Lj have Lk: "P ∧ Q" ..

And-Intro (backwards)

have Lk: "P ∧ Q"proof

...· · · show "P" · · ·

next...· · · show "Q" · · ·

Forward and Backwards Reasoning, cont’d

Or-Intro (1) (forwards) If Li proves P , then write

Or-Intro (1) (backwards)

have Lk: "P ∨ Q"proof (rule disjI1)

...· · · show "P" · · ·

Forward and Backwards Reasoning, cont’d

Or-Intro (2) (forwards) If Li proves Q, then write

Or-Intro (2) (backwards)

have Lk: "P ∨ Q"proof (rule disjI2)

...· · · show "Q" · · ·

Strategy

I Let the proposition you’re trying to prove guide your proof.I Find the top-most logical connective.I Apply the introduction rule, backwards, for that connective.I Keep doing that until what you need to prove no longer contains

any logical connectives.I Then work forwards from your assumptions (using elimination

rules) until you’ve proved what you need.

ConclusionAssumption

BackwardsReasoning

ForwardsReasoning

Assumption

Making Mistakes

I To err is human.I Isabelle will catch your mistakes.I Unfortunately, Isabelle is bad at describing your mistake.I Consider the following attempted proof

proof -

show "p −→ (p ∧ p)"

proofassume 1: "p"

from 1 show "p ∧ p"

I When Isabelle gets to from 1 show "p ∧ p" (adding .. at theend), it gives the following response:

Failed to finish proofAt command "..".

Making Mistakes, cont’d

I In this case, the mistake was a missing label in the from clause.Conjuction introduction requires two premises, not one. Here’sthe fix:

proof -

show "p −→ (p ∧ p)"

proofassume 1: "p"

from 1 1 show "p ∧ p" ..qed

I When Isablle says “no”, double check the inference rule. If thatdoesn’t work, get a classmate to look at it. If that doesn’t work,email the instructor with the minimal Isabelle file that exhibitsyour problem.

Making Mistakes, cont’d

I Here’s another proof with a typo:

theorem "p −→ p"proof

assume 1: "p"from 1 show "q" .

qedI Isabelle responds with:

Local statement will fail to refine any pending goal

Failed attempt to solve goal by exported rule:

(p) =⇒ qAt command "show".

I The problem here is that the proposition in the show "q", doesnot match what we are trying to prove, which is p.

Stuff to Rememeber

I How to write Isabelle/Isar proofs of tautologies in PropositionalLogic.

I The introduction and elimination rules.I Forwards and backwards reasoning.

1. Overview of First-Order Logic

2. Beyond Booleans: natural numbers, integers, etc.

3. Universal truths: “for all”

4. Existential truths: “there exists”

Overview of First-Order Logic

I First-order logic is an extension of propositional logic, adding theability to reason about well-defined entities and operations.

I Isabelle provides many entities, such as natural numbers,integers, and lists.

I Isabelle also provides the means to define new entities and theiroperations.

I First-order logic adds two new kinds of propositions, “for all” (∀)and “there exists” (∃), that enable quantification over theseentities.

I For example, first-order logic can express ∀x :: nat. x = x.

Beyond Booleans

I Natural numbers: 0, 1, 2, . . .

I Integers: . . . ,−1, 0, 1, . . .

I How does Isabelle know the difference between 0 (the naturalnumber) and 0 (the integer)?

I Sometimes it can tell from context, sometimes it can’t. (When itcan’t, you’ll see things like 0::’a)

I You can help Isabelle by giving a type annotation, such as 0 or 0.I We use natural numbers a lot, integers not so much.

Natural Numbers

I There’s only two ways to construct a natural number:I 0I If n is a natural number, then so is Suc n.

(Suc is for successor. Think of Suc n as n + 1.)I Isabelle provides shorthands for numerals:

I 1 = Suc 0I 2 = Suc (Suc 0)I 3 = Suc (Suc (Suc 0))

Arithmetic on Natural Numbers

I Isabelle provides arithmetic operations and many other functionson natural numbers.

I Warning: arithmetic on naturals is sometimes similar andsometimes different than integers. See/Isabelle/src/HOL/Nat.thy.

I For example,

1 + 1− 2 = 01− 2 + 1 = 1

Universal Truths

I How do we express that a property is true for all naturalnumbers?

I Let P be some proposition that may mention n, then the followingis a proposition:

∀ n. P

I Example:I ∀ i j k. i + (j + k) = i + j + kI ∀ i j k. i = j ∧ j = k −→ i = k

Introduction and Elimination Rules

For all-Intro

have Lk: "∀ n. P"proof

fix n...· · · show "P" · · ·

For all-Elim If Li proves ∀ n. P, then write

from Li have Lk: "[n7→m]P" ..

where m is any entity of the same type as n.

The notation [n7→m]P (called substitution) refers to the propositionthat is the same as P except that all free occurences of n in P arereplaced by m.

Substitution

I [x 7→ 1]x = 1

I [x 7→ 1]y = y

I [x 7→ 1](x ∧ y) = (1 ∧ y)I [x 7→ 1](∀y. x) = (∀y. 1)I [x 7→ 1](∀x. x) = (∀x. x) (The x under ∀x is not free, it is bound

by ∀x.)I [x 7→ 1]((∀x.x) ∧ x) = ((∀x. x) ∧ 1)

Substitution