mathematical preliminaries and notationsclab.yonsei.ac.kr/courses/05am_old/lecture1.pdf ·...

Formal Languages and Automata - Yonsei CS 1

Mathematical Preliminaries

And Notation


Structure of Chapter 1

1.1 The main ideas from mathematics that will be required

1.2 Central concepts of languages

1.3 Simple applications of the concepts


}2,1,0{=S

A set is a collection of elements• Specified by enclosing some description of its elements

in curly braces

SETS

},,,{ airplanebicyclebustrainB =

A∈1Bship∉

We write• 1 is an element of a set A

• ship is not in a set B


Set Representations

C = { a, b, c, d, e, f, g, h, i, j, k }

C = { a, b, …, k }

S = { 2, 4, 6, … }

S = { i : i is nonnegative and even }

S = { i : i > 0, i= 2k for some k>0 }

finite set

infinite set

• When the meaning is clear

• More explicit notation


A = { 1, 2, 3, 4, 5 }

Universal Set: all possible elements

U = { 1 , … , 10 }

1 2 34 5

A

U

6

78

910


Set Operations

A = { 1, 2, 3 } B = { 2, 3, 4, 5}

• Union

A U B = { 1, 2, 3, 4, 5 }

• Intersection

A B = { 2, 3 }

• Difference

A - B = { 1 }

B - A = { 4, 5 }

U

A B23

14

5

23

1

Venn diagrams


A

• Complement of A consists of all elements not in A

Universal set = {1, …, 7}

A = { 1, 2, 3 } A = { 4, 5, 6, 7}

12

3

4

5

6

7

A

A = A


024

6

1

3

5

7

even

{ even integers } = { odd integers }

odd

Integers

• Another example of complementation


DeMorgan’s Laws

A U B = A BU

A B = A U BU


Empty, Null Set:

= { }

S U = S

S =

S - = S

- S =

U

= Universal Set

• The set with no elements

• From the definition of a set, it is obvious


Subset

A = { 1, 2, 3} B = { 1, 2, 3, 4, 5 }A B

U

Proper Subset:

A BU

A

B

If every element of S1 is also an element of S,a set S1 is a subset of S

•B contains an element not in A


Disjoint Sets

A = { 1, 2, 3 } B = { 5, 6}

A B = U

A B

If S1 and S2 have no common element,


Set CardinalityFor finite sets

A = { 2, 5, 7 }

|A| = 3

•The cardinality of a finite set: the number of elements in the set.

•We will say that the cardinality of an infinite set is infinity (written as ∞).


PowersetsPowerset of S = the set of all the subsets of S

S = { a, b, c }2S = { , {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c} }

Observation: | 2S | = 2|S| ( 8 = 23 )


Cartesian Product

A = { 2, 4 } B = { 2, 3, 5 }

A X B = { (2, 2), (2, 3), (2, 5), ( 4, 2), (4, 3), (4, 5) }

|A X B| = |A| |B|

The set of all possible ordered pairs whose first component is a member of X and whose second component is a member of Y.

X × Y = { (x, y) | x ∈ X and y ∈ Y }

Generalizes to more than two sets:The collection of all ordered n-tuples that can be formed so that they contain one element of the first set, one element of the second,..., and one element of the n-th set. This collection can be seen as constituting an n-dimensional space in which each n-tupledesignates a cell. A X B X … X Z


FUNCTIONSdomain

123

a

bc

f : A -> B

A Brange

If the domain of f is all of Athen f is a total functionotherwise f is a partial function

f(1) = a4

5

• Function is a rule to assign to elements of one setto a unique element of another set


RELATIONS• For a function, each element of the domain has

exactly one associated element in the range.• In a relation, there can be several such elements

in the range.

R = {(x1, y1), (x2, y2), (x3, y3), …}

xi R yi

e. g. if R = ‘>’: 2 > 1, 3 > 2, 3 > 1


GRAPHS (Relation)

A directed graph

• Nodes (Vertices)

V = { a, b, c}

• Edges

E = { (a,b), (b,c),(c,a), }

bnode

edge

a

c

A graph is a construct consisting of 2 finite sets

• If we associate a direction with each edge


Labeled Graph

a

b

c

d

e

1 35 6

26

2


Walk

Walk is a sequence of adjacent edges

(e, d), (d, c), (c, a)

a

b

c

d

e


Path and Cycle

Path is a walk where no edge is repeatedSimple path: no node is repeated

(a,c), (c,b)

Cycle: a walk from a node (base) to itself with norepeated edges

(a,c), (c,c), (c,a)Loop: egde from a node to itself

a b c


Treesroot

leaf parent

childTree is a directed graph that has no cycles• root, leaf• parent, child: If there is an edge

from one node to another


root

leaf

Level 0

Level 1

Level 2

Level 3

Height 3

• Level: number of edges in the path from root to the vertex

• Height: largest level


PROOF TECHNIQUES

• Proof by induction

• Proof by contradiction

Two special proof techniques.


Induction

We have a seq. of statements P1, P2, P3, …

If we know

• for some k that P1, P2, …, Pk are true

• for any n >= k that the truths of P1, P2, …, Pn

imply the truth of Pn+1

Then

Every Pi is true


Proof by Induction• Inductive basis

Find P1, P2, …, Pk which are true

• Inductive hypothesisLet’s assume P1, P2, …, Pn are true, for any n >= k

• Inductive step (We will know that the patternis clear)

Show that Pn+1 is true


ExampleTheorem: A binary tree* of height n

has at most 2n leaves.Proof by induction:

let L(i) be the maximum number of leaves of any subtree at height i

* Binary tree is a tree in which no parent can have more than twochildren


We want to show: L(i) <= 2i

• Inductive basis

L(0) = 1 (the root node)

• Inductive hypothesis

Let’s assume L(i) <= 2i for all i = 0, 1, …, k

• Induction step

we need to show that L(k + 1) <= 2k+1


Induction Step

From Inductive hypothesis: L(k) <= 2k

height

k

k+1


L(k) <= 2k

L(k+1) <= 2 * L(k) <= 2 * 2k = 2k+1

Induction Step

height

k

k+1

(we add at most two nodes for every leaf of level k)


Proof by Contradiction

We want to prove that a statement P is true

• we assume that P is false

• then we arrive at an incorrect conclusion

• therefore, statement P must be true


Example

Theorem: is not rational

Proof:

Assume by contradiction that it is rational

= n/m

n and m have no common factors

We will show that this is impossible

2

2

* A rational number: expressed as the ration of two integers n and m so that n and m have no common factor.


= n/m 2 m2 = n2

Therefore, n2 is evenn is even

n = 2 k

2 m2 = 4k2 m2 = 2k2m is even

m = 2 p

Thus, m and n have common factor 2

Contradiction!

2


Languages


A language is a set of strings

String: A sequence of letters

Examples: “cat”, “dog”, “house”,…

Defined over an alphabet:

Three fundamental ideas: languages, grammars, and automata


Alphabets and StringsWe will use a finite, nonempty set of

symbols:

Strings: finite sequences of symbols from the alphabet

abbawbbbaaavabu

===

{ }ba,=Σ

baaabbbaabababaabbaaba

Stringnames

Specificvalue


String Operations

m

n

bbbvaaaw

L

L

21

21

==

bbbaaaabba

mn bbbaaawv LL 2121=

ConcatenationThe string obtained by appending the symbols of vto the right end of w.

abbabbbaaa

12aaaw nR L=

naaaw L21= ababaaabbbReverse

bbbaaababaObtained by writing the symbols in reverse order


String Length

Length:

Examples:

naaaw L21=

nw =

12

4

=

=

=

aaaabba


Empty StringA string with no letters:

Observations:

λ

abbaabbaabba

www

==

==

=

λλ

λλ

λ 0


SubstringSubstring of string:

a subsequence of consecutive characters

Prefixes Suffixesabbab

abbababbaabbabaλ

λbabbabbbababbab

uvw =

a prefix

a suffix


Another Operation

Example:

Definition for a special case:

43421 Ln

n wwww =

( ) abbaabbaabba =2

ww ∀= λ0

( ) λ=0abba

The string obtained by repeating w n times


The *, + Operation: the set of all possible strings fromalphabet

: the set of all possible strings fromalphabet except

λ−Σ=Σ+ *

*ΣΣ

{ }{ }K,,,,,,,,,*

,aabaaabbbaabaaba

baλ=Σ

=Σ

{ }K,,,,,,,, aabaaabbbaabaaba=Σ+

+ΣΣ λ


LanguagesA language is any subset of

Example:

These sets are languages on the alphabet :

*Σ

{ }{ }K,,,,,,,,*

,aaabbbaabaaba

baλ=Σ

=Σ

{ }{ }

},,,,,{,,

aaaaaaabaababaabbaaabaaa

λ

λ


Another Example

An infinite language }0:{ ≥= nbaL nn

aaaaabbbbbaabbabλ

L∈ Labb∉


Operations on LanguagesThe usual set operations since languages are

sets

Complement:

{ } { }{ } { }{ } { } { }aaaaaabbbaaaaaba

ababbbaaaaabaaaaabbabaabbbaaaaaba

,,,,}{,,,

},,,{,,,

=−==

I

U

LL −Σ= *

{ } { }K,,,,,,, aaabbabaabbaa λ=


Reverse

Definition:

The concatenation of two languages L1 and L2 is the set of all strings obtained by concatenating any element of L1 with any element of L2

Definition:

}:{ LwwL RR ∈=

The reverse of a lang. is the set of all stringreversals

{ }2121 ,: LyLxxyLL ∈∈=


Another OperationDefinition:

L concatenated with itself n times

Special case:

321Ln

n LLLL =

{ } { }{ }{ }{ }bbbbbababbaaabbabaaabaaa

babababa,,,,,,,

,,,, 3 ==

{ }{ } { }λ

λ=

=0

0

,, aaabbaaL


Star-Closure of a Language

Definition:

Example:

LUU 210* LLLL =

{ }

⎪⎪⎭

⎪⎪⎬

⎫

⎪⎪⎩

⎪⎪⎨

⎧

=

K,,,,,,,,

,,,

*,

abbbbabbaaabbaaabbbbbbaabbaa

bbabba

λ


Positive Closure

Definition:

{ }λ−==+

*

21

LLLL LUU

{ }⎪⎭

⎪⎬

⎫

⎪⎩

⎪⎨

⎧=+

K,,,,,,,,

,,,

abbbbabbaaabbaaabbbbbbaabbaa

bbabba


Note that:

}{}{ λ≠=∅

0}{ =∅=

1}{ =λ

0=λ

Sets

Set size

Set size

String length

mathematical preliminaries and notationsclab.yonsei.ac.kr/courses/05am_old/lecture1.pdf ·...

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