Automata and Formal Languages - · PDF fileAutomata and Formal Languages ... Introduction, basic tools, ... • Formal languages – Regular languages – Context free languages. 6

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Automata and Formal Languages

Winter 2009-2010

Yacov Hel-Or

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What this course is all about

This course is about mathematical modelsof computation

Well study different machine models (finite automata, pushdown automata, Turing machine)

and characterize what they can compute

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Why should I care ?

To understand the limits of computationsSome problems require more resources to compute

and others cannot be computed at all.

To learn some programming tools Automata show up in many different settings:

compilers, text editors, hardware design, communication protocols, program proofing,

To learn to think in a formal way about computing

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Computability and Complexity

Are there any problems that can not be solved by a (very powerful) computer?

What makes some problems computationally hard and other easy?

Can we partition the problems into classes such that problems in one class share the same computational properties?

Complete answers: next semester.

Introduction, basic tools, models, intuition: this semester.

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This Course: Formal computational models

The basic computational model : Finite State Automata

Additional models: Pushdown automaton

Turing machine

Formal languages Regular languages

Context free languages

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Administration

Course Home page: http://www1.idc.ac.il/toky/Automata

Lecture notes will appear at the course web-page the day before the lecture (the latest).

Proofs, examples, technical details will usually be presented on the board.

No recitations, part of lecture time will be dedicated to solving problems.

Grade calculation: Exam 70%, HW 30%

Exam: Must pass the exam (60) in order to have the HW component.

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Administration (cont.)

HW: Will be given every week.

Submission: Wednesdays before your lecture.

Appeals: No more than two weeks after return.

Grader + TA: Ilit Raz ([email protected])

Newsgroup: news://news.idc.ac.il/automata

Book: Sipser (see web-page for details).

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Unit 1

IntroductionMathematical Background

Reading: Sipser, chapter 1

Today

Set theory (review?)

Logic, proofs (review?)

Words and their operations:

Languages and their operations:

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wwwww i ,,, *21 LLLLL i ,,, *21

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Set TheoryA Set is a group of objects.The objects are called elements.

Notation:

1. By listing the elements. Examples: {3,5,7} ,{Alice, Bob, {1,2} } (for finite sets only)

2. By providing a rule. Examples:

{x | x is an odd integer between 2 and 8} ;

{x | name of a student in the Automata class}

A name of a set is usually a capital letter of theEnglish alphabet (A, B, C) or a capital letter withan index (X1 , X2 , X3)

,

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The Empty Set

A set with no elements is called an empty set.

Notation or { }

Example:

S - a set of all odd numbers that can be dividedby two without any remainder.

S is an empty set.

S = {xN | x is odd and x mod 2=0}=

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Membership in a set

s S , means that an element s is a memberof the set S

b S, means that an element b is not a member of the set S

Examples: 7 {21,7,30} and 8 {21,7,30}

Let N be the set of natural numbers.2 N and 3.2 N

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Subsets

Given two sets A and B we say that A is asubset of B, if each element of A is also anelement of B.

The notation: A B Formally: A B x A x B

Example:A set of natural numbers N is a subset of a

set of all real numbers R.N R

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Proper Subset

Given two sets A and B we say that A is aproper subset of B if each element of A isalso an element of B and there exists atleast one element in B that does not belongto A.

The notation: A B

Example:{1,3} {1, 2, 3, 8}

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Some Facts

Each set is a subset of itself:

A A

The empty set is a subset of every set:

A

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More on Sets

Two groups of elements that have the same elements but in different order form the same set.

Example: If A = { 1,2,3,4}, B = { 2,1,4,3} then A = B

Repetitions in a set are irrelevant

Example: {1,2,3,4,2} = {1,2,3,4}

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More on Sets

The cardinality of a set is the number ofelements in the set. Notation: |A|

Example: Let A = { 1,2,4,8,16}, then |A|=5

A set can be:

Finite A={Even integers smaller than 100}

Infinite A={Even integers dividing 7 with no reminder}

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Operations on sets union

intersection

complement

difference

A visual model, called Venn diagram can be used.

A

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Venn Diagram

Start-t End-z Start-j

terrifictheory topaz jazz

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Union

A binary operation

The notation : A B

Formally:

A B = { x | xA or xB }

A B

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Intersection

A binary operation

The notation: A B

Formally:

A B = { x | xA and xB }

A B

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Complement

An unary operation

The notation : ~A

Formally:

~A = { x | x A}

A

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Set Difference

An binary operation

The notation : A-B

Formally: A-B = { x | xA and x B}

Example: A = { 1,2,3 }, B={3,4,5}, A-B={1,2}

A B

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Power Set

The power set is the set of all subsets of a given set.

Notation: P(S) or 2S is a power set of S

Note, that sets may appear as elements of other sets.

The cardinality of a power set is: 2|S|

(why?)

Example:S = { 1,2 }P(S) = 2S = { ,{1},{2},{1,2}}

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Sequences and Tuples

A sequence of objects is a list of objects in some order.

The notation : (7,21,57,)

Unlike sets, the order and repetitions in the sequence do matter, thus

(7,21,57) (7,57,21) and (7,21,57,57) (7,21,57)

A sequence with k elements is a k-tuple

Example: (7,21,57) is a 3-tuple.

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Cartesian product

A binary operation

The Cartesian product (or cross product) is a set of all pairs (2-tuple) where the first element of the pair is in A and the second element of the pair is in B.

The notation: AB

A Cartesian product of a set with itself:

AAA (k times)=Ak

Example:

A = {2,3}, B = {b,c}, AB ={ (2,b), (2,c), (3,b), (3,c) }

Summary: sets

,{ } empty set

a A membership

|A| cardinality

AB subset

AB union

AB intersection

~A complement

A-B set difference

2A power set

(a,b,..) a sequence (k-tuple)

AxB cartesian product 27

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Letters and Alphabet

Any finite set of letters (symbols) is called an alphabet.

Notation: alphabet- ; Letter (symbol)-

Examples:

1 = { 0,1 } , 1=0 ; 2=1;

2 = { a,b,c,d,e, }, 1=a ; 2=b ; 3=c ;

3 = { 0,1,x,y,z}

,

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Strings (and words)

A string (word) over some alphabet is a finite sequence of letters from the alphabet.

Example: = {0, 1}, w = 101

The length of a word, w, denoted |w|, is the number of letters in it.

Example:

w1 = abracadabra; |w1| = 11

w2 = 001011 ; |w2| = 6

, ,

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More on Strings An empty word is a string without letters.

The notation of an empty word is

|| = 0

The number of occurrences of some letter in word w is denoted by #(W)

Example:

Let w=aaba, then #a(w)=3, #b(w)=1, #c(w)=0

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Operations on Strings

reverse

sub-string

prefix

suffix

concatenation

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Reversing strings

A reverse string is a string in which all letters are written in the opposite order

Notation: wR

Examples:w = 10, wR = 01

s = abcb, sR = bcba

A palindrome: a string w such that w=wR.

Examples: aba, 010010,

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Sub-strings

A sub-string is a subsequence of consecutiveletters from a string

Example: Let w = 101. All sub-strings of w are:

B(w) = { , 1, 0, 10, 01, 101 }.

Note: 11 B(w)

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Prefixes A prefix is a sub-string which starts from

the first letter of the word (or an empty word).

A proper prefix of a string is a prefix that is not equal to the complete word.

Example: Let w = acdb. The prefixes of w are { , a,ac,acd,acdb}. acdb is not a proper prefix.

,

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Suffixes A suffix is a substring ending with the last

letter of the word or an empty word.

A proper suffix of the string is a suffix that is not the whole word.

Example: Let w = acdb. The suffixes of w are { ,b, db,cdb,acdb}. acdb is not a proper suffix

Note: The prefixes of w are the reversed suffixes of wR.

,

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Concatenation

A binary operation (over two words)

The concatenation of two words x and yplaces them one after the other such thatthe first word is a prefix and a second oneis a suffix.

Notation: concatenation of two words x andy: xy

Example: Let x= 01, y=012, z=10

Then xy=01012, yx=01201, xyz=0101210

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Concatenation (Cont.)

The result of concatenating a word with anempty word is the string itself. Forexample