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Formal Languages and Automata Theory 1

CSC 3130

Formal Languages and Automata Theory Lecturer

Evangeline Young (SHB 916, 26098401, [email protected] )

Guest lecturerProf. C.K. Cheng from UCSD

Lecture HourMon 4:30pm - 6:15pmFri 9:30am - 10:15am(In September, we will only meet once a week: Mon 4:30pm 6:30pm)

Tutorial Hours No tutorial in the first week

Text bookIntroduction to Automata Theory, Languages and Computation, 3 rd editionJohn E. Hopcroft, Rajeev Motwani and Jeffrey D. UllmanAddison Wesley
mailto:[email protected]:[email protected]

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Formal Languages and Automata Theory

Planning ScheduleWeek 1-4 : Introduction, Finite Automata, Regular Expression

Different Kinds of FA, Properties of Regular Sets

Qui z 1 Week 5-8 : Context-Free Grammars, Normal Forms

Pushdown Automata, Properties of Context-Free LanguagesQui z 2

Week 9-12 : Turing Machines, UndecidabilityChomsky Hierarchy, LR(k) Grammars, Complexities

F inal Examin ation

(Chap. 2-4)

(Chap. 5-7)

(Chap. 8-10)

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Formal Languages and Automata Theory

Assessment Scheme Quiz 1 (12.5%) Quiz 2 (12.5%) Final Examination (40%) Homework (25%) Class Participation (10%)

Plagiarism in homework, quiz or examination will result in failing of thewhole course.

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Topics

Automata Theory

Grammars and Languages Complexities

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Why Automata Theory?

To study abstract computing devices which areclosely related to todays computers. A simple

example of finite state machine :

off onstart

1

1

There are many different kinds of machines.

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Another Example

start

0

1on

0

0

1

offoff

1

When will this be on?Try 100, 1001, 1000, 111, 00,

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Grammar and Languages

Grammars and languages are closely relatedto automata theory and are the basis ofmany important software components like: Compilers and interpreters Text editors and processors

Text searching System verification

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Complexities

Study the limits of computations. Whatkinds of problems can be solved with a

computer? What kinds of problems can besolved efficiently ?

Can you write a program in C that cancheck if another C program will terminate?

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Preliminaries

Alphabets Strings Languages Problems

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Alphabets

An alphabet is a finite set of symbols. Usually, use to represent an alphabet. Examples:

= {0,1}, the set of binary digits. = {a, b, , z}, the set of all lower -case letters. = {(, )}, the set of open and close parentheses.

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Strings

A string is a finite sequence of symbolsfrom an alphabet.

Examples: 0011 and 11 are strings from = {0,1} abc and bbb are strings from = {a, b, , z}

(()(())) and )(() are strings from = {(, )}

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Strings

Empty string : Length of string: |0010| = 4, |aa| = 2, | |=0 Prefix of string: aaabc, aaabc, aaabc Suffix of string: aaabc, aaabc, aaabc

Substring of string: aaabc, aaabc, aaabc

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Strings

Concatenation : =abd, =ce, =abdce Exponentiation : =abd, 3=abdabdabd, 0=

Reversal : =abd, R = dba k = set of all k-length strings formed by the

symbols in

e.g., ={a,b}, 2={ab, ba, aa, bb}, 0={ }What is 1? Is 1 different from ? How?

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Strings

Kleene Closure * = 0 1 2 = k 0 k e.g., ={a, b}, * = { , a, b, ab, aa, ba, bb, aaa,aab, abb, } is the set of all strings formed

by as and bs. + = 1 2 3 = k>0 k

i.e., * without the empty string.

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Languages A language is a set of strings over an alphabet. Examples:

={(, )}, L 1={(), )(, (())} is a language over . Theset L 2 of all strings with balanced left and right

parentheses is also a language over . ={a, b, c, , z}, the set L of all legal English

words is a language over . The set { } is a language over any alphabet.

What is the difference between and { }?

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Languages

Other Examples: ={0, 1}, L={0 n1n | n 1} is a language over

consisting of the strings {01, 0011, 000111, } ={0, 1}, L = {0 i1 j | j i 0} is a language over

consisting of the strings with some 0s (possiblynone) followed by at least as many 1s.

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Finite Automata

( or Finite State Machines) This is the simplest kind of machine.

We will study 3 types of Finite Automata: Deterministic Finite Automata (DFA) Non-deterministic Finite Automata (NFA)

Finite Automata with -transitions ( -NFA)

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Deterministic Finite Automata

(DFA)We have seen a simple example before:

off onstart

1

1

There are some states and transitions (edges) between the states. The edge labels tell whenwe can move from one state to another.

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Definition of DFA

A DFA is a 5-tuple (Q, , , q0, F) where

Q is a finite set of states is a finite input alphabet is the transition function mapping Q to Q

q0 in Q is the initial state (only one)F Q is a set of final states (zero or more)

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Definition of DFAFor example:

off onstart

1

1

Q is the set of states: { on, off } is the set of input symbols: {1} is the transitions: off 1 on; on 1 off

q0 is the initial state: offF is the set of final states (double circle): { on}

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Definition of DFAAnother Example:

start

0

1q2

0

0

1

q1 q0

1

What are Q, , , q 0 and F in this DFA?

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Transition TableWe can also use a table to specify the transitions.

For the previous example, the DFA is (Q, , ,q0,F)where Q = {q 0,q1,q2}, = {0,1}, F = {q 2} and issuch that

StatesInputs

0 1q0

q1q2

q1 q0

q2 q0q1q0

Note that there is one transition only for each inputsymbol from each state.

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DFA ExampleConsider the DFA M=(Q, , ,q0,F) where Q ={q0,q1,q2,q3}, = {0,1}, F = {q 0} and is:

StatesInputs

0 1q0q1q2q3

q2

q2q1

q1q3

q3q0

q0

q0

q2 q3

q111

1

1

0 000

Start

We can use a transition table or a transition diagram to specifythe transitions. What input can take you to the final state in M?

OR

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Language of a DFA

Given a DFA M, the language accepted (orrecognized) by M is the set of all stringsthat, starting from the initial state, will reachone of the final states after the whole stringis read.

For example, the language accepted by the previous example is the set of all 0 and 1strings with even number of 0s and 1s.

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Class Discussion

Start q0 q1

Start q0 q1

Start q0 q1 q2

0 011

0 110

1 0

0 0,11

What are the languages accepted by these DFA?

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Class Discussion

Construct a DFA that accepts a language Lover = {0, 1} such that:(a) L contains 010 and 1 only. (b) L is the set of all strings ending with 101. (c) L is the set of all strings containing no

consecutive 1s nor consecutive 0s.

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Non-deterministic FA (NFA)

For each state, zero, one or more transitionsare allowed on the same input symbol.

An input is accepted if there is a pathleading to a final state.

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An Example of NFA

1

01

1

0Start

q0

q1

q2

In this NFA(Q, , ,q0,F), Q = {q 0,q1,q2}, = {0,1},F = {q 2} and is:

States Inputs0 1q0q1q2

{q1,q2}{q1 }{q

0}

{q2 }OR

Note that each transition can lead to a set of states,which can be empty.

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Language of an NFA

Given an NFA M, the language recognized byM is the set of all strings that, starting fromthe initial state, has at least one path reaching

a final state after the whole string is read.Consider the previous example:

For input 101, one path is q 0 q1 q1 q2 and theother one is q 0 q2 q0 q1. Since q 2 is a final state,so 101 is accepted. For input 1010, none of its

paths can reach a final state, so it is rejected.

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More Examples of NFA

q0 q1 q2a

b

b b

q0

q1

q2

q3

q4

q5

q6

q7

0-9

0-9

0-90-9

0-9

+,- +,-

E.

0-9 0-9 0-9

0-9 0-9

Start

Start

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Class Discussion

Consider the language L that consists of allthe strings over = {0, 1} such that the

third last symbol is a 1, (a) Construct a DFA for L.(b) Construct an NFA for L.

Is NFA more powerful than DFA?

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DFA and NFA

Is NFA more powerful than DFA? NO! NFA is equivalent to DFA.

DFA NFATrivial

ConstructiveProof

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Constructing DFA from NFAGiven any NFA M=(Q, , ,q0,F) recognizing alanguage L over , we can construct a DFA

N=(Q, , ,q 0,F) which also recognizes L: Q = set of all subsets of Q

e.g., if Q = {q 0, q1}, Q = {{ }, {q 0}, {q 1}, {q 0, q1}} q0 = {q 0}

F = set of all states in Q containing a final state of M ({q 1,q2, q i}, a) = (q1,a) (q2,a) ... (q i,a)

a state in N a state in N

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An Example of NFA DFAConsider a simple NFA:

Construct a corresponding DFA:

0

1

1Start

0 1

q0 q1

{q0}

{q0, q1}

{q1}

{}

Start 1

01

0

1,0 1,0

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