Finite Automata and Regular Languages

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Finite Automata and Regular Languages

• Jianhua Yang
• Department of Math and Computer Science
• Bennett College

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Goals

• To understand the concepts of automata
• To know how to establish an automata
• To know how to recognize regular languages by
  using finite automata

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1.1 Lexical Analysis

• One problem in Compiler

To detect whether or not a given string within a
source program represents an acceptable variable
name.
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1. Transition Diagrams

• State diagram

is a finite collection of circles ,which
represent states, connected by arcs.
accept state
initial state
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Instruction sequence

• State1
• Read the next symbol from input
• While not end-of-string do
• case State of
• 1 if the symbol is a letter, then State3,
• else if it is a digit, then
  State2 ,
• else exit to error routine
• 2 exit to error routine
• 3 if the current symbol is a letter then
  State3,
• else if it is a digit, then State3,
• else exit to error routine
• Read the next symbol from the input
• End while
• If State is not 3, then exit to error routine

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2. Transition table
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A lexical analyzer

• State1
• Repeat
• Read the next symbol from the input stream
• case symbol of
• letter inputletter
• digit inputdigit
• end of string marker inputEOS
• none of the above exit to error routine
• StateTablestate, input
• if Stateerror then exit to error routine
• Until Stateaccept

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One more example

• Design a state diagram to recognize a real number
• 1.27E04, 0.345E11
• 1.29, 1239.
• 8E01
• 1.274E-10
• 2.871E10

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The transition diagram
2
5
3
6
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The transition table

• Please write its transition table
• Please write its lexical analyzer

Home work 1
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Home work 2

• Draw the state diagram
• Write the transition table
• Write the lexical analyzer

When considering abnormal processing
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1.2 Deterministic finite automata

• We want to ask one question
• Do transition diagrams provide a tool powerful
  enough to develop programs for recognizing
  syntactic structures of arbitrary complexity?

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1. Basic Definitions

• Alphabet
• Input stream

A nonempty, finite set of symbols from which the
strings to be analyzed are constructed is called
an alphabet.
Each string to be analyzed is received as a
sequence of symbols, one symbol at a time, we
refer to the source of this sequence as the input
stream.
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Deterministic finite automaton

• Device
• Input stream
• Control mechanism

It can be in any one of a finite number of
states, of which one is an initial state, and at
least one is an accept state.
On a input stream, symbols from a given alphabet
arrive sequentially.
To compute the next state of the device based on
the current state and each received input symbol
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Make more sense

• Deterministic
• finite

not ambiguous
The machine has only a finite number of states
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DFA model
Input tape
Head moves in this direction
tape head
1
8
2
7
3
4
6
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Control mechanism
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DFA model

• DFA(S, ? ,d, i , F)
• S is a finite set of states
• ? is the machines alphabet
• d is a function from S X ? into S
• i is the initial state
• F is the set of accept states

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A DFA accepting empty string

• Empty string
• DFA

A string containing no symbols, represented by ?
If and only if its initial state is also an
accept state
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2. Deterministic Transition Diagrams

• deterministic means

1. Each state in such a diagram must have only
one arc leaving it for each symbol in the
alphabet.
2. The diagram must be fully defined.
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Example of DFA

• Consider a typical vending machine that dispenses
  a persons choice of candy after it has received
  a total of 30 cents in nickels, dimes, and
  quarters

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Example of DFA
0
5
Q
10
15
20
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1.3 The limits of DFA

• Question
• Does the use of transition table provide enough
  flexibility for general string processing?

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1. DFA as Language Accepters

• The length of string w
• Full language
• A language
• Regular language

is denoted by w
?, ? is the alphabet

• is defined as the subset of ?

• We use M to represent a DFA(S, ? ,d, i , F), the
  strings that can be accepted by M is called
  regular language, denoted by L(M)

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Questions?

• What is the alphabet of language English?
• Does that alphabet constitute only one language
  English?
• How to prove a given language is regular?

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Examples

• To prove that the following language is a regular
  language
• A language is composed of the strings of xs and
  ys that contain an even number of xs and any
  number of ys. ?x,y

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Test 1

• To prove that the following language is a regular
  language
• A language is composed of the strings of xs and
  ys that contain an even number of xs and even
  number of ys. ?x,y

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Test 2

• To prove that the following language is a regular
  language
• A language is composed of the strings of xs and
  ys that contain an even number of xs and odd
  number of ys. ?x,y

Answer this language is not regular, we do not
have a DFA to accept it
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Example

• A language ?, empty language F

Containing no strings
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Theory about regular language

• Theorem 1.1
• For any alphabet ?, there is a language that is
  not equal to L(M) for any deterministic finite
  automaton M

• L(M) is countable because of countable states
• The number of subsets of ? is uncountable

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2. A nonregular language

• Think about an expression
• (ab)/c, or ((ab)/c-d)2

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Theorem 1.2

• If a regular language contains of the form xnyn
  for arbitrarily large integers n, then it must
  contain strings of the form xmyn where m and n
  are not equal

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Proof

• We assume L(M) contains xnyn
• There are u states for M
• There must exist positive integer k gtu
• and that xkyk is in L(M)
• In reading xs, there must exist j, so that xkjyk
  is accepted by M
• mkj, such that xmyn is in L(M)

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Consequence of Theorem 1.2

• 1. xnyn n?N is not regular language
• 2. DFA is not powerful enough to analyze
  nonregular language

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1.4 Nondeterministic Finite Automata (NFA)

• How to increase the power of DFA?

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An Example
1
2
3
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NFA

• NFA(S, ? ,?, i , F)
• a. S is a finite set of states
• b. ? is the machines alphabet
• c. ? is a subset of S ? S
• d. i (en element of ) is the initial state
• e. F (a subset of S) is the collection of
  accept states.

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Accepted by a NFA

• We say that a string is accepted by a NFA if it
  is possible for its analysis to leave the machine
  in an accept state

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L(M)

• The set of all the strings accepted by a NFA M is
  a language that we denote by L(M) and refer to as
  the language accepted by M.

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And More

• A NFA(S, ? ,?, i , F) accepts the non empty
  string x1x2xn?? if and only if there is a
  sequence of states s0, s1,, sn such that s0 i,
  sn ?F, and for each integer j from 1 to n, (sj-1,
  xj, sj) is in ?.

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The problem of NFA

• Cost too much time to accept a string

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Converting a NFA to a DFA
y
S1
x
y
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The DFA
s0,s1,s2
s0
empty
x
s0,s1
s1
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Theorem 1.3

• For each NFA, there is a DFA that accepts exactly
  the same language.

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Proof

• Suppose that M is a NFA(S, ? ,?, i , F), our
  purpose is to demonstrate there exists a DFA that
  accepts exactly the same string as M.

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Proof (Cont.)

• We define (S, ? ,d, i , F )as the
  following
• SP(S)
• ii
• F is the collection of subsets of S that contain
  at least one state in F.
• d S ?? S

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Proof (Cont.)
is a DFA
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Proof (Cont.)

• We need to prove
• For each path in M from state i to state sn
  traversing arcs labeled w1, w2,, wn, there is a
  path in from state i to state
  traversing arcs labeled w1, w2,, wn, so that
  and visa versa.

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Proof (Cont.) Induction

• n0 it is correct
• We assume that the statement holds for some n
• Then we need to prove that for n1, the statement
  still holds.

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Theorem 1.4

• For any alphabet ?, L(M) M is a DFAL(M) M
  is a NFA.

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1.5 Regular Grammars

• A grammar describing a small subset of the
  English language

ltsentencegt? ltsubjectgtltpredicategtltperiodgt
ltsubjectgt?ltnoungt ltnoungt?John ltnoungt?Mary
ltpredicategt?ltintransitive verbgt
ltpredicategt?lttransitive verbgtltobjectgt
ltintransitive verbgt?skates lttransitive verbgt?
hit lttransitive verbgt ?likes ltobjectgt? ltnoungt
ltperiodgt ? .
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Some definitions

• Terminals
• Nonterminals
• Start symbol
• Rewrite rule
• Grammar
• ?-rule

Terms that dont appear in brackets are called
terminals.
The terms enclosed in brackets are called
nonterminals.
The first nonterminal
each line consisting of a left and right side
connected by an arrow is called a rewrite rule
Such a finite collection of nonterminals and
terminals together with a start symbol and a
finite set of rewrite rules is called a grammar,
or phrase-structure grammar.
if a nonterminal is defined by an empty string,
this rule is a ?-rule .
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For convenience

• We denote nonterminals by upper case letters and
  terminals by lower case letters.

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Derivation

• To generate a particular string of terminals if,
  by starting with the start symbol, one can
  produce the string by successively replacing
  patterns found on the left of the grammars
  rewrite rules with the corresponding expressions
  on the right, until only terminals remain.

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A derivation example

• ltsentencegt ?ltsubjectgtltpredicategtltperiodgt
• ?ltnoungtltpredicategtltperiodgt
• ?Maryltpredicategtltperiodgt
• ?Marylttransitive verbgtltobjectgtltperiodgt
• ?Mary likes ltobjectgtltperiodgt
• ?Mary likes ltnoungtltperiodgt
• ?Mary likes John ltperiodgt
• ?Mary likes John.

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Grammar example

• S?XSZ
• S?Y
• Y?yY
• Y??
• X?x
• Z?z

Please determine if string xyz can be generated
by this grammar.
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L(G)

• If the terminals of a grammar G are symbols in an
  alphabet ?, G is a grammar over the alphabet?.

• The strings generated by G is a subset of ?.

• We define L(G) as the strings generated by G over
  ?.

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Regular grammar

• It is a grammar whose rewrite rules conform to
  the following restrictions.
• The left side of any rewrite rule must consist of
  a single nonterminal
• The right side must be a terminal followed by a
  nonterminal, or a single terminal, or the empty
  string.

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Examples

• Z?yX
• Z?x
• W??

A Regular grammar
yW?X X?xZy YX?WvZ
Not a Regular grammar
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From regular grammar G to DFA

• S?xX
• S?yY
• X?yY
• X??
• Y?xX
• Y? ?

S
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Theorem 1.5

• A language generated by a regular grammar G is a
  regular language. This language can be accepted
  by a DFA.

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Proof

• Converting if G is a regular grammar, we can
  convert it to a regular grammar G ?, which
  generates the same language but does not contain
  the rewrite rules whose right sides consist of a
  single terminal.

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Proof (cont.)

• Construct NFA M(S, ? ,?, i , F) as the following
• S is the collection of nonterminals in G ?.
• i is the start symbol in G ?.
• F is the nonterminals of ?-rules.
• And ? consists of the triples (P, x, Q) which
  corresponds to a rewrite rule P?xQ in G ?.

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Proof (cont.)

• Generate a regular grammar from a NFA M(S, ? ,?,
  i , F) as the following
• The start symbol is i.
• Nonterminals are the states in S .
• If Q is in F, the we have a rule Q??.
• If (P, x, Q) is in ?, then we have a rule P?xQ.

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Proof (cont.)

• In either case, we have L(M)L(G ?) .

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1.6 Regular Expressions

• Another characterization of regular languages
• Providing more insight into the composition of
  regular languages.

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Definition of regular expressions

• A regular expression (over an alphabet ?) is
  defined as follows
• F is a regular expression
• Each member of ? is a regular expression
• If p and q are regular expressions, then so are
  (p U q), (p ?q), and (p), or (q).

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1. Operation Union U

• The union of any two regular languages is regular.

L1x, xy, and L2yx, yy LL1 U L2x, xy,
yx, yy
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Construct a NFA for Union

• Draw a new initial state, and declare it is an
  accept state if and only if one of the original
  initial states was also an accept state.
• To each state that is the destination of arc from
  one of the original initial states, draw an arc
  with the same label from the new initial state.
• Cancel the initial status of the original initial
  states.

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Example about Union operation
L2
L1
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2. Concatenation

• The concatenation of any two regular languages is
  regular.

L1x, xy, and L2yx, yy LL1 ? L2 xyx, xyy,
xyyx, xyyy
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Construct a NFA for concatenation

• From each accept state of T1, draw an arc to each
  state of T2 that is the destination of an arc
  from the initial state of T2
• Label each arc with the label of the
  corresponding arc in T2
• Allow the accept states in T1 to remain accept
  states if and only if the initial states in T2 is
  also an accept state
• Remove the initial state from T2.

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Example
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Example (Cont.)
x
x
1
2
y
y
x
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3. Kleene star operation

• The Kleene star of any regular language is
  regular.

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The rules to construct a NFA

• To each state that was the destination of an arc
  from the initial state, we draw an arc from the
  new initial state with the same label.
• To add an arc from each accept state to each
  state that is the destination of an arc from the
  initial state.
• We designate the new initial state as an accept
  state.

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Example
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4. Theorem 1.6

• Given an alphabet ? , the regular languages over
  ? are exactly the languages that are represented
  by regular expressions over ?.

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Proof

• Study by yourself

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1. 7 Summary

• DFA, NFA
• Regular languages
• Regular grammar
• Regular expressions

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Example 1

• (ab), construct its NFA, and convert it to a DFA.

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NFA
e
a
b
e
e
e
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DFA
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Example 2

• Expression (b(ab)a(ab))(ab)

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NFA
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DFA
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