1 Language Recognition (12.4) and Turing Machines (12.5)

1

Language Recognition (12.4)and Turing Machines (12.5)

2

Three Equivalent Representations

Finite automata

Regularexpressions

Regular languages

Each can

describethe others

Kleene’s Theorem:

For every regular expression, there is a deterministic finite-state automaton that defines the same language, and vice versa.

3

NFAs Regular grammarsThus, the language recognized by FSA

is a regular language

Every NFA (=FSA) can be converted into a corresponding regular grammar and vice versa.

Each symbol A of the grammar is associated with a non-terminal node of the NFA sA, in particular, start symbol

S is associated with the start state sS.

Every transition is associated with a grammar production:

T(sA,a) = sB A aB.

Every production B is associated with final state sB.See p. 823.

4

Kleene’s Theorem

LanguagesGenerated byRegular Expressions

LanguagesRecognizedby FSA

5





We will show:

6

Proof - Part 1

r)(rL

For any regular expression

the language is recognized by FSA (= is a regular language)



Proof by induction on the size of r

7

Induction BasisPrimitive Regular Expressions: , , a

NFAs

)()( 1 LML

)(}{)( 2 LML

)(}{)( 3 aLaML

regularlanguages

a

8

Inductive Hypothesis

Assume for regular expressions andthat and are regular languages (i.e., L(FSA))

1r 2r

)( 1rL )( 2rL

9

Inductive StepWe will prove:

1

1

21

21

*

rL

rL

rrL

rrL

Are regular Languages

10

By definition of regular expressions:

11

11

2121

2121

**

rLrL

rLrL

rLrLrrL

rLrLrrL

p. 818

11

)( 1rL )( 2rLBy inductive hypothesis we know: and are regular languages

Regular languages are closed under:

*1

21

21

rL

rLrL

rLrL Union

Concatenation

Star

We need to show:

Can be shown via defn. of regular grammar.,

Note: alt. Proof Thm. 1 p 819, directly constructs FSA. See Fig. 2 and Fig. 3.

12

Therefore:

** 11

2121

2121

rLrL

rLrLrrL

rLrLrrL

Are regularlanguages

And trivially: ))(( 1rL is a regular language

13

Proof - Part 2



Lr LrL )(

For any regular language there is a regular expression with

Proof by construction of regular expression

14

Since is regular take the NFA that accepts it

LM

LML )(

Single final state

15

From construct the equivalentGeneralized Transition Graph in which transition labels are regular

expressions

M

Example:

a

ba,

cM

a

ba

c

16

Another Example:

ba a

b

b

0q 1q 2q

ba,a

b

b

0q 1q 2q

b

b

17

Reducing the states:

ba a

b

b

0q 1q 2q

b

0q 2q

babb*

)(* babb

18

Resulting Regular Expression:

0q 2q

babb*

)(* babb

*)(**)*( bbabbabbr

LMLrL )()(

19

In GeneralRemoving states:

iq q jqa b

cde

iq jq

dae* bce*dce*

bae*

20

The final transition graph:

0q fq

1r

2r

3r4r

*)*(* 213421 rrrrrrr

LMLrL )()(

The resulting regular expression:

21

DFA - regular languages

Push down automata - Context-free

Bounded Turing M’s - Context sensitive

Turing machines - Phrase-structure

Models of computing

22

Foundations

The theory of computation and the practical application it made possible — the computer — was developed by an Englishman called Alan Turing.

24

The Decision ProblemIn 1928 the German mathematician, David Hilbert (1862-1943), asked whether there could be a mechanical way (i.e. by means of a fully specifiable set of instructions) of determining whether some statement in a formal system like arithmetic was provable or not.In 1936 Turing published a paper the aim of which was to show that there was no such method. “On computable numbers, with an application to the Entscheidungs problem.” Proceedings of the London Mathematical Society, 2(42):230-265).

29

FunctionsIt is essential to the idea of a Turing machine that it is not a physical machine, but an abstract one — a set of procedures.

It makes no difference whether the machine is embodied by a person in a boxcar on a track, or a person with a paper and pencil, or a smart and well-trained flamingo.

31

First computers: custom computing machines

1950 -- Eniac: the control is hardwired manually foreach problem.

Control

Input tape (read only)

Output tape (write only)

Work tape (memory)

1940: VON NEUMANN: DISTINCTION BETWEEN DATA AND INSTRUCTIONS

32

Can Machines Think?In “Computing machinery and intelligence,” written in 1950, Turing asks whether machines can think.He claims that this question is too vague, and proposes, instead, to replace it with a different one.That question is: Can machines pass the “imitation game” (now called the Turing test)? If they can, they are intelligent. Turing is thus the first to have offered a rigorous test for the determination of intelligence quite generally.

33

The Turing TestThe game runs as follows. You sit at a computer terminal and have an electronic conversation. You don’t know who is on the other end; it could be a person or a computer responding as it has been programmed to do.

If you can’t distinguish between a human being and a computer from your interactions, then the computer is intelligent.

Note that this is meant to be a sufficient condition of intelligence only. There may be other ways to be intelligent.

35

The Church-Turning Thesis

Turing, and a logician called Alonzo Church (1903-1995), independently developed the idea (not yet proven by widely accepted) that whatever can be computed by a mechanical procedure can be computed by a Turing machine.This is known as the Church-Turing thesis.

37

Turing Machines

38

The Language Hierarchy

*aRegular Languages

Context-Free Languagesnnba Rww

nnn cba ww?

**ba

?

39

*aRegular Languages


nnn cba ww

**ba

Languages accepted byTuring Machines

40

A Turing Machine

............Tape

Read-Write head

Control Unit

41

The Tape

............

Read-Write head

No boundaries -- infinite length

The head moves Left or Right

42

............

Read-Write head

The head at each time step:

1. Reads a symbol 2. Writes a symbol 3. Moves Left or Right

43

............

Example:Time 0

............Time 1

1. Reads

2. Writes

a a cb

a b k c

a

k3. Moves Left

44

............Time 1

a b k c

............Time 2

a k cf

1. Reads

2. Writes

bf

3. Moves Right

45

The Input String

............

Blank symbol

head

a b ca

Head starts at the leftmost positionof the input string

Input string

46

............

Blank symbol

head

a b ca

Input string

Remark: the input string is never empty

47

States & Transitions

1q 2qLba ,

Read Write Move Left

1q 2qRba ,

Move Right

48

Example:

1q 2qRba ,

............ a b ca

Time 1

1qcurrent state

49

............ a b caTime 1

1q 2qRba ,

............ a b cbTime 2

1q

2q

50

............ a b caTime 1

1q 2qLba ,

............ a b cbTime 2

1q

2q

Example:

51

............ a b caTime 1

1q 2qRg,

............ ga b cbTime 2

1q

2q

Example:

52

Determinism

1q

2qRba ,

Allowed Not Allowed

3qLdb ,

1q

2qRba ,

3qLda ,

Turing Machines are deterministic

53

Partial Transition Function

1q

2qRba ,

3qLdb ,

............ a b ca

1q

Example:

No transitionfor input symbol c

Allowed:

54

Halting

The machine halts if there areno possible transitions to follow

55

Example:

............ a b ca

1q

1q

2qRba ,

3qLdb ,

No possible transition

HALT!!!

56

Final States

1q 2q Allowed

1q 2q Not Allowed

• Final states have no outgoing transitions

• In a final state the machine halts

57

Acceptance

Accept Input If machine halts in a final state

Reject Input

If machine halts in a non-final state or If machine enters an infinite loop

58

Turing Machine Example

A Turing machine that accepts the language:

*aa

0q

Raa ,

L,1q

59

aaTime 0

0q

a

0q

Raa ,

L,1q

60

aaTime 1

0q

a

0q

Raa ,

L,1q

61

aaTime 2

0q

a

0q

Raa ,

L,1q

62

aaTime 3

0q

a

0q

Raa ,

L,1q

63

aaTime 4

1q

a

0q

Raa ,

L,1q

Halt & Accept

64

Rejection Example

0q

Raa ,

L,1q

baTime 0

0q

a

65

0q

Raa ,

L,1q

baTime 1

0q

a

No possible Transition

Halt & Reject

66

Infinite Loop Example

0q

Raa ,

L,1q

Lbb ,

A Turing machine for language *)(* babaa

67

baTime 0

0q

a

0q

Raa ,

L,1q

Lbb ,

68

baTime 1

0q

a

0q

Raa ,

L,1q

Lbb ,

69

baTime 2

0q

a

0q

Raa ,

L,1q

Lbb ,

70

baTime 2

0q

a

baTime 3

0q

a

baTime 4

0q

a

baTime 5

0q

a

Infinite

loop

71

Because of the infinite loop:

•The final state cannot be reached

•The machine never halts

•The input is not accepted

72

Another Turing Machine Example

Turing machine for the language }{ nnba

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

73

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

ba

0q

a bTime 0

74

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

bx

1q

a b Time 1

75

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

bx

1q

a b Time 2

76

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

yx

2q

a b Time 3

77

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

yx

2q

a b Time 4

78

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

yx

0q

a b Time 5

79

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

yx

1q

x b Time 6

80

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

yx

1q

x b Time 7

81

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

yx x y

2q

Time 8

82

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

yx x y

2q

Time 9

83

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

yx

0q

x y Time 10

84

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

yx

3q

x y Time 11

85

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

yx

3q

x y Time 12

86

0q 1q 2q3qRxa ,

Raa ,Ryy ,

Lyb ,

Laa ,Lyy ,

Rxx ,

Ryy ,

Ryy ,4q

L,

yx

4q

x y

Halt & Accept

Time 13

87

If we modify the machine for the language }{ nnba

we can easily construct a machine for the language }{ nnn cba

Observation:

88

*aRegular Languages


nnn cba ww

**ba

Languages accepted byTuring Machines

89

More details in Chapter 12 of Rosen.

End of intro to Languages (sets of strings), Grammars, Automata, andTuring Machines (= computation).

91

Formal Definitionsfor

Turing Machines

92

Transition Function

1q 2qRba ,

),,(),( 21 Rbqaq

93

1q 2qLdc ,

),,(),( 21 Ldqcq

Transition Function

94

Turing Machine:

),,,,,,( 0 FqQM

States

Inputalphabet

Tapealphabet

Transitionfunction

Initialstate

blank

Finalstates

95

Configuration

ba

1q

a

Instantaneous description:

c

baqca 1

96

yx

2q

a b

Time 4

yx

0q

a b

Time 5

A Move: aybqxxaybq 02

97

yx

2q

a b

Time 4

yx

0q

a b

Time 5

bqxxyybqxxaybqxxaybq 1102

yx

1q

x b

Time 6

yx

1q

x b

Time 7

98

bqxxyybqxxaybqxxaybq 1102

bqxxyxaybq 12

Equivalent notation:

99

Initial configuration: wq0

ba

0q

a b

w

Input string

100

The Accepted Language

For any Turing Machine M

}:{)( 210 xqxwqwML f

Initial state Final state

101

Standard Turing Machine

• Deterministic

• Infinite tape in both directions

•Tape is the input/output file

The machine we described is the standard:

102

Computing Functionswith

Turing Machines

103

A function )(wf

Domain: Result Region:

has:

D

Dw

S

Swf )()(wf

104

A function may have many parameters:

yxyxf ),(

Example: Addition function

105

Integer Domain

Unary:

Binary:

Decimal:

11111

101

5

We prefer unary representation:

easier to manipulate with Turing machines

106

Definition:

A function is computable ifthere is a Turing Machine such that:

fM

Initial configuration Final configuration

Dw Domain

0q

w

fq

)(wf

final stateinitial state

For all

107

)(0 wfqwq f

Initial Configuration

FinalConfiguration

A function is computable ifthere is a Turing Machine such that:

fM

In other words:

Dw DomainFor all

108

Example

The function yxyxf ),( is computable

Turing Machine:

Input string: yx0 unary

Output string: 0xy unary

yx, are integers

109

0

0q

1 1 1 1

x y

1 Start

initial state

The 0 is the delimiter that separates the two numbers

110

0

0q

1 1 1 1

x y

1

0

fq

1 1

yx

11

Start

Finish

final state

initial state

111

0

fq

1 1

yx

11Finish

final state

The 0 helps when we usethe result for other operations

112

0q

Turing machine for function

1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

yxyxf ),(

113

Execution Example:

11x

11y 0

0q

1 1 1 1

Time 0

x y

Final Result

0

4q

1 1 1 1

yx

(2)

(2)

114

0

0q

1 1Time 0

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

1 1

115

0q

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

01 11 1Time 1

116

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

0

0q

1 1 1 1Time 2

117

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

1q

1 11 11Time 3

118

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

1q

1 1 1 11Time 4

119

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

1q

1 11 11Time 5

120

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

2q

1 1 1 11Time 6

121

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 11 01Time 7

122

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 1 1 01Time 8

123

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 11 01Time 9

124

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 1 1 01Time 10

125

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

3q

1 11 01Time 11

126

0q 1q 2q 3qL, L,01

L,11

R,

R,10

R,11

4q

R,11

4q

1 1 1 01

HALT & accept

Time 12

127

Another Example

The function xxf 2)( is computable

Turing Machine:

Input string: x unary

Output string: xx unary

x is integer

128

0q

1 1

x

1

1

fq

1 1

x2

11

Start

Finish

final state

initial state

129

Turing Machine Pseudocode for xxf 2)(

• Replace every 1 with $

• Repeat:

• Find rightmost $, replace it with 1

• Go to right end, insert 1

Until no more $ remain

130

0q 1q 2q

3q

R,1$

L,1

L,

R$,1 L,11 R,11

R,

Turing Machine for xxf 2)(

131

0q 1q 2q

3q

R,1$

L,1

L,

R$,1 L,11 R,11

R,

Example

0q

1 1

3q

1 11 1

Start Finish

132

Another Example

The function ),( yxf

is computable0

1 yx

yx

if

if

133

Turing Machine for

Input: yx0

Output: 1 0or

),( yxf0

1 yx

yx

if

if

134

Turing Machine Pseudocode:

Match a 1 from with a 1 from x y

• Repeat

Until all of or is matchedx y

• If a 1 from is not matched erase tape, write 1 else erase tape, write 0

x)( yx

)( yx

135

Combining Turing Machines

136

Block Diagram

TuringMachine

input output

137

Example:

),( yxf0

yx yx

yx

if

if

Comparer

Adder

Eraser

yx,

yx,

yx

yx

yx

0

1 Language Recognition (12.4) and Turing Machines (12.5)

Documents

Transcript of 1 Language Recognition (12.4) and Turing Machines (12.5)