Formal Languages

Models of Computation

2

Computation

CPU memory

3

CPU

input memory

output memory

Program memory

temporary memory

4

CPU

input memory

output memoryProgram memory

temporary memory

3)( xxf

compute xx

compute xx 2

Example:

5

CPU

input memory

output memoryProgram memory

temporary memory

3)( xxf

compute xx

compute xx 2

2x

6

CPU

input memory

output memoryProgram memory

temporary memory3)( xxf

compute xx

compute xx 2

2x

42*2 z82*)( zxf

7

CPU

input memory

output memoryProgram memory

temporary memory3)( xxf

compute xx

compute xx 2

2x

42*2 z82*)( zxf

8)( xf

Automata theory

Automata theory studies the laws of computation.

In reality, the laws of computation are not quite understood, but automata theory is a good start.

A simple computer

BATTERY

SWITCH

input: switch

output: light bulb

actions: flip switch

states: on, off

A simple “computer”

BATTERY

SWITCH

off onstartf

f

input: switch

output: light bulb

actions: f for “flip switch”

states: on, off

bulb is on if and only if there was an odd number of flips

Another “computer”

BATTERY

off offstart1

inputs: switches 1 and 2

actions: 1 for “flip switch 1”actions: 2 for “flip switch 2”

states: on, off

bulb is on if and only if both switches were flipped an odd number of times

1

2

1

off on1

1

2 2 2 2

12

Automaton

CPU

input memory

output memory

Program memory

temporary memory

Automaton

13

Different Kinds of Automata

Automata are distinguished by the temporary memory

• Finite Automata: no temporary memory

• Pushdown Automata: stack

• Turing Machines: random access memory

14

input memory

output memory

temporary memory

Finite

Automaton

Finite Automaton

Example: Vending Machines

(small computing power)

15

input memory

output memory

Stack

Pushdown

Automaton

Pushdown Automaton

Example: Compilers for Programming Languages

(medium computing power)

Push, Pop

16

input memory

output memory

Random Access Memory

Turing

Machine

Turing Machine

Examples: Any Algorithm

(highest computing power)

17

Finite

Automata

Pushdown

Automata

Turing

Machine

Power of Automata

Less power More power

Solve more

computational problems

18

Languages

19

A language is a set of strings

String: A sequence of letters

Examples: “cat”, “dog”, “house”, …

Defined over an alphabet: zcba ,,,,

20

Alphabets and StringsWe will use small alphabets:

Strings

abbaw

bbbaaav

abu

ba,

baaabbbaaba

baba

abba

ab

a

21

String Operations

m

n

bbbv

aaaw

21

21

bbbaaa

abba

mn bbbaaawv 2121

Concatenation

abbabbbaaa

22

12aaaw nR

naaaw 21 ababaaabbb

Reverse

bbbaaababa

23

String Length

Length:

Examples:

naaaw 21

nw

1

2

4

a

aa

abba

24

Length of Concatenation

Example:

vuuv

853

8

5,

3,

vuuv

aababaabuv

vabaabv

uaabu

25

Empty StringA string with no letters:

Observations:

abbaabbaabba

www

0

26

SubstringSubstring of string:

a subsequence of consecutive characters

String Substring

bbab

b

abba

ab

abbab

abbab

abbab

abbab

27

Prefix and Suffix

Prefixes Suffixesabbab

abbab

abba

abb

ab

a

b

ab

bab

bbab

abbab uvw

prefix

suffix

28

Another Operation

Example:

Definition:

n

n wwww

abbaabbaabba 2

0w

0abba

29

The * Operation : the set of all possible strings from alphabet

*

,,,,,,,,,*

,

aabaaabbbaabaaba

ba

30

The + Operation : the set of all possible strings from alphabet except

,,,,,,,,,*

,

aabaaabbbaabaaba

ba

*

,,,,,,,, aabaaabbbaabaaba

31

LanguagesA language is any subset of

Example:

Languages:

*

,,,,,,,,*

,

aaabbbaabaaba

ba

},,,,,{

,,

aaaaaaabaababaabba

aabaaa

32

Note that:

}{}{

0}{

1}{

0

Sets

Set size

Set size

String length

33

Another Example

An infinite language }0:{ nbaL nn

aaaaabbbbb

aabb

ab

L Labb

34

Operations on LanguagesThe usual set operations

Complement:

aaaaaabbbaaaaaba

ababbbaaaaaba

aaaabbabaabbbaaaaaba

,,,,

}{,,,

},,,{,,,

LL *

,,,,,,, aaabbabaabbaa

35

Reverse

Definition:

Examples:

}:{ LwwL RR

ababbaabababaaabab R ,,,,

}0:{

}0:{

nabL

nbaL

nnR

nn

36

Concatenation

Definition:

Example:

2121 ,: LyLxxyLL

baaabababaaabbaaaab

aabbaaba

,,,,,

,,,

37

Another OperationDefinition:

Special case:

n

n LLLL

bbbbbababbaaabbabaaabaaa

babababa

,,,,,,,

,,,, 3

0

0

,, aaabbaa

L

38

More Examples

}0:{ nbaL nn

}0,:{2 mnbabaL mmnn

2Laabbaaabbb

39

Star-Closure (Kleene *)

Definition:

Example:

210* LLLL

,,,,

,,,,

,,

,

*,

abbbbabbaaabbaaa

bbbbbbaabbaa

bbabba

40

Positive Closure

Definition:

*

21

L

LLL

,,,,

,,,,

,,

,

abbbbabbaaabbaaa

bbbbbbaabbaa

bba

bba