Basic idea about Automata Theory

1. 1
Automata Theory
Dr. Muhammad Saleem Vighio (PhD, Denmark)
Associate Professor,
Information Technology, Quaid-e-Awam University of
Engineering, Science & Technology, Nawabshah
Lecture 01 and 02
Monday, April 08th 2017

2. • Course C.Hs:
• 04 C.Hs:
- Theoretical work
- Practical work
• Recommended Text Books:
• Introduction to Automata Theory, Languages and Computation
J.E. Hopcroft, R. Motwani, J.D. Ullman
3rd Edition (Addison Wesley/Pearson)
• Introduction to the Theory of Computation, 2nd Ed.
Michael Sipser
• Introduction to Computer Theory
Daniel I.A. Cohen
• Theory of Automata, Formal Languages and Computation
S. P. Eugene Xavier
• Online Literature
Course Introduction

3. • Teaching plan:
• 02 tests:
- At the end of 20th and 45th lectures respectively
• Class exercises
• Short presentations
• Practical exercises
Course Introduction

4. Lecture slides/announcements/results can be found at
course home page:
http://saleem.quest.edu.pk/
Course Introduction

5. • During the lecture:
• Be punctual/regular
• Concentrate on the lecture
• Remain active and interactive
• Try to find/ask (research) questions
• Think in terms of “why”, “how”
During the Course

6. • In studying this subject, seek to determine:
• What can and cannot be computed ?
• How quickly ?
• With how much memory ? and
• On which type of computational model ?
What to Seek?

7. • Introduction
• Regular Languages and their descriptors
- Finite state automata
- Non-deterministic automata
- Regular expressions
- Algorithms to decide questions about regular languages,
e.g., is it empty?
- Closure properties of regular languages.
• Context Free Languages and their descriptors
- Context free grammars
- Pushdown automata
• Turing Machines and TM Variants
• Decidable and Undecidable Languages
• Time and Space Complexities
Course Outline

8. • Introduction
• Models of Computation
• Languages
Lecture Outline

9. 9
What is Automata Theory?
• Study of abstract computing devices, or “machines”
• Automaton = an abstract computing device
Note: A “device” need not even be a physical hardware!
• A fundamental question in computer science:
 Find out what different models of machines can do and
cannot do
 The theory of computation
• Computability vs. Complexity

10. Computation
• Computation is a general term for any type of
information processing that can be
represented as an algorithm precisely
(mathematically).
Examples:
• Adding two numbers in our brains, on a piece of
paper or using a calculator.
• Converting a decimal number to its binary
presentation or vice versa.
• Finding the greatest common divisors of two
numbers.
• …

11. • The theory of computation studies
• whether (computability theory), and
• how efficiently (complexity theory)
certain problems can be solved on a computer, or rather on a model
of a computer.
• Several equivalent models of computational devices can be used:
• Finite automata.
• Pushdown automata
• Simple while programming language (pseudo-code).
• Turing machines.
• ...
Theory of Computation

12. Theory of Computation: A
Historical Perspective
1930s • Alan Turing studies Turing machines
• Decidability
• Halting problem
1940-1950s • “Finite automata” machines studied
• Noam Chomsky proposes the
“Chomsky Hierarchy” for formal
languages
1969 Cook introduces “intractable” problems
or “NP-Hard” problems
1970- Modern computer science: compilers,
computational & complexity theory evolve

13. 13
Models of Computation

14. 14
Computation
CPU memory

15. 15
CPU
input memory
output memory
Program memory
temporary memory

16. 16
CPU
input memory
output memory
Program memory
temporary memory
3
)( xxf 
compute xx
compute xx 2
Example:

17. 17
CPU
input memory
output memory
Program memory
temporary memory
3
)( xxf 
compute xx
compute xx 2
2x

18. 18
CPU
input memory
output memory
Program memory
temporary memory
3
)( xxf 
compute xx
compute xx 2
2x
42*2 z
82*)(  zxf

19. 19
CPU
input memory
output memoryProgram memory
temporary memory
3
)( xxf 
compute xx
compute xx 2
2x
42*2 z
82*)(  zxf
8)( xf

20. 20
Automaton
CPU
input memory
output memory
Program memory
temporary memory
Automaton

21. 21
Different Kinds of Automata
Automata are distinguished by the temporary memory
• Finite Automata: no temporary memory
• Pushdown Automata: stack
• Turing Machines: random access memory

22. 22
input memory
output memory
temporary memory
Finite
Automaton
Finite Automaton
Example: Vending Machines
(small computing power)

23. 23
input memory
output memory
Stack
Pushdown
Automaton
Pushdown Automaton
Example: Compilers for Programming Languages
(medium computing power)
Push, Pop

24. 24
input memory
output memory
Random Access Memory
Turing
Machine
Turing Machine
Examples: Any Algorithm
(highest computing power)

25. 25
Finite
Automata
Pushdown
Automata
Turing
Machine
Power of Automata
Less power More power
Solve more
computational problems

26. 26
Languages

27. 27
A language is a set of strings
String: A sequence of letters
Examples: “cat”, “dog”, “house”, …
Defined over an alphabet:
 zcba ,,,, 

28. 28
Alphabets and Strings
We will use small alphabets:
Strings
abbaw
bbbaaav
abu



 ba,
baaabbbaaba
baba
abba
ab
a

29. 29
String Operations: Concatenation
m
n
bbbv
aaaw


21
21


bbbaaa
abba
mn bbbaaawv  2121
Concatenation
abbabbbaaa

30. 30
12aaaw n
R

naaaw 21 ababaaabbb
Reverse
bbbaaababa
String Operations: Reverse

31. 31
String Operations: String Length
Length:
naaaw 21
nw 
1
2
4



a
aa
abbaExamples

32. 32
String Operations:
Length of Concatenation
vuuv 
853
8
5,
3,




vuuv
aababaabuv
vabaabv
uaabuExamples

33. 33
String Operations: Empty String
A string with no letters: 
abbaabbaabba
www





 0Observations:

34. 34
String Operations: Substring
Substring of string:
a subsequence of consecutive characters
String Substring
bbab
b
abba
ab
abbab
abbab
abbab
abbab

35. 35
String Operations: Prefix and Suffix
abbab
abbab
abba
abb
ab
a


b
ab
bab
bbab
abbab uvw 
prefix
suffix
prefixes Suffixes
Example:

36. 36
String Operations: Power
 
n
n
wwww 
  abbaabbaabba 2
0
w
  0
abba
Example:
Definition:

37. 37
String Operations: The * Operation
: the set of all possible strings from
alphabet
*

 
 ,,,,,,,,,*
,
aabaaabbbaabaaba
ba



38. 38
String Operations: The + Operation
: the set of all possible strings from
alphabet except



 
 ,,,,,,,,,*
,
aabaaabbbaabaaba
ba



*
 ,,,,,,,, aabaaabbbaabaaba


39. 39
Languages
A language is any subset of *
 
 ,,,,,,,,*
,
aaabbbaabaaba
ba


 
 
},,,,,{
,,
aaaaaaabaababaabba
aabaaa


Example:
Languages:

40. 40
Note that:
}{}{ 
0}{ 
1}{ 
0
Sets
Set size
Set size
String length

41. 41
Another Language Example
An infinite language }0:{  nbaL nn
aaaaabbbbb
aabb
ab

L Labb

42. 42
Operations on Languages
The usual set operations
Complement:
   
   
     aaaaaabbbaaaaaba
ababbbaaaaaba
aaaabbabaabbbaaaaaba
,,,,
}{,,,
},,,{,,,





LL  *
   ,,,,,,, aaabbabaabbaa 

43. 43
Operations on Languages: Reverse
Definition: }:{ LwwL RR

   ababbaabababaaabab R
,,,, 
}0:{
}0:{


nabL
nbaL
nnR
nn
Examples:

44. 44
Operations on Languages: Concatenation
Definition:  2121 ,: LyLxxyLL 
  
 baaabababaaabbaaaab
aabbaaba
,,,,,
,,,

Example:

45. 45
Operations on Languages: Power
Definition:

n
n
LLLL 
     
 bbbbbababbaaabbabaaabaaa
babababa
,,,,,,,
,,,, 3

 
   



0
0
,, aaabbaa
L
Example:
Special Case:

46. 46
More Examples
}0:{  nbaL nn
}0,:{2
 mnbabaL mmnn
2
Laabbaaabbb

47. 47
Operations on Languages:
Star-Closure (Kleene *)
Definition:
Example:
 210
* LLLL 
 















,,,,
,,,,
,,
,
*,
abbbbabbaaabbaaa
bbbbbbaabbaa
bba
bba


48. 48
Operations on Languages:
Positive Closure
Definition:
 

*
21
L
LLL 
 











,,,,
,,,,
,,
,
abbbbabbaaabbaaa
bbbbbbaabbaa
bba
bba

49. • Recommended Tools:
• UPPAAL
http://www.uppaal.org/
• Tool for the modelling, simulation and verification of
different types of computational algorithms: real-
time, embedded, and concurrent systems e.g.,
embedded processors in mobile phones, microwave
ovens, sensor nodes communicating with the
environment, traffic control systems and so on.
ToC: Course Introduction

50. INPUT
UPPAAL