CS 208: Computing Theory

1. CS 208: Computing Theory Assoc. Prof. Dr. Brahim Hnich Faculty of Computer Sciences Izmir University of Economics

2. Automata Theory Finite Sate Machines

3. Motivation • Question: What is a computer? • Too complex to allow a precise mathematical model • We introduce an abstract idealized computer • Computational model • One such a model is called finite automaton

4. Key ideas • Formal definition of a finite automaton • Deterministic vs. nondeterministic finite automaton • Regular languages • Operations on regular languages • Regular expressions • Pumping Lemma

5. Automata Theory Deterministic Finite Automaton

6. Front pad Rear pad door Example: Automatic door Opens when someone approaches Holds open until person clears Doesn’t open when someone is standing behind the door

7. Automatic door Sensors • The door has 4 sensors • FRONT: someone on rear pad • BEHIND: someone on the rear pad • BOTH: someone on both pads • NEITHER: no one on either pad

8. Automatic door states • The door has two states • OPEN • CLOSE

9. Automatic door: state diagram CLOSED OPEN

10. Automatic door: state diagram FRONT REAR BOTH CLOSED OPEN

11. Automatic door: state diagram REAR BOTH NEITHER FRONT REAR BOTH CLOSED OPEN

12. Automatic door: state diagram REAR BOTH NEITHER FRONT REAR BOTH FRONT CLOSED OPEN

13. REAR BOTH NEITHER FRONT REAR BOTH FRONT CLOSED OPEN NEITHER Automatic door: state diagram

14. REAR BOTH NEITHER FRONT REAR BOTH FRONT CLOSED OPEN NEITHER Automatic door: state transition table NEITHER FRONT REAR BOTH CLOSED OPEN CLOSED OPEN CLOSED CLOSED CLOSED OPEN OPEN OPEN

15. Informal definition 0 1 1 0 q1 q2 q3 0,1 • The machine M has • States: q1,q2, and q3 • Start state: q1 (incoming arrow from nowhere) • Accept state: q2 (thick circle) • State transitions: arrows

16. Informal definition 0 1 1 0 q1 q2 q3 0,1 • The machine M on input string • begins in start stateq1 • after reading each symbol, 1 takes transition with matching symbol

17. Informal definition 0 1 1 0 q1 q2 q3 0,1 • After processing all the input string machine M produces output • accept if M is in an accepting state • reject otherwise

18. Informal definition 0 1 1 0 q1 q2 q3 0,1 • What happens on input string? • 1101 • 0010 • 01100

19. Informal definition 0 1 1 0 q1 q2 q3 0,1 • What happens on input string? • 1101

20. Informal definition 0 1 1 0 q1 q2 q3 0,1 • What happens on input string? • 1101

21. Informal definition 0 1 1 0 q1 q2 q3 0,1 • What happens on input string? • 1101

22. Informal definition 0 1 1 0 q1 q2 q3 0,1 • What happens on input string? • 1101 • Accept

23. Informal definition 0 1 1 0 q1 q2 q3 0,1 • What happens on input string? • 0010

24. Informal definition 0 1 1 0 q1 q2 q3 0,1 • What happens on input string? • 0010

25. Informal definition 0 1 1 0 q1 q2 q3 0,1 • What happens on input string? • 0010

26. Informal definition 0 1 1 0 q1 q2 q3 0,1 • What happens on input string? • 0010 • Reject

27. Informal definition 0 1 1 0 q1 q2 q3 0,1 • Can you describe the language consisting of all string that M actually accepts?

28. Informal definition 0 1 1 0 q1 q2 q3 0,1 • M actually accepts • any string that ends with a 1 • 1,01,11,01010101010101,… • any string the ends with an even number of 0’s after the last 1 • 100, 101010000,…

29. Formal Definitions • A finite automaton is a 5-tuple (Q,∑,ξ,q0,F), where • Q is a finite set of states • ∑ is a finite set of symbols called the alphabet • ξ : Q x ∑  Q is the transition function • q0 Є Q is the start state • F ⊆ Q is the set of accept states or final states

30. Back to M 0 1 1 0 q1 q2 q3 0,1 • M is a 5-tuple (Q,∑,ξ,q1,F), where • Q is a finite set of states {q1,q2,q3} • ∑ is {0,1} • ξ : Q x ∑  Q is the transition function • q1 Є Q is the start state • F ⊆ Q is {q2} ξ 0 1 q1 q2 q3 q1 q2 q3 q2 q2 q2

31. Another example: M1 0 1 1 q1 q2 0 • M1 is a 5-tuple (Q,∑,ξ,q1,F), where • Q is a finite set of states {q1,q2} • ∑ is {0,1} • ξ : Q x ∑  Q is the transition function • q1 Є Q is the start state • F ⊆ Q is {q2} ξ 0 1 q1 q2 q1 q2 q1 q2

32. Another example: M1 0 1 1 q1 q2 0 • What language does M1 accept?

33. Another example:M1 0 1 1 q1 q2 0 • What language does M1 accept? • M1 accepts all the strings that end with a 1

34. Another example:M2 s a b b a q1 r1 a b a b q2 r2 b a

35. Another example:M2 s a b b a q1 r1 a b a b q2 r2 • What language does M2 accept? b a

36. Another example:M2 s a b b a q1 r1 a b a b q2 r2 • What language does M2 accept? • M1 accepts all the strings that start and end with the same symbol! b a

37. Languages • The language of machine M is the set of strings A that M accepts • L(M) = A • Note that • M can accept many strings • … but M accepts only one language • We say that M recognizeslanguage A if • A= {w | M accepts w}

38. Regular language • A language is called a regular language if some finite automaton recognizes it

39. Formal model of computation • Let M= (Q,∑,ξ,q0,F) be a Deterministic Finite Automaton (DFA) • Let w=w1w2…wn be a string over ∑ • M accepts w if a sequence of states r0,…,rnexist such that: • r0=q0 • ξ(ri,wi+1)=ri+1, i<n • rnЄ F

40. Formal model of computation • Let M= (Q,∑,ξ,q0,F) be a Deterministic Finite Automaton (DFA) • Let w=w1w2…wn be a string over ∑ • M accepts w if a sequence of states r0,…,rnexist such that: • r0=q0 : the machine starts in the start state • ξ(ri,wi+1)=ri+1, i<n: the machines goes from state to state according to the transition function • rnЄ F: machine accepts its input if it ends up in an accept state

41. The regular operations • Union operation • A U B = {x | x Є A or x Є B} • It takes all string and A and B and puts them together into one language

42. The regular operations • Union operation • A U B = {x | x Є A or x Є B} • It takes all string in A and B and puts them together into one language • Let A={good, bad} and B ={boy, girl} •  A U B ={good, bad, boy, girl}

43. The regular operations • Concatenation operation • A o B = {xy| x Є A and y Є B} • It attaches a string from B to the end of a string from A in the new language

44. The regular operations • Concatenation operation • A o B = {xy| x Є A and y Є B} • It attaches a string from B to the end of a string from A in the new language • Let A={good, bad} and B ={boy, girl} • A o B = {goodboy, goodgirl, badboy, badgirl}

45. The regular operations • Star operation • A*={x1x2…xk| k is positive and each xiЄ A} • It applies to a single language • Attaches any number of string from A to get a string in the new language

46. The regular operations • Star operation • A*={x1x2…xk| k is positive and each xiЄ A} • It applies to a single language • Attaches any number of string from A to get a string in the new language • Let A={good, bad} •  A* = {Ø, good,bad, goodgood, goodbad, badbad, badgood, gooodgooodgood, goodgoodbad, goodbadgood,…}

47. “Closed under” • The set of natural number N={1,2,3,….} is closed under multiplication • For any a and b in N, a x b is also in N • 3 x 5 = 15: 3, 5, and 15 are all naturals • In general, a collection of objects is closed under some operation if applying that operation to members of the collection returns and object still in the collection • Note; the set of natural numbers is not closed under division • 3 / 5 is not a natural number

48. Regular languages closed under union • Theorem: The class of regular languages is closed under the union operation • If A1 and A2 are regular languages, so is A1 U A2

49. Proof idea • How to prove such a theorem? • Remember that the only result we have seen so far is • “A language is called a regular language if some finite automaton recognizes it”

50. Proof idea • “A language is called a regular language if some finite automaton recognizes it” • So, let us look at our theorem from this perspective: • If A1 and A2 are regular languages, so is A1 U A2 M1