Languages and Finite Automata

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Albert McKinney
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1 Languages and Finite Automata or how to talk to machines... Costas Busch - RPI 1

2 Languages A language is a set of strings String: A sequence of letters (a word) Examples: cat, dog, house, Defined over an alphaet: set of symols (letters) a,, c,, z Costas Busch - RPI 2

3 Alphaets and Strings We will use small alphaets Strings a a aa aa aaaaaa u a v aaa w aa Costas Busch - RPI 3

4 w w String Operations a 1 a 2 an v 1 2 aa v aaa m Concatenation wv wv a1a2 an 1 2 aaaaa m Reverse v v R R m 2 1 aaa Costas Busch - RPI 4

5 String Length w a a 1 2 a n Length: w n Examples: aa aa 2 a 1 4 Costas Busch - RPI 5

6 Recursive Definition of Length For any letter: a 1 For any string wa : wa w 1 Example: aa a 1 a 1 1 a Costas Busch - RPI 6

7 Length of Concatenation uv u v Example: u aa, u 3 v aaa, v 5 uv aaaaa 8 uv u v Costas Busch - RPI 7

8 Proof of Concatenation Length Claim: uv u v Proof: By induction on the length v Induction asis: v 1 v is only one symol From definition of length: uv u 1 u v Costas Busch - RPI 8

9 Inductive hypothesis: uv u v for all v with v 1,2,, n Inductive step: we will prove uv u v for v n 1 Costas Busch - RPI 9

10 Inductive Step Write v wa, where w n, a 1 From definition of length: uv uwa uw 1 wa w 1 From inductive hypothesis: uw u w Thus: uv u w 1 u wa u v Costas Busch - RPI 10

11 Empty String A string with no letters: Oservations: 0 w w w aa aa aa Costas Busch - RPI 11

12 Definition: Sustring w A sustring of a string consecutive characters is any sequence of Example: aa Sustrings a aa a Costas Busch - RPI 12

13 Prefix and Suffix aa Prefixes a a a aa aa Suffixes aa a a a prefix w uv suffix Costas Busch - RPI 13

14 Another Operation n w ww w n Example: 2 aa aaaa Definition w for any : 0 w 0 aa Costas Busch - RPI 14

15 * The * Operation : the set of all possile strings from alphaet Example: * a,, a,, aa, a, a,, aaa, aa, * a,, aa, a, a,, aaa, aa, Costas Busch - RPI 15

16 Language A language is any suset of * Examples: * a,, a,, aa, a, a,, aaa, aa,, a, aa, aa {, aa, aa, aa, a, aaaaaa } A string is called sentence Costas Busch - RPI 16

17 An infinite language Another Example L a n n : n 0 a aa aaaaa L a L Costas Busch - RPI 17

18 Operations on Languages The usual set operations a, a, aaaa a, a, aaaa, a, a { a, a,, aaaa } { a } a, a, aaaa, a a, aaaa Complement: L * L a, a,, aa, a,, aaa, Costas Busch - RPI 18

19 Reverse Definition: L R R w : w L Examples: R a, aa, aa a, aa, aa L a n n : n 0 L R n a n : n 0 Costas Busch - RPI 19

20 Concatenation Definition: L 1 L2 xy x L1, : y L 2 Example: a, a, a, aa a, aaa, a, aaa, a, aaa Costas Busch - RPI 20

21 Definition: Another Operation n L LL L n Example: a, aaa, aa, aa, a, aa, a, a, 3 a, a, a, Special case: L 0 0 a, a, aaa Costas Busch - RPI 21

22 Example L a n n : n 0 L 2 n n m m a a : n, m 0 2 aaaaa L Costas Busch - RPI 22

23 Star-Closure (Kleene *) Definition: 0 2 L* L L L 1 Example: a, *, a,, aa, a, a,, aaa, aa, aa, a, Costas Busch - RPI 23

24 Positive Closure Definition: L 1 L L 2 L * a, a,, aa, a, a,, aaa, aa, aa, a, Costas Busch - RPI 24

25 Finite Automata Costas Busch - RPI 25

26 Finite Automaton Input String Finite Automaton Output String Costas Busch - RPI 26

27 Finite Accepter Input String Finite Automaton Output Accept or Reject Costas Busch - RPI 27

28 Transition Graph Aa -Finite Accepter a a a a q 5 q0 q3 q4 initial state state transition final state accept Costas Busch - RPI 28

29 Initial Configuration Input String a a q 0 a a a a q 5 q3 q4 Costas Busch - RPI 29

30 Reading the Input a a a a a a q 5 q0 q3 q4 Costas Busch - RPI 30

31 a a a a a a q 5 q0 q3 q4 Costas Busch - RPI 31

32 a a a a a a q 5 q0 q3 q4 Costas Busch - RPI 32

33 a a a a a a q 5 q0 q3 q4 Costas Busch - RPI 33

34 a a Input finished a a a a q 5 q0 q3 q4 Output: accept Costas Busch - RPI 34

35 Rejection a a q 0 a a a a q 5 q3 q4 Costas Busch - RPI 35

36 a a a a a a q 5 q0 q3 q4 Costas Busch - RPI 36

37 a a a a a a q 5 q0 q3 q4 Costas Busch - RPI 37

38 a a a a a a q 5 q0 q3 q4 Costas Busch - RPI 38

39 Input finished a a Output: a a a a q 5 q0 q3 q4 reject Costas Busch - RPI 39

40 Another Rejection q 0 a a a a q 5 q3 q4 Costas Busch - RPI 40

41 q 0 a a a a q 5 q3 q4 Output: reject Costas Busch - RPI 41

42 Another Example a a a a, q0 Costas Busch - RPI 42

43 a a a a, q0 Costas Busch - RPI 43

44 a a a a, q0 Costas Busch - RPI 44

45 a a a a, q0 Costas Busch - RPI 45

46 Input finished a a a Output: accept a, q0 Costas Busch - RPI 46

47 Rejection a a a, q0 Costas Busch - RPI 47

48 a a a, q0 Costas Busch - RPI 48

49 a a a, q0 Costas Busch - RPI 49

50 a a a, q0 Costas Busch - RPI 50

51 Input finished a a a, q0 Output: reject Costas Busch - RPI 51

52 Formalities Deterministic Finite Accepter (DFA) M Q,,, q0, F Q q 0 F : set of states : input alphaet : transition function : initial state : set of final states Costas Busch - RPI 52

53 Input Alphaet a a a a q 5 q0 q3 q4 Costas Busch - RPI 53

54 Set of States Q Q q 0 1, 2, 3, 4,, q q q q q 5 a a a a q 5 q0 q3 q4 Costas Busch - RPI 54

55 Initial State q0 q 0 a a a a q 5 q3 q4 Costas Busch - RPI 55

56 Set of Final States F F q 4 a a a a q0 q3 q 5 q 4 Costas Busch - RPI 56

57 Transition Function : Q Q a a a a q 5 q0 q3 q4 Costas Busch - RPI 57

58 q 0, a q 1 a a a q a q0 1 q 5 q3 q4 Costas Busch - RPI 58

59 q 0, q 5 q 0 a a a a q 5 q3 q4 Costas Busch - RPI 59

60 3 2, q q a a a a q 5 q0 q3 q4 Costas Busch - RPI 60

61 a q q 0 q 1 q 2 q 3 5 q5 Transition Function q q5 3 q q4 5 q 4 q 5 q 5 q 5 q 5 q 5 a a a a q 5 q0 q3 q4 Costas Busch - RPI 61

62 Extended Transition Function * *: Q * Q a a a a q 5 q0 q3 q4 Costas Busch - RPI 62

63 q 0, a 2 * q a a a q a q0 2 q 5 q3 q4 Costas Busch - RPI 63

64 q 0, aa 4 * q a a a a q 5 q0 q3 q4 Costas Busch - RPI 64

65 q 0, aaa 5 * q q 0 a a a a q 5 q3 q4 Costas Busch - RPI 65

66 Oservation: There is a walk from with lael w q to q * q, w q q w q q 1 2 w 1 2 k k q Costas Busch - RPI 66

67 Example: There is a walk from with lael q 0 aaa to q 5 q 0, aaa 5 * q q 0 a a a a q 5 q3 q4 Costas Busch - RPI 67

68 Costas Busch - RPI 68 Recursive Definition ) ),, *( (, *, * w q w q q q q q w q 1 q q q w q ), (, * 1 1 1, * ), (, * q w q q w q ) ),, *( (, * w q w q

69 Costas Busch - RPI 69 0 q 1 q 2 q 3 q 4 q a a q 5 a a ,,,,,, * ),, * (, * q q a q a q a q a q

70 Languages Accepted y DFAs Take DFA M Definition: The language L contains all input strings accepted y M M L M M = { strings that drive to a final state} Costas Busch - RPI 70

71 M aa Example L M a a a a q 5 q0 q3 q4 accept Costas Busch - RPI 71

72 Another Example M, a aa L, M a a a a q 5 q0 q3 q4 accept accept accept Costas Busch - RPI 72

73 Formally For a DFA M Q,,, q0, F M Language accepted y : L M w : * q, w * 0 F alphaet transition function initial state final states q0 w q q F Costas Busch - RPI 73

74 Oservation Language rejected y M : L M w : * q, w * 0 F q0 w q q F Costas Busch - RPI 74

75 More Examples L n M { a : n 0} a a, q0 accept trap state Costas Busch - RPI 75

76 L M = { all strings with prefix a } a q0 a accept q 3 Costas Busch - RPI 76

77 L M = { all strings without sustring 001 } , 1 Costas Busch - RPI 77

78 Regular Languages A language a DFA L is regular if there is M L LM such that All regular languages form a language family Costas Busch - RPI 78

79 Examples of regular languages: aa,a,aa { a n : n 0} { all strings with prefix } a { all strings with prefix } a { all strings without sustring } 001 There exist automata that accept these Languages (see previous slides). Costas Busch - RPI 79

80 Another Example The language is regular: L L M L awa a : w a, * a q0 q3 a q 4 Costas Busch - RPI 80

81 There exist languages which are not Regular: Example: L n n { a : n 0} There is no DFA that accepts such a language Costas Busch - RPI 81

8 ε. Figure 1: An NFA-ǫ

8 ε. Figure 1: An NFA-ǫ 0 1 2 3 4 a 6 5 7 8 9 10 LECTURE 27 Figure 1: An FA-ǫ 12.1 ǫ Transitions In all automata that we have seen so far, every time that it has to change from one state to another, it must use one input symol.