1. 1 Languages

2. 2 A language is a set of strings String: A sequence of letters Examples: “cat”, “dog”, “house”, … Defined over an alphabet: Languages

3. 3 Alphabets and Strings We will use small alphabets: Strings

4. 4 String Operations Concatenation

5. 5 Reverse

6. 6 String Length Length: Examples:

7. 7 Recursive Definition of Length For any letter: For any string : Example:

8. 8 Length of Concatenation Example:

9. 9 Proof of Concatenation Length Claim: Proof: By induction on the length Induction basis: From definition of length:

10. 10 Inductive hypothesis: for Inductive step: we will prove for

11. 11 Inductive Step Write, where From definition of length: From inductive hypothesis: Thus:

12. 12 Empty String A string with no letters: Observations:

13. 13 Substring Substring of string: a subsequence of consecutive characters String Substring

14. 14 Prefix and Suffix Prefixes Suffixes prefix suffix

15. 15 Another Operation Example: Definition:

16. 16 The * Operation : the set of all possible strings from alphabet

17. 17 The + Operation : the set of all possible strings from alphabet except

18. 18 Language A language is any subset of Example: Languages:

19. 19 Another Example An infinite language

20. 20 Operations on Languages The usual set operations Complement:

21. 21 Reverse Definition: Examples:

22. 22 Concatenation Definition: Example:

23. 23 Another Operation Definition: Special case:

24. 24 More Examples

25. 25 Star-Closure (Kleene *) Definition: Example:

26. 26 Positive Closure Definition:

27. 27 Finite Automata

28. 28 Finite Automaton Input String Output String Finite Automaton

29. 29 Finite Accepter Input “Accept” or “Reject” String Finite Automaton Output

30. 30 Transition Graph initial state final state “accept” state transition Abba -Finite Accepter

31. 31 Initial Configuration Input String

32. 32 Reading the Input

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36. 36 Output: “accept” Input finished

37. 37 Rejection

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41. 41 Output: “reject” Input finished

42. 42 Another Example

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46. 46 Output: “accept” Input finished

47. 47 Rejection

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51. 51 Output: “reject” Input finished

52. 52 Formalities Deterministic Finite Accepter (DFA) : set of states : input alphabet : transition function : initial state : set of final states

53. 53 Input Alphabet

54. 54 Set of States

55. 55 Initial State

56. 56 Set of Final States

57. 57 Transition Function

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61. 61 Transition Function

62. 62 Extended Transition Function

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66. 66 Observation: There is a walk from to with label

67. 67 Recursive Definition

68. 68

69. 69 Languages Accepted by DFAs Take DFA Definition: The language contains all input strings accepted by = { strings that drive to a final state}

70. 70 Example accept

71. 71 Another Example accept

72. 72 Formally For a DFA Language accepted by : alphabet transition function initial state final states

73. 73 Observation Language accepted by : Language rejected by :

74. 74 More Examples accept trap state

75. 75 = { all substrings with prefix } accept

76. 76 = { all strings without substring }

77. 77 Regular Languages A language is regular if there is a DFA such that All regular languages form a language family

78. 78 Example The language is regular: