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1 Languages and Finite Automata or how to talk to machines...

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1 1 Languages and Finite Automata or how to talk to machines...

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

3 3 Alphabets and Strings We will use small alphabets Strings

4 4 String Operations Concatenation Reverse

5 5 String Length Length: Examples:

6 6 Recursive Definition of Length For any letter: For any string : Example:

7 7 Length of Concatenation Example:

8 8 Proof of Concatenation Length Claim: Proof: By induction on the length Induction basis: is only one symbol From definition of length:

9 9 Inductive hypothesis: for all with Inductive step: we will prove for

10 10 Inductive Step Write, where From definition of length: From inductive hypothesis: Thus:

11 11 Empty String A string with no letters: Observations:

12 12 Substring Definition: A substring of a string is any sequence of consecutive characters Example: Substrings

13 13 Prefix and Suffix Prefixes Suffixes prefix suffix

14 14 Another Operation Example: Definition for any :

15 15 The * Operation : the set of all possible strings from alphabet Example:

16 16 Language A language is any subset of Examples: A string is called “sentence”

17 17 Another Example An infinite language

18 18 Operations on Languages The usual set operations Complement:

19 19 Reverse Definition: Examples:

20 20 Concatenation Definition: Example:

21 21 Another Operation Definition: Example: Special case:

22 22 Example

23 23 Star-Closure (Kleene *) Definition: Example:

24 24 Positive Closure Definition:

25 25 Finite Automata Input String Output String Finite Automaton

26 26 Finite Accepter Input “Accept” or “Reject” String Finite Automaton Output

27 27 Transition Graph initial state final state “accept” state transition Abba -Finite Accepter

28 28 Initial Configuration Input String

29 29 Reading the Input

30 30

31 31

32 32 Output: “accept”

33 33 Rejection

34 34

35 35

36 36 Output: “reject”

37 37 Formalities Deterministic Finite Accepter (DFA) : set of states : input alphabet : transition function : initial state : set of final states

38 38 Input Aplhabet

39 39 Set of States

40 40 Initial State

41 41 Set of Final States

42 42 Transition Function

43 43 Transition Function

44 44 Extended Transition Function

45 45 Recursive Definition

46 46 Languages Accepted by DFAs Take DFA Definition: The language accepted by contains all input strings accepted by In other words: = { strings that drive to a final state}

47 47 Example accept

48 48 Another Example accept

49 49 Formally For a DFA Language accepted by : alphabet transition function initial state final states

50 50 Observation Language accepted by : Language rejected by :

51 51 More Examples accept trap state

52 52 = { all substrings with prefix } accept

53 53 = { all strings without substring }

54 54 Regular Languages Definition: A language is regular if there is a DFA such that All the regular languages form a family

55 55 Example The language is regular:


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