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

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 Alphabets and Strings We will use small alphabets Strings

4 String Operations Concatenation Reverse

5 String Length Length: Examples:

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

7 Length of Concatenation Example:

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

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

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

11 Empty String A string with no letters: Observations:

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

13 Prefix and Suffix Prefixes Suffixes prefix suffix

14 Another Operation Example: Definition for any :

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

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

17 Another Example An infinite language

18 Operations on Languages The usual set operations Complement:

19 Reverse Definition: Examples:

20 Concatenation Definition: Example:

21 Another Operation Definition: Example: Special case:

22 Example

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

24 Positive Closure Definition:

25 Finite Automata Input String Output String Finite Automaton

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

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

28 Initial Configuration Input String

29 Reading the Input

30

31

32 Output: “accept”

33 Rejection

34

35

36 Output: “reject”

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

38 Input Aplhabet

39 Set of States

40 Initial State

41 Set of Final States

42 Transition Function

43 Transition Function

44 Extended Transition Function

45 Recursive Definition

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 Example accept

48 Another Example accept

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

50 Observation Language accepted by : Language rejected by :

51 More Examples accept trap state

52 = { all substrings with prefix } accept

53 = { all strings without substring }

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

55 Example The language is regular: