Languages and Finite Automata

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

2. Languages • 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)

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 Finite Automaton String

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

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

28. Initial Configuration Input String

29. Reading the Input

32. Output: “accept”

33. Rejection

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 accept accept

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

50. Observation • Language accepted by : • Language rejected by :