Finite Automata (Finite State Machine)

1. Finite Automata(Finite State Machine) Lecture 4-5 Ref. Handout p18-24

3. Controlling Things ... If x has happened then do something If in state y then do something change to new state If x happened while in state y do something change to new state

4. State Transition Table

5. State Diagrams a state transition A x B when in state A if x happens then we change to state B

6. cancelled start tomorrow Happy Anxious done next week State Transition Diagram

7. cancelled tomorrow Happy Anxious done next week A Finite Automata

8. Components input symbol starting state (one per machine) d transition halting state (one or more) non-halting state

9. Sequence of Input Symbols What input sequences leave this FA in its halt state? a S1 S2 b ac ab abc abac yes no no yes c S3 { (ab)nac | n>=0 } { ac, abac, ababac, abababac, ... }

10. cancelled tomorrow Happy Anxious done next week The Original Example ... Some acceptable inputs nnnntc ntd Λ nntcntd

11. Recognising Strings the set of all strings accepted/recognised by an FA = the language accepted/recognised by the FA

12. Rejecting Strings An FA rejects a string if with some symbols remaining there is no transition for the next symbol or with no symbols remaining the FA is not in a halt state

13. c t Happy Anxious d n Rejected / Accepted Strings tdnnn yes/no tdnnt yes/no n yes/no ntc yes/no ntcd yes/no

14. { Λ, a, aa, aaa, aaaa, ...} = { an | n >= 0 } { a, aa, aaa, aaaa, ...} = { an | n > 0 } { 00, 01, 10, 11 } Recall some simple languages ...Can you draw FA to recognise them?

15. An Assignment Related Example Draw an FA to recognise non-negative integers Extend it to all integers (i.e. allowing +,- signs)

16. Memo for In-class test 4 [ /5]

17. Non-Deterministic Automata a 1 What happens here? 2 b c d cd cdacd 3 4 d

18. Non-Deterministic Automata • In deterministic automata, there can never be two transitions to different state for the same input symbol. • If there are more than one transition to different states for the same input symbol, we call it a non-deterministic automata.

19. Deterministic FAs and Non-Deterministic FAs For every non-deterministic FA (NFA) there is a deterministic FA (DFA) which accepts the same language

20. A Simple Example NFA 2 a b 3 1 a DFA a b

21. Step by Step (1) NFA 2 a b 3 1 a after ‘a’, must be in state 2 or 3 a { 1 } { 2,3 }

22. Step by Step (2) NFA 2 a b 3 1 a from state 2, read ‘b’, move to state 3 a b {3} { 1 } { 2,3 }

23. Last Step NFA 2 a b 3 1 a a b { 1 } { 2,3 } {3} 3 is a halt state

24. A General Algorithm • Write down {1} as the start state of the DFA • Choose a state in the DFA which hasn’t been processed • Work out all transitions from it adding new states • Repeat from step 2 until all states have been processed • Mark halting states

25. Building DFAs Machine A Language L start halt Machine B Language K start halt

26. What does this do? Machine A Language L halt start Machine B Language K halt

27. What does this do? Machine A Language L start Machine B Language K ?? halt

28. What does this do? Machine A Language L start halt

29. Using Building Blocks For example, you are asked to define a format for a collection of pictures’ file names G = { a1.gif, a2.gif, a3.gif, ... a33.gif, .... }

30. Using Building Blocks Now you are asked to define a format for another collection of pictures’ file names P = { b1.jpg, b2.jpg, b3.jpg, ... b33.jpg, .... }

31. What if we need an FA for the union of G and Pi.e. G U P a 1-9 1-9 . f i g b 1-9 1-9 . g p j

32. Using Building Blocks Let’s change the language P to P = { a1.jpg, a2.jpg, a3.jpg, ... a33.jpg, .... } (replace ‘b’ by ‘a’) a 1-9 1-9 . g p j

33. What happens now? If we need an FA for the union of G and P. i.e. G U P a 1-9 1-9 . f i g a 1-9 1-9 . It’s a NFA! g p j

34. Limitations of FAs Can you draw an FA to accept this? { anbn | n > 0 }

35. Memo for In-class test 5 [ /5]