1. finite_automata_new.ppt
- 2. Slide 2 Finite Automata • A Finite automaton (FA) is a simple idealized machine used to recognize patterns within input taken from some character (or alphabet). • The job of an FA is to accept or reject an input depending on whether the pattern defined by the FA occurs in the input. • FA consists of a finite set of transitions from state to state that occur on input symbols chosen from alphabet ∑
- 3. Slide 3 Finite Automata (FA) CONTROL UNIT INPUT TAPE TAPE HEAD Finite set of states Move to the right one cell at a time Store input of the DFA Can be of any length Start state Final state(s) is in exactly one state at a time
- 4. Slide 4 Finite Automata Model •Finite automata can be represented by input tape, read only head and finite control. •Input tape: It is a linear tape having some number of cells. Each input symbol is placed in each cell. •Finite control: The finite control decides the next state on receiving particular input from input tape. •The tape reader reads the cells one by one from left to right, and at a time only one input symbol is read.
- 5. Finite Automata • Finite automata are used to recognize patterns. • It takes the string of symbol as input and changes its state accordingly. • When the desired symbol is found, then the transition occurs. • At the time of transition, the automata can either move to the next state or stay in the same state. • Finite automata have two states, Accept state or Reject state. • When the input string is processed successfully, and the automata reached its final state, then it will accept. a b a b b a 0 1 2 n Input tape Control unit Tape head
- 6. Slide 6 What does FA do? •Read an input string from tape •Determine if the input string is in a language •Determine if the answer for the problem is “YES” or “NO” for the given input on the tape
- 7. Slide 7 How does an FA work? •At the beginning, – FA is in the start state (initial state) – its tape head points at the first cell • For each move, FA – reads the symbol under its tape head – changes its state (according to the transition function) to the next state determined by the symbol read from the tape and its current state – move its tape head to the right one cell
- 8. Slide 8 When does an FA stop working? •When it reads all symbols on the tape •Then, it gives an answer if the input is in the specific language: –Answer “YES” if its last state is a final state –Answer “NO” if its last state is not a final state
- 9. Slide 9 Formal Definition • A finite automate is a collection of 5 tuple (Q, ∑, δ , q0 , F), Where: Q: finite set of states ∑: finite set of the input symbol q0ϵ Q: initial state F ⊆ Q : Set of final states δ : Transition function which maps Q X ∑ into Q δ : Q X ∑ Q
- 10. Finite automata • Finite automata can be represented by transition diagram (directed graph). • The steps to construct the transition diagrams – The vertices in the transition diagram are the states of the finite automata – The edges represents the transitions. • If there is a transition from state q0 to q1 on an input symbol “1” then draw an edge from q0 to q1 labelled “1”. • The initial state will be q0 • The final states will be denoted by double circles q0 q1 0 1 0 1
- 11. Slide 11 Types of Finite Automata •There are two types of finite automata: –Deterministic Finite Automata (DFA) –Non deterministic Finite Automata (NFA)
- 12. Slide 12 DFA (Deterministic Finite Automata) • The finite automata is called DFA if there is only one path for a specific input from current state to next state. • A finite automate can be defined as a 5 tuple (Q, ∑, δ , q0 , F), Where: Q: non empty finite set of states ∑: non empty finite set of symbols q0ϵ Q: initial state F ⊆ Q : finite set of final states δ : Transition function, which maps Q X ∑ into Q δ : Q X ∑ Q
- 13. Slide 13 q2 q0 q1 a a a/b b Start M= (Q, ∑, δ, q0, F) Tuple notation for the above figure M= {{ q0, q1, q2}, { a, b}, δ ,q0 , q2} δ : transition function, which maps Q x ∑ Q
- 14. Slide 14 • Let M = (Q, ∑ , δ , q0, F) • Where Q = {q0, q1, q2} ∑ = {0,1} F= {q0} • The transition function δ is defined as State Input 0 1 q0 q0 q1 q1 q0 q1 q2 q0 q2 q0 q2 q1 0 0 1 1 Start 0 1
- 15. Slide 16 How an FA works s f 0 1 0 1 Input: 001101 Input: 01001 0 0 1 1 0 1 0 1 0 0 1 ACCEPT REJECT
- 16. Slide 17 • Deterministic refers to the uniqueness of the computation. • In DFA, there is only one path for specific input from the current state to the next state. • DFA does not accept the null move i.e., DFA cannot change state without any input character. • DFA can contain multiple final states.
- 17. Slide 18 Acceptance of Language • A language acceptance is defined by if a string w is accepted by the machine M i.e if it is reaching the final state F by taking the string w. • Not accepted if it is not reached the final state • Eg., L = {φ} => (no string) • L = { ɛ } => • DFA to accept “a” => • No. of state =|min string| +1 (states depends on length of string) q0 q0 q0 q0 q0 q1 a
- 18. Slide 19 • DFA to accept zero or more ‘a’ L= {ϵ,a,aa,aaa, ….} q0 q0 a
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- 21. Slide 22 Basis: OP(r) = 0 Then r is either Ø, ε, or a, for some symbol a in Σ For Ø: For ε: For a: qf q0 qf qf q0 a
- 22. Slide 23 Conversion of RE into NFA
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- 26. Slide 27 • Example: r = 0(0+1)* r = r1r2 r1 = 0 r2 = (0+1)* r2 = r3* r3 = 0+1 r3 = r4 + r5 r4 = 0 r5 = 1 q0 1 q1
- 27. Slide 28 • Example: r = 0(0+1)* r = r1r2 r1 = 0 r2 = (0+1)* r2 = r3* r3 = 0+1 r3 = r4 + r5 r4 = 0 r5 = 1 q0 1 q2 0 q1 q3
- 28. Slide 29 • Example: r = 0(0+1)* r = r1r2 r1 = 0 r2 = (0+1)* r2 = r3* r3 = 0+1 r3 = r4 + r5 r4 = 0 r5 = 1 q4 q0 q1 1 q2 q3 0 ε ε ε ε q5
- 29. Slide 30 • Example: r = 0(0+1)* r = r1r2 r1 = 0 r2 = (0+1)* r2 = r3* r3 = 0+1 r3 = r4 + r5 r4 = 0 r5 = 1 q6 q5 q4 q0 q1 1 q2 q3 0 ε ε ε ε ε qf ε ε ε
- 30. Slide 31 • Example: r = 0(0+1)* r = r1r2 r1 = 0 r2 = (0+1)* r2 = r3* r3 = 0+1 r3 = r4 + r5 r4 = 0 r5 = 1 q8 q9 q6 q5 q4 q0 q1 1 q2 q3 0 ε ε ε ε ε qf ε ε ε 0
- 31. Slide 32 • Example: r = 0(0+1)* r = r1r2 r1 = 0 r2 = (0+1)* r2 = r3* r3 = 0+1 r3 = r4 + r5 r4 = 0 r5 = 1 q8 q9 q6 q5 q4 q0 q1 1 q2 q3 0 ε ε ε ε ε qf ε ε ε 0 ε
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- 33. Slide 34 (ε|a*b)
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- 37. Slide 38 Draw NFA for the Regular Expression • a(a+b)*ab
- 39. Slide 40 Draw NFA for the Regular Expression • a + b + ab
- 40. Slide 41 Draw NFA for the Regular Expression • letter (letter+digit)*
- 43. Slide 44 a + b + ab