Non Deterministic Automata. Formal Languages and Automata - Yonsei CS 1
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1 Non Deterministic Automt Forml Lnguges nd Automt - Yonsei CS 1
2 Nondeterministic Finite Accepter (NFA) We llow set of possible moves insted of A unique move. Alphbet = {} Two choices q 1 q2 Forml Lnguges nd Automt - Yonsei CS 2
3 Nondeterministic Finite Accepter (NFA) Alphbet ={} Two choices q 1 q2 No trnsition No trnsition Forml Lnguges nd Automt - Yonsei CS 3
4 First Choice q 1 q2 Forml Lnguges nd Automt - Yonsei CS 4
5 First Choice q 1 q2 Forml Lnguges nd Automt - Yonsei CS 5
6 First Choice q 1 q2 Forml Lnguges nd Automt - Yonsei CS 6
7 First Choice All input is consumed q 1 q2 ccept Forml Lnguges nd Automt - Yonsei CS 7
8 Second Choice q 1 q2 Forml Lnguges nd Automt - Yonsei CS 8
9 Second Choice q 1 q2 Forml Lnguges nd Automt - Yonsei CS 9
10 Second Choice q 1 q2 No trnsition: the utomton hngs Forml Lnguges nd Automt - Yonsei CS 10
11 Second Choice Input cnnot be consumed q 1 q2 reject Forml Lnguges nd Automt - Yonsei CS 11
12 An NFA ccepts string: when there is computtion of the NFA tht ccepts the string AND ll the input is consumed nd the utomton is in finl stte Forml Lnguges nd Automt - Yonsei CS 12
13 Exmple is ccepted by the NFA: ccept q 1 q2 q 1 q2 becuse this computtion ccepts Forml Lnguges nd Automt - Yonsei CS 13 reject
14 Rejection exmple q 1 q2 Forml Lnguges nd Automt - Yonsei CS 14
15 First Choice q 1 q2 Forml Lnguges nd Automt - Yonsei CS 15
16 First Choice reject q 1 q2 Forml Lnguges nd Automt - Yonsei CS 16
17 Second Choice q 1 q2 Forml Lnguges nd Automt - Yonsei CS 17
18 Second Choice q 1 q2 Forml Lnguges nd Automt - Yonsei CS 18
19 Second Choice q 1 q2 reject Forml Lnguges nd Automt - Yonsei CS 19
20 An NFA rejects string: when there is no computtion of the NFA tht ccepts the string: All the input is consumed nd the utomton is in non finl stte OR The input cnnot be consumed Forml Lnguges nd Automt - Yonsei CS 20
21 Exmple is rejected by the NFA: q 1 q2 reject q 1 q2 reject All possible computtions led to rejection Forml Lnguges nd Automt - Yonsei CS 21
22 Rejection exmple q 1 q2 Forml Lnguges nd Automt - Yonsei CS 22
23 First Choice q 1 q2 Forml Lnguges nd Automt - Yonsei CS 23
24 First Choice q 1 q2 No trnsition: the utomton hngs Forml Lnguges nd Automt - Yonsei CS 24
25 First Choice Input cnnot be consumed q 1 q2 reject Forml Lnguges nd Automt - Yonsei CS 25
26 Second Choice q 1 q2 Forml Lnguges nd Automt - Yonsei CS 26
27 Second Choice q 1 q2 Forml Lnguges nd Automt - Yonsei CS 27
28 Second Choice q 1 q2 No trnsition: the utomton hngs Forml Lnguges nd Automt - Yonsei CS 28
29 Second Choice Input cnnot be consumed q 1 q2 reject Forml Lnguges nd Automt - Yonsei CS 29
30 is rejected by the NFA: reject q 1 q2 q 1 q2 reject All possible computtions led to rejection Forml Lnguges nd Automt - Yonsei CS 30
31 Lnguge ccepted: L = {} q 1 q2 Forml Lnguges nd Automt - Yonsei CS 31
32 Lmbd Trnsitions nf cn mke trnsition without consuming n input symbol q0 q 1 q2 Forml Lnguges nd Automt - Yonsei CS 32
33 q0 q 1 q2 Forml Lnguges nd Automt - Yonsei CS 33
34 (red hed does not move) q0 q 1 q2 Forml Lnguges nd Automt - Yonsei CS 34
35 q0 q 1 q2 Forml Lnguges nd Automt - Yonsei CS 35
36 ll input is consumed q0 q 1 q2 ccept String is ccepted Forml Lnguges nd Automt - Yonsei CS 36
37 Rejection Exmple q0 q 1 q2 Forml Lnguges nd Automt - Yonsei CS 37
38 q0 q 1 q2 Forml Lnguges nd Automt - Yonsei CS 38
39 (red hed doesn t move) q0 q 1 q2 Forml Lnguges nd Automt - Yonsei CS 39
40 q0 q 1 q2 Forml Lnguges nd Automt - Yonsei CS 40 No trnsition: the utomton hngs
41 Input cnnot be consumed q0 q 1 q2 reject String is rejected Forml Lnguges nd Automt - Yonsei CS 41
42 Lnguge ccepted: L = {} q0 q 1 q2 Forml Lnguges nd Automt - Yonsei CS 42
43 Another NFA Exmple b q b q 2 q0 1 Forml Lnguges nd Automt - Yonsei CS 43
44 b q b q 2 0 q 1 Forml Lnguges nd Automt - Yonsei CS 44
45 b q q b 1 0 q 2 Forml Lnguges nd Automt - Yonsei CS 45
46 b q q b 1 0 ccept q 2 Forml Lnguges nd Automt - Yonsei CS 46
47 Another String b b q b q q3 0 q1 2 Forml Lnguges nd Automt - Yonsei CS 47
48 b b q b q q3 0 q1 2 Forml Lnguges nd Automt - Yonsei CS 48
49 b b q b q q3 0 q1 2 Forml Lnguges nd Automt - Yonsei CS 49
50 b b q b q q3 0 q1 2 Forml Lnguges nd Automt - Yonsei CS 50
51 b b q b q q3 0 q1 2 Forml Lnguges nd Automt - Yonsei CS 51
52 b b q b q q3 0 q1 2 Forml Lnguges nd Automt - Yonsei CS 52
53 b b q b q q3 0 q1 2 Forml Lnguges nd Automt - Yonsei CS 53
54 b b q b q q3 0 ccept q1 2 Forml Lnguges nd Automt - Yonsei CS 54
55 L = Lnguge ccepted { b, bb, bbb,...} = { b} + q b q 2 q0 1 Forml Lnguges nd Automt - Yonsei CS 55
56 Another NFA Exmple 0 q0 1 1,1 q2 Forml Lnguges nd Automt - Yonsei CS 56
57 Lnguge ccepted L(M ) = = { }, 10, { 10}* 1010, ,... 0 q0 1 1,1 q2 (redundnt stte) Forml Lnguges nd Automt - Yonsei CS 57
58 Remrks: The symbol never ppers on the input tpe Simple utomt: M 1 M 2 L(M 1 ) = {} (M ) = {} L 2 Forml Lnguges nd Automt - Yonsei CS 58
59 NFAs re interesting becuse we cn express lnguges esier thn DFAs NFA M 1 DFA M 2 q 0 q1 q 2 q 1 L( M1) = { } L( M 2) = { } Forml Lnguges nd Automt - Yonsei CS 59
60 Forml Definition of NFAs ( Q q F ) M =, Σ, δ, 0, Q : Set of sttes, i.e. { q, q q } 0 1, 2 Σ Input lphbet, i.e. {, b} : δ : Trnsition function : Initil stte F : Finl sttes Forml Lnguges nd Automt - Yonsei CS 60
61 Trnsition Function δ δ ( q, ) = { q } q0 1 1,1 q2 Forml Lnguges nd Automt - Yonsei CS 61
62 δ ( q 1,0) = { q0, q 2 } 0 1,1 1 q2 Forml Lnguges nd Automt - Yonsei CS 62
63 δ (, ) = { q0, q 2 } 0 1,1 1 q2 Forml Lnguges nd Automt - Yonsei CS 63
64 δ ( q 2,1) = 0 1,1 1 q2 Forml Lnguges nd Automt - Yonsei CS 64
65 Extended Trnsition Function δ * δ ( q ) = { } * q 0, 1 q 4 q 5 q 1 b q 2 Forml Lnguges nd Automt - Yonsei CS 65
66 δ ( q ) = { } * q q 0, 4, 5 q 4 q 5 q 1 b q 2 Forml Lnguges nd Automt - Yonsei CS 66
67 δ ( q b) = { } * q q q 0, 2, 3, 0 q 4 q 5 q 1 b q 2 Forml Lnguges nd Automt - Yonsei CS 67
68 Formlly q ( q w) j δ * i, : there is wlk from qi to with lbel w q j qi w q j w = σ 1 σ 2 Lσ k qi σ1 σ 2 σ k q j Forml Lnguges nd Automt - Yonsei CS 68
69 The Lnguge of n NFA M F = { },q 5 q 4 q 5 q 1 b q 2 δ ( q ) = { } * q q 0, 4, 5 F L(M ) Forml Lnguges nd Automt - Yonsei CS 69
70 F = { },q 5 q 4 q 5 q 1 b q 2 ( q b) = { } δ * q q q b L( M ) 0, 2, 3, 0 F Forml Lnguges nd Automt - Yonsei CS 70
71 F = { },q 5 q 4 q 5 q 1 b q 2 δ ( q b) = { } * q q 0, 4, 5 F b L(M ) Forml Lnguges nd Automt - Yonsei CS 71
72 F = { },q 5 q 4 q 5 q 1 b q 2 δ *( q b) = { q } b L( M ) 0, 1 F Forml Lnguges nd Automt - Yonsei CS 72
73 q 4 q 5 q 1 b q 2 L ( M ) = { } { b} * { } Forml Lnguges nd Automt - Yonsei CS 73
74 Formlly The lnguge ccepted by NFA is: M ( M ) {,,...} L = w1 w w, 2 3 where δ *(, w ) = { q, q,..., q, K} m i j k nd there is some q k F (finl stte) Forml Lnguges nd Automt - Yonsei CS 74
75 w L( M ) δ *(, w) w q i q0 w qk q k F w q j Forml Lnguges nd Automt - Yonsei CS 75
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