Formal Languages The Pumping Lemma for CFLs
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1 Forl Lguges The Pupig Le for CFLs
2 Review: pupig le for regulr lguges
3 Tke ifiite cotext-free lguge Geertes ifiite uer of differet strigs Exple: 3
4 I derivtio of log strig, vriles re repeted derivtio: 4
5 Derivtio tree strig 5
6 Derivtio tree strig repeted 6
7 * 7
8 Repeted Prt * 8
9 other possile derivtio fro * * * 9
10 * * ( ) ( ) ( ) ( ) * 10
11 Derivtio fro * * * 11
12 * * * 0 ( ) ( ) 0 1
13 * * * 0 ( ) ( ) 0 ( ) 0 ( ) 0 L( G) 13
14 Derivtio fro * * * 14
15 * * * * 15
16 16 * * ) ( ) ( ) ( ) ( * *
17 17 * * ) ( ) ( ) ( ) ( * *
18 * * * ( ) ( ) ( ) ( ) L( G) 18
19 Derivtio fro * * * ( ) ( ) 19
20 0 3 3 * * ) ( ) ( ) ( ) ( * *
21 * ) ( ) ( ) ( ) ( * *
22 * * * 3 ( ) ( ) 3 ( ) 3 ( ) 3 L( G)
23 I Geerl: * * * ( ) i ( ) i i ( ) ( ) L( G) i i 0 3
24 Cosider ow ifiite cotext-free lguge L Let G e the grr of L {} Tke G so tht it hs o uit-productios o -productios 4
25 Let p = (Nuer of productios) x (Lrgest right side of productio) Let p 1 Exple : G p 43 1 p 113 5
26 Tke strig with legth wl(g) w We will show: i the derivtio of w vrile of is repeted G 6
27 * w v 1 v v k w v 1 7
28 v 1 v v k w vi vi 1 f xiu right hd side of y productio w k f w k f p k f 8
29 v 1 v v k w p k f k p f Nuer of productios i grr 9
30 30 Nuer of productios i grr k oe productio ust e repeted w v v v k 1 1 r r r Repeted vrile w v
31 w L(G) w Derivtio of strig w 1 34 w oe vrile is repeted 31
32 Derivtio tree of strig w u Lst repeted vrile w uvxy v repeted y u, v, x, y, : trigs of terils x 3
33 Possile derivtios: u u vy v y x x 33
34 We kow: u vy x This strig is lso geerted: u * ux uv 0 xy 0 34
35 We kow: u vy x This strig is lso geerted: * uuvyuvxy * The origil w uv 1 xy 1 35
36 We kow: u vy x This strig is lso geerted: * * uuvyuvvyy uvvxyy * uv xy 36
37 We kow: u vy x This strig is lso geerted: * u uvy * uvvyy uvvvyyyuvvvxyyy uv * 3 xy 3 * 37
38 We kow: u vy x This strig is lso geerted: * u uvy uvvvyyy * * * * * uvvvvyyyy * uvvvvxy yyy uvvyy * * uv i xy i 38
39 Therefore, y strig of the for uv i xy i i 0 is geerted y the grr G 39
40 Therefore, kowig tht uvxy L(G) i i we lso kow tht uv xy L(G) L( G) L { } uv i xy i L 40
41 u v y x Oservtio: vxy ice is the lst repeted vrile 41
42 u v y Oservtio: x vy 1 ice there re o uit or -productios 4
43 The Pupig Le: For ifiite cotext-free lguge L there exists iteger such tht for y strig we c write w L, w w uvxy with legths vxy d vy 1 d it ust e: uv i xy i L, for ll i 0 43
44 pplictios of The Pupig Le 44
45 No-cotext free lguges { c : 0} Cotext-free lguges { : 0} 45
46 Theore: The lguge L { c : 0} is ot cotext free Proof: Use the Pupig Le for cotext-free lguges 46
47 L { c : 0} ssue for cotrdictio tht is cotext-free L ice L is cotext-free d ifiite we c pply the pupig le 47
48 L { c : 0} Pupig Le gives gic uer such tht: Pick y strig w L with legth w We pick: w c 48
49 L { c : 0} w c We c write: w uvxy with legths vxy vy 1 d 49
50 L { c : 0} w c w uvxy vxy vy 1 Pupig Le sys: uv i xy i L for ll i 0 50
51 L { c : 0} w c w uvxy vxy vy 1 We exie ll the possile loctios of strig vxy i w 51
52 L { c : 0} w c w uvxy vxy vy 1 Cse 1: vxy is withi ccc... ccc u vxy 5
53 L { c : 0} w c w uvxy vxy vy 1 Cse 1: v d y oly coti ccc... ccc u vxy 53
54 L { c : 0} w c w uvxy vxy vy 1 Cse 1: Repetig v d y k 1 k ccc... ccc u xy v 54
55 L { c : 0} w c w uvxy vxy vy 1 Cse 1: Fro Pupig Le: uv xy L k 1 k ccc... ccc u xy v 55
56 L { c : 0} w c w uvxy vxy vy 1 Cse 1: Fro Pupig Le: uv xy L k 1 However: uv xy k c L Cotrdictio!!! 56
57 L { c : 0} w c w uvxy vxy vy 1 Cse : vxy is withi ccc... ccc u vxy 57
58 L { c : 0} w c w uvxy vxy vy 1 Cse : e lysis s i cse ccc... ccc u vxy 58
59 L { c : 0} w c w uvxy vxy vy 1 Cse 3: vxy is withi c ccc... ccc u vxy 59
60 L { c : 0} w c w uvxy vxy vy 1 Cse 3: e lysis s i cse ccc... ccc u vxy 60
61 L { c : 0} w c w uvxy vxy vy 1 Cse 4: vxy overlps d ccc... ccc u vxy 61
62 L { c : 0} w c w uvxy vxy vy 1 Cse 4: ucse 1: v y cotis oly cotis oly ccc... ccc u vxy 6
63 L { c : 0} w c w uvxy vxy vy 1 Cse 4: ucse 1: k 1 k k 1 1 v y k cotis oly cotis oly ccc... ccc u xy v 63
64 L { c : 0} w c w uvxy vxy vy 1 Cse 4: Fro Pupig Le: uv xy L k 1 k k 1 1 k ccc... ccc u xy v 64
65 L { c : 0} w c w uvxy vxy vy 1 Cse 4: Fro Pupig Le: uv xy L k1 k 1 However: uv xy k1 k c L Cotrdictio!!! 65
66 L { c : 0} w c w uvxy vxy vy 1 Cse 4: ucse : v cotis d y oly cotis ccc... ccc u vxy 66
67 L { c : 0} w c w uvxy vxy vy 1 Cse 4: k1 k k v y ucse : cotis d oly cotis 1 k1 k k ccc... ccc u xy v 67
68 L { c : 0} w c w uvxy vxy vy 1 Cse 4: Fro Pupig Le: uv xy L k1 k k 1 k1 k k ccc... ccc u xy v 68
69 L { c : 0} w c w uvxy vxy vy 1 Cse 4: Fro Pupig Le: uv xy L However: k1 k k 1 uv xy k 1 k k c L Cotrdictio!!! 69
70 L { c : 0} w c w uvxy vxy vy 1 Cse 4: ucse 3: v y oly cotis cotis d ccc... ccc u vxy 70
71 L { c : 0} w c w uvxy vxy vy 1 Cse 4: ucse 3: v y oly cotis cotis d e lysis s for sucse 71
72 L { c : 0} w c w uvxy vxy vy 1 Cse 5: vxy overlps d c ccc... ccc u vxy 7
73 L { c : 0} w c w uvxy vxy vy 1 Cse 5: e lysis s i cse ccc... ccc u vxy 73
74 There re o other cses to cosider vxy (sice, strig cot vxy overlp, d c t the se tie) 74
75 I ll cses we otied cotrdictio Therefore: The origil ssuptio tht L { c : 0} is cotext-free ust e wrog Coclusio: L is ot cotext-free 75
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