Review of CFGs and Parsing I Context-free Languages and Grammars. Winter 2014 Costas Busch - RPI 1
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1 Review of CFGs d Prsig I Cotext-free Lguges d Grmmrs Witer 2014 Costs Busch - RPI 1
2 Cotext-Free Lguges { b : { ww } 0} R Regulr Lguges *b* ( b) * Witer 2014 Costs Busch - RPI 2
3 Cotext-Free Lguges Cotext-Free Grmmrs Pushdow Automt stck utomto Witer 2014 Costs Busch - RPI 3
4 Cotext-Free Grmmrs Witer 2014 Costs Busch - RPI 4
5 Forml Defiitios Grmmr: G = ( V, T,, P) et of Vribles/ No-termil et of termil trt ymbol et of productios symbols Witer 2014 Costs Busch - RPI 5
6 Cotext-Free Grmmr: G = ( V, T,, P) All productios i P re of the form A s No-termil trig of No-termil d termils Witer 2014 Costs Busch - RPI 6
7 xmple of Cotext-Free Grmmr b λ productios P = { b, λ} G = ( V, T,, P) V = {} vribles T = {, b} termils strt vrible Witer 2014 Costs Busch - RPI 7
8 Lguge of Grmmr: For grmmr G with strt vrible * L( G) = { w : w, w T*} trig of termils or λ Witer 2014 Costs Busch - RPI 8
9 xmple equece of termils d o-termils Grmmr: b λ No-termils The right side my be λ Witer 2014 Costs Busch - RPI 9
10 Grmmr: b λ Derivtio of strig : b b b b λ Witer 2014 Costs Busch - RPI 10
11 Grmmr: b λ bb Derivtio of strig : b bb bb b λ Witer 2014 Costs Busch - RPI 11
12 Grmmr: b λ Lguge of the grmmr: L = { b : 0} Witer 2014 Costs Busch - RPI 12
13 A Coveiet Nottio We write: * bbb for zero or more derivtio steps Isted of: b bb bbb bbb Witer 2014 Costs Busch - RPI 13
14 I geerl we write: w * 1 w If: w 1 w 2 w 3 w i zero or more derivtio steps Trivilly: w * w Witer 2014 Costs Busch - RPI 14
15 Cotext-Free Lguge: A lguge is cotext-free if there is cotext-free grmmr with L L = L(G) G Witer 2014 Costs Busch - RPI 15
16 xmple: L = b { : 0} is cotext-free lguge sice cotext-free grmmr : G b λ geertes L ( G) = L Witer 2014 Costs Busch - RPI 16
17 Aother xmple Cotext-free grmmr G : bb λ xmple derivtios: bb bb bb bb bb L(G) = { ww R : w {, b}*} Plidromes of eve legth Witer 2014 Costs Busch - RPI 17
18 Aother xmple Cotext-free grmmr G : b λ xmple derivtios: b b b b b bb bb L(G) = Describes mtched pretheses: { w : d ( w) = i y ( v) ( w), prefix ( v) v} Witer 2014 Costs Busch - RPI 18 b b () ((( ))) (( )) = (, b = )
19 Derivtio Order d Derivtio Trees Witer 2014 Costs Busch - RPI 19
20 Derivtio Order Cosider the followig exmple grmmr with 5 productios: 1. AB 2. A A 4. B Bb 3. A λ 5. B λ Witer 2014 Costs Busch - RPI 20
21 1. AB 2. A A 4. B Bb 3. A λ 5. B λ Leftmost derivtio order of strig : b ABABBBbb 4 5 At ech step, we substitute the leftmost vrible Witer 2014 Costs Busch - RPI 21
22 1. AB 2. A A 4. B Bb 3. A λ 5. B λ Rightmost derivtio order of strig : b AB ABb AbAbb At ech step, we substitute the rightmost vrible Witer 2014 Costs Busch - RPI 22
23 1. AB 2. A A 4. B Bb 3. A λ 5. B λ Leftmost derivtio of : b ABABBBbb 4 5 Rightmost derivtio of b : AB ABb 3 AbAbb Witer 2014 Costs Busch - RPI 23
24 Derivtio Trees Cosider the sme exmple grmmr: AB A A λ B Bb λ Ad derivtio of : b AB AB ABb Bb b Witer 2014 Costs Busch - RPI 24
25 AB A A λ B Bb λ AB A B yield AB Witer 2014 Costs Busch - RPI 25
26 AB A A λ B Bb λ AB AB A B A yield AB Witer 2014 Costs Busch - RPI 26
27 AB A A λ B Bb λ AB AB ABb A B A B b yield ABb Witer 2014 Costs Busch - RPI 27
28 AB A A λ B Bb λ AB AB ABb Bb A B A B b λ yield λbb = Bb Witer 2014 Costs Busch - RPI 28
29 AB A A λ B Bb λ AB AB ABb Bb b Derivtio Tree (prse tree) A B A B b λ yield λλb = b Witer 2014 Costs Busch - RPI 29 λ
30 ometimes, derivtio order does t mtter Leftmost derivtio: AB AB B Bb b Rightmost derivtio: AB ABb Ab Ab b Give sme derivtio tree A B A B b Witer 2014 Costs Busch - RPI 30 λ λ
31 Ambiguity Witer 2014 Costs Busch - RPI 31
32 Grmmr for mthemticl expressios ( ) xmple strigs: ( ) ( ( )) Deotes y umber Witer 2014 Costs Busch - RPI 32
33 Witer 2014 Costs Busch - RPI 33 A leftmost derivtio for * ) (
34 Witer 2014 Costs Busch - RPI 34 Aother leftmost derivtio for ) (
35 Witer 2014 Costs Busch - RPI 35 Two derivtio trees for ) (
36 tke = 2 = Witer 2014 Costs Busch - RPI 36
37 Good Tree Bd Tree = = Compute expressio result usig the tree Witer 2014 Costs Busch - RPI 37
38 Two differet derivtio trees my cuse problems i pplictios which use the derivtio trees: vlutig expressios I geerl, i compilers for progrmmig lguges Witer 2014 Costs Busch - RPI 38
39 Ambiguous Grmmr: A cotext-free grmmr if there is strig G w L(G) is mbiguous which hs: two differet derivtio trees or two leftmost derivtios (Two differet derivtio trees give two differet leftmost derivtios d vice-vers) Witer 2014 Costs Busch - RPI 39
40 Witer 2014 Costs Busch - RPI 40 strig hs two derivtio trees ) ( this grmmr is mbiguous sice xmple:
41 Witer 2014 Costs Busch - RPI 41 strig hs two leftmost derivtios * ) ( this grmmr is mbiguous lso becuse
42 Aother mbiguous grmmr: IF_TMT if XPR the TMT if XPR the TMT else TMT Vribles Termils Very commo piece of grmmr i progrmmig lguges Witer 2014 Costs Busch - RPI 42
43 If expr1 the if expr2 the stmt1 else stmt2 IF_TMT if expr1 the TMT if expr2 the stmt1 else stmt2 IF_TMT Two derivtio trees if expr1 the TMT else stmt2 if expr2 the stmt1 Witer 2014 Costs Busch - RPI 43
44 I geerl, mbiguity is bd d we wt to remove it ometimes it is possible to fid o-mbiguous grmmr for lguge But, i geerl we cot do so Witer 2014 Costs Busch - RPI 44
45 A successful exmple: Ambiguous Grmmr ( ) quivlet No-Ambiguous Grmmr T T T T F F F ( ) geertes the sme lguge Witer 2014 Costs Busch - RPI 45
46 Witer 2014 Costs Busch - RPI 46 F F F F T T T F T T T T T F F T F F F F T T T T ) ( Uique derivtio tree for
47 A u-successful exmple: L = { b c m } { b m c m }, m 0 L is iheretly mbiguous: every grmmr tht geertes this lguge is mbiguous Witer 2014 Costs Busch - RPI 47
48 Witer 2014 Costs Busch - RPI 48 } { } { m m m c b c b L = λ 1 1 Ab A A c λ 2 2 bbc B B 1 2 xmple (mbiguous) grmmr for : L
49 Delig with Ambiguity There re severl wys to hdle mbiguity Most direct method is to rewrite grmmr umbiguously ' ' id ʹ id () ʹ () forces precedece of * over ' Professor Alex Aike Lecture #5 49
50 Ambiguity No geerl techiques for hdlig mbiguity Impossible to covert utomticlly mbiguous grmmr to umbiguous oe Used with cre, mbiguity c simplify the grmmr. ometimes llows more turl defiitios. However, we still eed to evetully elimite mbiguity. How? 50
51 Precedece d Associtivity Declrtios Isted of rewritig the grmmr Use the more turl (mbiguous) grmmr Alog with dismbigutig declrtios Most prser-geertors llow precedece d ssocitivity declrtios to dismbigute grmmrs 51
52 Associtivity Declrtios Ambiguous two prse trees of it it it Left ssocitivity declrtio: %left it it it it it it 52
53 Precedece Declrtios Cosider the strig it it * it Precedece declrtios: %left %left * * it it * it it it it 53
54 Properties of Cotext-Free lguges Witer 2014 Costs Busch - RPI 54
55 Uio Cotext-free lguges re closed uder: Uio L1 is cotext free L2 is cotext free L1 L 2 is cotext-free Witer 2014 Costs Busch - RPI 55
56 xmple Lguge Grmmr L = 1 { b } 1 1b λ L = 2 R { ww } 2 2 b2b λ Uio L = { b } { ww R } 1 2 Witer 2014 Costs Busch - RPI 56
57 I geerl: For cotext-free lguges with cotext-free grmmrs d strt vribles L 1, L 2 G 1, G 2 1, 2 The grmmr of the uio hs ew strt vrible 1 L 2 L d dditiol productio 1 2 Witer 2014 Costs Busch - RPI 57
58 Coctetio Cotext-free lguges re closed uder: Coctetio L1 is cotext free L2 is cotext free L 1 L 2 is cotext-free Witer 2014 Costs Busch - RPI 58
59 xmple Lguge Grmmr L = 1 { b } 1 1b λ L = 2 R { ww } 2 2 b2b λ L = { b Coctetio }{ ww R } 1 2 Witer 2014 Costs Busch - RPI 59
60 I geerl: For cotext-free lguges with cotext-free grmmrs d strt vribles L 1, L 2 G 1, G 2 1, 2 The grmmr of the coctetio hs ew strt vrible L 1 L 2 d dditiol productio 1 2 Witer 2014 Costs Busch - RPI 60
61 tr Opertio Cotext-free lguges re closed uder: tr-opertio L is cotext free * L is cotext-free Witer 2014 Costs Busch - RPI 61
62 xmple Lguge Grmmr L = { b } b λ tr Opertio L = { b }* 1 1 λ Witer 2014 Costs Busch - RPI 62
63 I geerl: For cotext-free lguge with cotext-free grmmr d strt vrible L G The grmmr of the str opertio hs ew strt vrible 1 d dditiol productio 1 1 L* λ Witer 2014 Costs Busch - RPI 63
64 Cotext-free Lguges do t hve the followig Properties Witer 2014 Costs Busch - RPI 64
65 Itersectio Cotext-free lguges re ot closed uder: itersectio L1 L2 is cotext free is cotext free L1 L 2 ot ecessrily cotext-free Witer 2014 Costs Busch - RPI 65
66 L = { 1 b c m Cotext-free: } xmple L = { 2 b m c m Cotext-free: } AC AB A Ab λ A A λ C cc λ B bbc λ Itersectio L L = 1 2 { b c } NOT cotext-free Witer 2014 Costs Busch - RPI 66
67 Complemet Cotext-free lguges re ot closed uder: complemet L is cotext free L ot ecessrily cotext-free. I.e., the complemet of cotext-free lguge my or my ot be cotext-free. Witer 2014 Costs Busch - RPI 67
68 L = { 1 b c m Cotext-free: } xmple L = { 2 b m c m Cotext-free: } AC AB A Ab λ A A λ C cc λ B bbc λ 1 Complemet L2 L1 L2 { NOT cotext-free L = = Witer 2014 Costs Busch - RPI 68 b c }
69 Itersectio of Cotext-free lguges d Regulr Lguges Witer 2014 Costs Busch - RPI 69
70 The itersectio of cotext-free lguge d regulr lguge is cotext-free lguge L1 cotext free L2 regulr L1 L 2 cotext-free Witer 2014 Costs Busch - RPI 70
71 Applictios of Regulr Closure Witer 2014 Costs Busch - RPI 71
72 The itersectio of cotext-free lguge d regulr lguge is cotext-free lguge L1 L2 cotext free regulr Regulr Closure L1 L 2 cotext-free Witer 2014 Costs Busch - RPI 72
73 A Applictio of Regulr Closure Prove tht: L = { b : 100, 0} is cotext-free Witer 2014 Costs Busch - RPI 73
74 We kow: { b : 0} is cotext-free Witer 2014 Costs Busch - RPI 74
75 We lso kow: 100 L 1 = { b 100 } is regulr * L1 = {( b) } { b } is regulr Witer 2014 Costs Busch - RPI 75
76 { b } cotext-free 100 L1 = {( b) } { b * regulr 100 } (regulr closure) { b } L 1 cotext-free { b } L1 = { b : 100, 0} = L is cotext-free Witer 2014 Costs Busch - RPI 76
77 Aother Applictio of Regulr Closure Prove tht: L = { w: = b = c } is ot cotext-free Witer 2014 Costs Busch - RPI 77
78 If L = { w: = b = c } is cotext-free The (regulr closure) L { * b* c*} = { b c } cotext-free regulr cotext-free Impossible!!! Therefore, L is ot cotext free Witer 2014 Costs Busch - RPI 78
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