Course Notes on Formal Languages and Compilers

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10 5 bX→Xb aX→aab abcd→ε bc→XbcY Yc→cY Yd→cdd 10 15 We shall not prove that L(G 1 ) = { a n b n c n d n : n≥0} but will give examples of derivations of ε and a 3 b 3 c 3 d 3 from the start set. Incidentally abcd∈L(G 1 ) because abcd∈T 1 * and abcd∈S 1 . As S 1 has only one element that is, abcd, all derivations must start with abcd. In the following derivations we shall underline the part of the string which is being replaced at each derivation step. Derivation of ε: abcd ⇒ ε directly by a rule Derivation of a 3 b 3 c 3 d 3 : 20 25 30 35 abcd ⇒ aXbcYd ⇒ aabbcYd ⇒ aabbccdd ⇒ aabXbcYcdd ⇒ aaXbbcYcdd ⇒ aaXbbccYdd ⇒ aaabbbccYdd ⇒ aaabbbcccddd = a 3 b 3 c 3 d 3 Exercise: Continue the above derivation to derive a 4 b 4 c 4 d 4 . Example of a Context Sensitive Grammar: The previous grammar G 1 is not a context sensitive grammar. An equivalent context sensitive grammar is G 2 = (N 2 , T 2 , P 2 , S 2 ) where N 2 ={X, Y} as before T 2 = {a,b,c,d} as before, S 2 ={abcd,ε} and P 2 consists of the rules bX→Xb aX→aab bc→XbcY Yc→cY Yd→cdd What we have done is to eliminate the rule abcd→ε which is not permitted in a context sensitive grammar and add ε to the starting set of G 2 . Exercises 40 45 1) Is G 2 a phrase structure grammar too 2) Write a context sensitive grammar for the language {a 2n b n c n d 2n : n≥1}. Present a derivation of "aaaabbccdddd" from the starting set of the grammar.
11 3) Write a context sensitive grammar and a context free grammar for {a m b m c n d n : m≥2, n≥3}. Present derivations of "aaabbbcccccddddd" from the starting sets of the grammars. 5 10 4) Let G = ({E}, {v, +, -, (, )}, {E→v, E→(E+E), E→(E-E)}, {E}). a) What types of grammars is G b) Present derivations of (v+v) and ((v+v)-v) from the starting set of the grammar. c) Can (v+v-v) be derived from the starting set of the grammar d) Describe L(G) in words. 5) Write a regular grammar for { a m b n : m,n≥0}. Present a derivation of "aab" from the starting set of the grammar. 15 20 25 30 35 40 45 Recognition of Context Sensitive Languages Let G=(N, T, P, S) be a context sensitive grammar and let t∈T*. Let m=|t| denote the length of t. Here is an algorithm for deciding whether or not t∈L(G). We shall construct a series of sets L 0 ,L 1 ,...., L m , where for each i such that 0 ≤ i ≤ m, where L i is the set of all strings of length i which are elements of S or derivable from S. Each L i is a subset of the set of all strings of (N ∪ T)* of length i and so must be finite. Initially L i ={z∈S:|z|=i} As G is context sensitive |x|>|y| impossible. So G has two types of rules: (a) x → y where |x|=|y| (b) x → y where |x||t|, since the rules never shorten a string when applied to it. Thus all strings of length greater than |t|=m can be ignored. This is the basis of the method. Do steps A) and B) for i=0,1,2,...., m. A) Apply all rules of type (a) to elements of L i , repeatedly adding all new strings derived to L i . Stop when nothing further can be added. As L i is finite (see above) infinite looping is impossible. B) Apply all rules of type (b) to elements of L i . The new strings derived will be increased in length by |y|-|x| where x→y is the rule being applied and |y|>|x|. Add the new strings to L i+|y|-|x| if i+|y|-|x|≤m. After L 0 ,L 1 ....,L m have all been generated we can test if t∈L(G) by testing whether or not t∈L m , a finite set. NOTE This method cannot be applied to phrase structure grammars because our observation above is invalid in the general case.
Page 1 and 2: 1 5 ב ‏"ה 10 Course</st
Page 3 and 4: 3 4. FINITE (STATE) AUTOMATA 28 Equ
Page 5 and 6: 5 14. TRANSLATION TO INTERMEDIATE C
Page 7 and 8: 7 Vocabulary and language 5 10 A vo
Page 9: 9 Type of Grammar Phrase Structure
Page 13 and 14: 13 Ambiguity 5 A context free gramm
Page 15 and 16: 15 Exercise: Let G = (N, T, P, S) b
Page 17 and 18: Derivation In Chomsky Normal Form A
Page 19 and 20: 19 Forward Deterministic Context Fr
Page 21 and 22: 21 3. REGULAR GRAMMARS AND LANGUAGE
Page 23 and 24: 23 S' = { U⊆N : U∩S ≠ ∅, th
Page 25 and 26: 25 5 10 15 20 (c) Again, let us arr
Page 27 and 28: 27 5 10 15 5) Let L 1 = {a 2n : n
Page 29 and 30: 29 that is δ(state, input) = new s
Page 31 and 32: 31 3) Similarly any finite automato
Page 33 and 34: Theorem: Converting regular express
Page 35 and 36: 35 The regular expressions for L k
Page 37 and 38: 37 6. SURVEY OF COMPILER AND INTERP
Page 39 and 40: 39 Bottom Up Stack (Top at right) I
Page 41 and 42: 41 Code generation (Option 1, in th
Page 43 and 44: 43 Peephole optimization in assembl
Page 45 and 46: 45 we wish to build L C B B for a n
Page 48 and 49: 48 Note that "#" means "otherwise g
Page 50 and 51: 50 Aids for the compiler writer 5 T
Page 52 and 53: 52 Basic data types and their opera
Page 54 and 55: 54 5 Call by reference is economica
Page 56 and 57: 56 5 So when B executes it can acce
Page 58 and 59: 58 Notes and quest
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60 Example Grammar Used in Descript
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62 different terminals an error is
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64 Exercises: 5 10 15 20 25 30 35 4
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69 The parse table is constructed b
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5 id 72 E E ** E E ** E id id 10 Al
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74 1) When t 1 and t 2 have the sam
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76 Exercises: 5 1) For the ambiguou
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78 For example the rules 5 10 C →
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80 Note: When a non terminal is rep
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82 4) Apply the methods of left rec
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84 Constructing FIRST and FOLLOW se
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86 14. TRANSLATION TO INTERMEDIATE
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88 end program; 5 10 15 20 25 30 35
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90 5 10 15 case CS of when name =>
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92 1) The source program is free of
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Structure of translation of express
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96 Exercises: 5 10 1) Write a proce
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Course Notes on Formal Languages and Compilers

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