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Unformatted text preview: l38 Chapter 4 Normal Forms for Context-Free Grammars Exercises 1 39 The derivation in the Chomsky normal form grammar generates six variables. Each B —> bb l C
of these is transformed to a terminal by a rule of the form A —> a. The Greibach normal I C —> cc 1 A
form derivation generates a terminal with each rule application. The derivation is completed '
using only six rule applications. El 9‘ G: S _) A l C
A —+ (IA 1 a | B
B —> 273 I b
C #4) CC 1 c l B Exercises 10 G- s —> AB 1 C For Exercises 1 through 5, construct an equivalent essentially nonconn'acting grammar ~ A a 0A l B ' ' - - . .. .- ' B —> [73 l C
GL With a nonrecursive start symbol. Give a regular expression for the language of each
grammar. C—+cCla|A ' ‘ ‘ ' h ' G of Exercise 1.
1. G: S ~> 05 | [75 | B 11. Eliminate the chain rules from t e grammar L B _’ bb l C l A 12. Eliminate the chain rules from the grammar GL of Exercise 4- C —> CC I A 13 Prove that Algorithm 442 generates the set of variables that derive terminal strings.
2. G: s —> ABC |A ' A —> aA | a t B _) b3 l A For Exercises 14 through 16, construct an equivalent grammar WlthOUl 11561683 symbols- C .9 CC 1 A Trace the generation of the sets of TERM and REACH used to construct GT and GU.
3. G: S —> BSA | A Describe the language generated by the grammar. A —> M 1 x B—>Bba|l 14.G:S—>AA[CleB
4. GzS—rABIBCS A—WAla A—>aA|C B—>bB|bC B —> bbB | b C —> 03 C —> CC I A D —> dB 1 d
5.G:S—>ABC!aBC 15. G:S—>uA|BD A—>aA|BC AeaAlaABlaD B—rbBI/‘t BaaB|aClBF C —> cC I )~ C —> Bb | aAC | E
6. Prove Lemma 43.2. D —-> [7D | bC l b
E —> (:3 l bC
For Exercises 7 through 10, constructan equivalent grammar GC that does not contain chain F _, a p 1 “G 1 a
rules. Give a regular expression for the language of each grammar. Note that these grammars G _, a I b do not contain Avrules. 16. G:S—>ACH|BB
7. st—sAsm AauAlaF
B.G:S—>A|B|C F—>bBlb A—>aalB H—>dH|d 140 Chapter 4 Normal Forms for Context-Free Grammars Exercises 141 17. Show that all the symbols of the grammar b) What is the maximum depth of a derivation tree for a string of length n in L(G)? c) What is the minimum depth of a derivation tree for a string of length n in L(G)? G: S —> A | CB
A _> C I D 25. Give the upper diagonal matrix produced by the CYK algorithm when run with the
Chomsky normal form grammar from Example 4.5.2 and the input strings abbb and
B —> M? | b aabbb.
C -) CC I c 26. Let G be the Chomsky normal form grammar
D —> dD | d -: S —> AX AY '
are useful. C(mstruct an equivalent grammar GC by removing the chain rules from G. I I a Show that GC contains useless symbols. X —’ AX l a
18. Convert the grammar Y —> BY la
A —+ AA la "
B _> AbB I bb Give t c upper diagonal matrix produced by the CYK algorithm when rrrn With the grammar G and the input strings buaa and abaaa. to Chomsky normal form. G already satisﬁes the conditions on the start symbol 5, A-rules, useless symbols. and chain rules. ' 27. Lethe the grammar
G: S—r A | B
A —> (103 l Aab l Aba
B—>bB|Bb|abu. 19. Convert the grammar G: S—>aAbB|ABCla A —> all a I a) Give a regular expression for MG)
3 _> bBCC I b b) Construct a grammar G’ that contains no left-recursive rules and is equivalent to G.
C —> abc a: 28. Construct a grammar G’ that contains no left—recursive rules and is equivalent to
to Chomsky normal form. G already satisﬁes the conditions on the start symbol S, . A-rules, useless symbols, and chain rules. G: S _) A l C
20. Convert the result of Exercise 9 to Chomsky nonnal form. A _) AaB I AaC I B I a
21. Convert the result of Exercise 1 l to Chomsky normal foml. B _’ Bb I Cb
C —> ('C l c. 22. Convert the result of Exercise 12 t0 Chomsky normal form. 23. Convert the grammar Give a leftmost derivation of the string aacmcb in the grammars G and G’. G: S __) A I A3“ I AbA 29. Construct a grammar G’ that contains no left-recursive rules and is equivalent to A~9Aa|k G:S—>AlB
C—aCBlCAle B—>BBb|b. m Chomsky normal form 30. Construct a Greibach normal form grammar equivalent to " 24. Let G be a grammar in Chomsky normal form. S —+ aAb | a a) What is the length of a derivation of a string of length n in L(G)? A —> 55 1 b_ ...
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This note was uploaded on 05/12/2011 for the course CS 4114 taught by Professor Ehrich,r during the Spring '08 term at Virginia Tech.
- Spring '08