CFG Simplification. (simplify) 1. Eliminate useless symbols 2. Eliminate -productions 3. Eliminate unit productions
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1 CFG Simplification (simplify) 1. Eliminate useless symbols 2. Eliminate -productions 3. Eliminate unit productions 1
2 Eliminating useless symbols 1. A symbol X is generating if there exists: X * w, for some w T* 2. A symbol X is reachable if there exists: S * X 3. For a symbol X to be useful, it has to be both reachable and generating S X * w, for some w T* reachable generating Algorithm to detect useless symbols 1. First, eliminate all symbols that are not generating 2. Next, eliminate all symbols that are not reachable 2
3 A production is useless, if it involves a useless variable. Note that the concept of 'useless variable' includes the case that the variable does not lead to a terminal string, and the case that the variable cannot be reached from the start symbol. The elimination of useless variables and productions from a grammar (or the selection of those variables and productions, which are useful) proceeds in two phases: A. Determine and select those variables, which can lead to a terminal string, and subsequently determine and select the related productions. B. Determine and select those variables, which can be reached from the start symbol, and select related productions. 3
4 Example1: Useless symbols S AB a A A b 1. A, S are generating 2. B is not generating (and therefore B is useless) 3. ==> Eliminating B (i.e., remove all productions that involve B) S a A, A b Example2: S asb λ A A aa nonterminal A is useless Example3:B B cannot be reached from S S A A aa B ba nonterminal B is useless 4
5 Example4 : Eliminate useless symbols from the grammar with productions S AB CA B BC AB A a C AB b Step 1: Eliminate non-generating symbols B N = {A, C, S} P1 = {S CA, A a, C b} Step 2: Eliminate symbols that are non reachable Sol. All nonterminal are reachable. 5 5
6 HW: Eliminate useless symbols from the grammar 1. P= {S aaa, A Sb bcc, C abb, E ac} 2. P= {S aba BC, A ac BCC, C a, B bcc, D E, E d} 3. P= {S aaa, A bbb, B ab, C ab} 4. P= {S as AB, A ba, B AA} 6
7 Eliminating -productions Any variable A such that there is a production in P: A ε is a nullable variable..if P contains A B1B2B3 BN and B1, B2, BN are nullable variables, then A is nullable. That mean A * ε 7
8 Example: Eliminating -productions Let L be the language represented by the following CFG G: S AB A aaa B bbb Goal: To construct G1, which is the grammar for L- { } Simplified grammar All variables are nullable,s, A and B are nullable since: A ε and B ε, S is nullable since S AB and A and B are nullable G 1 can be constructed from G as follows: B b bb bbb ==> B b bb bbb Similarly, A a aa aaa Similarly, S A B AB G 1 : S A B AB A a aa aaa B b bb bbb 8
9 Find out the grammar without - Productions G = S as AB A B D b Sol: Nullable variables = {S, A, B} New Set of productions: S as a AB A B D b HW: Eliminate - productions from the grammar 1. S ASB, A aas a, B SbS A bb 2. S a Xb aya, X Y, Y b X Eliminate - productions and useless symbols from the grammar S a aa B C, A ab, B aa, C acd, D ddd 9
10 Example: S [E] E E T E+T E-T T F T*F T/F F a b c The -free grammar constructed from G has productions S [E] E [ ] E T E+T E-T E+ E- +T -T + - T F T*F T/F T* T/ *F /F * / F a b c 10
11 Eliminating unit productions Unit production is one which is of the form A B, where both A & B are nonterminal example E T E+T T F T*F F I (E) I a b Ia Ib I0 I1 How to eliminate unit productions? Replace E T with E F T*F Then, upon recursive application wherever there is a unit production: E F T*F E+T (substituting for T) E I (E) T*F E+T (substituting for F) E a b Ia Ib I0 l1 (E) T*F E+T (substituting for I) Now, E has no unit productions Similarly, eliminate for the remainder of the unit productions 11
12 Eliminate unit productions S Aa B B A bb A a bc B S Aa B B a bc bb bb A a bc bb A a bc A bb A a bc bb S Aa B B a bc bb A a bc bb S Aa a bc bb B a bc bb A a bc bb 12
13 CFG Simplification S ACD AB A a Aa C D Dd E E e A e F ff 1) Delete: B useless because nothing derivable from B. 2) Delete C & also add S AD. 3) Replace D Dd E with D eae and delete E e A e 4) Delete: F useless because not derivable from S. 13
14 linear grammar A grammar is linear if it is context-free and all of its productions' right hand sides have at most one nonterminal. A linear language is a language generated by some linear grammar. Example A simple linear grammar is G with N = {S}, Σ = {a, b}, P with start symbol S and rules S asb It generates the language 14
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