4.4 Normal Forms

4.4 Normal Forms In this section, we shall introduce context-free grammars to two special forms, one is Chomsky normalform(CNF), the other is Greiback normalform(GNF). Definition 5: A CFG G = (V, T, P, S) is in Chomsky normal form if for each product of G is of the forms A  BC, or A  a. Here, A, B and C are in V, and a is in T. Definition 6: A CFG G = (V, T, P, S) is in Greiback normal form if for each product of G is of the form A  a, where A V, a T and  V*.

-productions An -production is of the form A  . But, if the language L generated by a grammar G does not include , we can modify the grammar G to an equivalent grammar G’ so that there are no -productionsin G’. The algorithm to modify a grammar G to G’ is as follows. while ( there is a an -production A in G ) do remove the-production A from G If there is a production B  xAy from G, then add a production B  xy to G

Example 20: Consider the grammar G S  0T0 | 1T1 T 0T0 | 1T1 |  Then L(G) = L = { x x R | x T*, and x  }. We can modify the above grammar to a grammar without -productions as the following and generating the same language L. S  0T0 | 1T1 | 00 | 11 T 0T0 | 1T1 | 00 | 11 The above grammar is equivalent to the following grammar. S  0S0 | 1S1 | 00 | 11

Useless variables A is a useful variable if A * w for some w T*. For a CFG G, we can modify the grammar G to an equivalent grammar G’ so that there are no useless variables in G’. The algorithm to modify a grammar G to G’ is as follows. First, let the set S  , flag  TRUE while ( flag ) do // get useful variables flag  FALSE If B  T*in G, or B  x 1A 1x 2A 2…x k A ky in G, where y, x i T* andA i S, i = 1,2, …, k, then S  S  {B}, flag  TRUE The set V \ S is a set of useless variables.

Next, if there is a useless production B  x 1A 1x 2A 2…x k A ky in G, where y, x i T* and one of B and A i, i = 1,2, …, k, is in V \ S, then remove the production B  x 1A 1x 2A 2…x k A ky from G. The algorithm to remove useless variables and useless productions from the grammar G to get an equivalent grammar G’ without useless variables and useless productions.

Unit productions A unit production is of the form AB, where A and B are variables. A unit production basically is a redundant production. Therefore, we can eliminate the unit productions by the following method: For each unit production AB, remove the production from the grammar, and add the following productions: For each non-unit production Bw in G, add the production A w to G.

By the above algorithms, we are able to obtain the following theorem. Theorem 5 : For each CFL L without , there is a CFG G with no useless variables, -productionsor unit productions such that L(G) = L.

Theorem 6 : For each CFL L without , there is a CFG G in Chomsky normal form such that L(G) = L, i.e., each production in G is of the form A  BC, or A  a ,where A, B and C are in V, and a is in T. Proof : For each CFL L without , there is a CFG G with no useless variables, -productionsor unit productions such that L(G) = L. First step, for each productionA in G, if || = 1, then  T. Otherwise, || > 1, say  = X 1X 2…X k, k>1. IfX i , say i =1and X1= a, is a terminal and there is a production B a, then replace X 1 by B to get a new production A BX 2…X k , and remove the production A .

IfX i , say i =1and X 1 = a, is a terminal and there is no production Ba, then add a new variable C, and replace X 1 by C to get a new production A CX 2…X k , remove the production A and add a new production C  a. After the first step, all productions are of the form either A a, or A , where || >1 and  V+ Second step, for each production of the formA , where || >1, we know that  V+. If || = 2, then we do not need to modify the production. If || > 2, say  = X 1X 2…X k, k>2, then we need to introduce new variables Y 1 ,Y 2 ,… , Y k-2, and add the new productions as follows.

A X 1Y 1 Y 1X 2Y 2 Y iX i+1Y i+1, i = 1, 2, …, k3, Y k-3X k-2Y k-2 Y k-2X k-1X k Therefore, we can modify the grammar to a new equivalent grammar in Chomsky normal form.

Theorem 7 : For each CFL L without , there is a CFG G in Greiback normal form such that L(G) = L. Proof : For each CFL L without , there is a CFG G’’ with no useless variables, -productionsor unit productions such that L(G’’) = L. By theorem 6, there is a CFG G’=(V’, T, P’, S’) in Chomsky normal form such that L(G’) = L(G’’) = L. Rewrite the k variables in V’ with indices 1, 2, …, k. Let it be V = {A 1, A 2, …, A k} and the start variable is A 1.

First step, modify the productions into the forms A ia, where a T and  V*, or A iA j , where j >i To achieve the result, we start from A 1. If there is a production A 1a, where a T and  V*, then we keep the production. If there is a production A 1A 1, then apply theorem 4 in section 4.2 to revise the left recursive production to a right recursive production until there is no A 1recursive production. Next we revise the A 2 productions until A k. A possible algorithm for this step is as follows.

for i=1 to k do for j=1 to i 1 do for each production A i A j do for each production A j do add production A i  remove production A i A j  for each production A i A i do add productions B i and B i  B i remove productions A i A i  for each production A i , where  does not begin with A i do add production A i  B i

After the first step, the productions are of the forms: A ia, where a T and  V*, A iA j , where k  j > i, or B i, where  ( V{B 1, B 2,…, B i1} )*. The A k production must be of the form A ka, where a T and  V*, The B i production must be of the form B iA j, where  (V {B 1, B 2,…, B i1}) *, or B ia, where a T and  (V {B 1, B 2,…, B i1}) *,

Second step, modify each A i production into the forms A ia, where a T and  V*. To achieve the result, we start from A k1to modify the A k1productions so that the right side of the production starting with a terminal. Then proceed the same process until A 1. A possible algorithm for this step is as follows. for i=k 1 to 1 do for j=k to i +1 do for each production A i A j do for each production A j do add production A i  remove production A i A j 

Third step, modify each B i production into the forms B ia, where a T and  V*. A possible algorithm for this step is as follows. for i=1 to k do for each production B i A j do for each production A j do add production A j  remove production B i A j 

Example 21: Consider the grammar G S  0S0 | 1S1 | 00 | 11 An equivalent CFG in GNF is S  0SZ | 1SY | 0Z | 1Y Y  1 Z  0 An equivalent CFG in CNF is S  ZX | YW | ZZ | YY X  SZ W  SY Y  1 Z  0

Second stage: Step 5: By A 3 productions A 3  1 | 1B 3 , modify the A 2 productions to the following productions: A 2 1A 1 A 3 | 1 B 3 A 1 A 3 | 1A 1 B 1 A 3 | 1 B 3 A 1 B 1 A 3 | 0 Step 6: Modify the A 1 productions to the following productions: A 1 1A 1 | 1 B 3 A 1 | 1A 1 B 1 | 1 B 3 A 1 B 1 After the second stage, we have the following productions: A 1 1A 1 | 1 B 3 A 1 | 1A 1 B 1 | 1 B 3 A 1 B 1 A 2 1A 1 A 3 | 1 B 3 A 1 A 3 | 1A 1 B 1 A 3 | 1 B 3 A 1 B 1 A 3 | 0 A 3  1 | 1B 3 B 1 A 2 | A 2 B 1 B 3  A 1 A 2 B 3| A 1 B 1 A 2 B 3 | A 1 A 2 | A 1 B 1 A 2

4.4 Normal Forms

4.4 Normal Forms

Presentation Transcript

Normal Forms through BCNF

Higher Normal Forms

Normal Forms Part 1

Normal Forms and Infinity

Normal Forms

Schema Refinement and Normal Forms

Normal Forms

Normal Forms

Normal forms

Normal Forms

4.4 - Indeterminate Forms and L’Hospital’s Rule

Normal Forms

Normal Forms (Part 1)

Normal Forms

Schema Refinement: Normal Forms

Normal Forms for Tables

Normal Forms and XML

Normal Forms for CFG’s

Schema Refinement: Normal Forms