1. 1 Chapter 6 Simplification of CFGs and Normal Forms

2.  2 6.1: Methods for Transforming Grammars (1) A Useful Substitution Rule  Theorem 6.1:  This intuitive theorem allows us to simplify grammars.  Let G = (NT, T, S, P) be a context-free grammar. Suppose that P contains a production rule of the form  A → xBz  Assume that A and B are different NT and that  B → y 1 | y 2 |... | y n is the set of all productions in P which have B as the left side.  Let G’ = (NT, T, S, P’) be the grammar in which P’ is constructed from P by replacing rule  A → xBz with A → x y 1 z | xy 2 z |... | xy n z  Then L(G’) = L(G)

3.  3 6.1: Methods for Transforming Grammars (2) A Useful Substitution Rule  Let G be  S → a | aaS | abBc  B → abbS | b  Applying theorem 6.1 results in  S → a | aaS | ababbSc | abbc  B → abbS | b  The rules B → abbS | b, which are still part of the grammar, no longer serve any purpose.  Both of these useless rules may be deleted without effectively changing the grammar.

4.  4 6.1: Methods for Transforming Grammars (3) Removing Useless Productions  A non-terminal A is useful (it occurs in at least one derivation.) if:  it is reachable: occurs in a sentential form S  *  A   it is live: generates a terminal string A  * w  T *  A non-terminal is useless if:  A non-terminal A is useless if:  A does not occur in any sentential form  It cannot be reached from start symbol OR  A does not generate any string of terminals.  It cannot derive a terminal string  A terminal is useful if it occurs in a sentence w  L(G)  Any production involving a useless symbol is a useless production.

5.  5 6.1: Methods for Transforming Grammars (4) Removing Useless Productions  To eliminate useless symbols:  First: Find the set TERM that contains all non- terminals that derive a terminal string  A  * w, where w  T *  Non-terminals NOT in TERM are useless, they cannot contribute to generate strings in L(G)  Second: Find the set REACH that contains all non-terminals A  TERM that are reachable from S  S  *  A 

6.  6 6.1: Methods for Transforming Grammars (5) Removing Useless Productions  Example 1  G:S → AC | BS | B A → aA | aF B → CF | b  C → cC | D D → aD | BD | C E → aA | BSA F → bB | b   L(G) is b +  B, F  TERM, since both generate terminals  S  TERM, since S → B and hence S  * b  A  TERM, since A → aF and hence A  * ab  E  TERM, since E → aA and hence E  * aab

7.  7 6.1: Methods for Transforming Grammars (6) Removing Useless Productions  C and D do not belong to TERM, so all rules containing C and D are removed  The new grammar is  G T :S → BS | B A → aA | aF B → b E → aA | BSA F → bB | b  All non-terminals in G T derive terminal strings  Now, we must remove the non-terminals that do not occur in sentential forms of the grammar  A set REACH is built that contains all non- terminals derivable from S  A set REACH is built that contains all non- terminals  TERM derivable from S

8.  8 6.1: Methods for Transforming Grammars (7) Removing Useless Productions  G T :S → BS | B A → aA | aF B → b E → aA | BSA F → bB | b  S  REACH, since it is the start symbol  B  REACH, since S → SB, and hence B is derivable from S  A, E, and F can not be derived from S or B, so all rules containing A, E and F are removed

9.  9 6.1: Methods for Transforming Grammars (8) Removing Useless Productions  The new grammar is  G U :S → BS | B B → b  L(G U ) = b +  The set of terminals of G U is {b}, a is removed since it does not occur in any string in the language of G U  The order is important:  Applying Second step (REACH) before First Step (TERM) may not remove all useless symbols.

10.  10 6.1: Methods for Transforming Grammars (9) Removing Useless Productions  Home exercise: Remove all useless productions.  S → AB | CD | ADF | CF | EA  A → abA | ab  B → bB | aD | BF | aF  C → cB | EC | Ab  D → bB | FFB  E → bC | AB  F → abbF | baF | bD | BB  G → EbE | CE | ba  Let G = ({S, A, B, C}, {a, b}, S, {S → aS | A | C, A → a, B → aa, C → aCb}) be a CFG.  Remove all useless productions  Final grammar is  G’ = ({S}, {a}, S, {S → aS | a})

11.  11 6.1: Methods for Transforming Grammars (10) Removing  - Productions  Let G be  Let G beS → SaB | aBB → bB |   A non-terminal symbol that can derive the null string (  is called nullable.  For example, in G above, B is nullable since B   For example, in G above, B is nullable since B →   A grammar without nullable non-terminals is called non-contracting  G, above, is not non-contracting, since it has one nullable non-terminal, which is B.

12.  12 6.1: Methods for Transforming Grammars (11) Removing  - Productions  How to find nullable non-terminals?  Mark all non-terminals A for which there exists a production of the form A →   Repeat  Mark non-terminal X for which there exists X →  and all symbols in  have been marked as nullable  Until no new non-terminal is marked  Read Theorem 6.3

13.  13 6.1: Methods for Transforming Grammars (12) Removing  - Productions  The set of nullable non-terminals of the grammar  S → ACA A → aAa | B | C B → bB | b C → cC |   is {S, A, C}  C is nullable  since C →  and hence C  *   A is nullable  since A → C, and C is nullable  S is nullable  since S → ACA, and A and C are nullable

14.  14 6.1: Methods for Transforming Grammars (13) Removing  - Productions  Find nullable non-terminals. S → aS | SS | bA A → BB B → CC | ab | aAbC C → 

15.  15 6.1: Methods for Transforming Grammars (14) Removing  - Productions  If   L(G), we can eliminate all productions A   If   L(G), we can eliminate all productions A →   For every B referring to A:  For example, if B  and A BABa  For example, if B →  and A → BABa  Then after eliminating the rule B , new rules for A will be added  Then after eliminating the rule B → , new rules for A will be added  A → BABa  A → ABa  A → BAa  A → Aa B →  A  | … A →  | … B →   |  A  | … A → …

16.  16 6.1: Methods for Transforming Grammars (15) Removing  - Productions  Let G be  S → SaB | aB B → bB |   After removing  -productions, the new grammar will be  S → SaB | Sa | aB | a B → bB | b  The removal of  -productions increases the number of rules but reduces the length of derivations.

17.  17 6.1: Methods for Transforming Grammars (16) Removing  - Productions  Let G  Let G S → ACA A → aAa | B | C B → bB | b C → cC |   The equivalent essentially non-contracting grammar G L is  G L :S → ACA | CA | AA | AC | A | C |  A → aAa | aa | B | C B → bB | b C → cC | c  Since S  *  in G, the rule S  is allowed in G L, but all other  -productions are replaced  Since S  *  in G, the rule S →  is allowed in G L, but all other  -productions are replaced  A grammar satisfying these conditions is called essentially non-contracting (only start symbol is nullable)

18.  18 6.1: Methods for Transforming Grammars (17) Removing  - Productions  Let G be  S → aS | SS | bA  A → BB  B → ab | aAbC | aAb | CC  C →   We eliminate C  by replacing:  We eliminate C →  by replacing:  B → CC into B → CC, B → C, and B →   B → aAbC into B → aAbC and B → aAb  Since C  is only C production  Since C →  is only C production  only B  and B aAb retained.  only B →  and B → aAb retained.  The new grammar:  S → aS | SS | bA  A → BB  B →  | ab | aAb

19.  19 6.1: Methods for Transforming Grammars (18) Removing  - Productions  The new grammar:  S → aS | SS | bA  A → BB  B →  | ab | aAb  We eliminate B  by replacing  We eliminate B →  by replacing  A → BB into A → BB, A → B, and A →   Since there are other B productions, these are all retained  The new grammar:  S → aS | SS | bA  A → BB | B |   B → ab | aAb

20.  20 6.1: Methods for Transforming Grammars (19) Removing  - Productions  The new grammar:  S → aS | SS | bA  A → BB | B |   B → ab | aAb  Finally we eliminate A  by replacing  Finally we eliminate A →  by replacing  B → aAb into B → aAb, B → ab  S → bA into S → bA | b  The final CFG is:  S → aS | SS | bA | b  A → BB | B  B → ab | aAb

21.  21 6.1: Methods for Transforming Grammars (20) Removing of Unit Rules  Rules having this form A B are called unit rules  Rules having this form A → B are called unit rules  Consider the rules  A → aA | a | B  B → bB | b | C  The unit rule A B indicates that any string derivable from B is also derivable from A  The unit rule A → B indicates that any string derivable from B is also derivable from A  The removal of unit rules increases the number of rules but reduces the length of derivations.

22.  22 6.1: Methods for Transforming Grammars (21) Removing of Unit Rules  To eliminate the unit rule, add A rules that directly generate the same strings as B  Add a rule A → u for each B → u and deleting A → B from the grammar  Read Theorem 6.4 A → B B →  |... A →  |... B →  |...

23.  23 6.1: Methods for Transforming Grammars (22) Removing of Unit Rules   Consider the rules  A → aA | a | B  B → bB | b | C   The new rules after eliminating the unit rule A → B  A → aA | a | bB | b | C  B → bB | b | C   We add new rules to A by replacing B in A with all its RHS rules

24.  24 6.1: Methods for Transforming Grammars (23) Removing of Unit Rules   G L :S → ACA | CA | AA | AC | A | C |  A → aAa | aa | B | C B → bB | b C → cC | c  The new equivalent grammar (without unit rules)   G C :S → ACA | CA | AA | AC | aAa | aa | bB | b | cC | c |  A → aAa | aa | bB | b | cC | c B → bB | b C → cC | c

25.  25 6.1: Methods for Transforming Grammars (24) Removing of Unit Rules  Remove unit rules:  S → T | S + T  T → F | F * T  F → a | (S)  S → T | S + T  T → a | (S) | F * T  F → a | (S)  S → a | (S) | F * T | S + T  T → a | (S) | F * T  F → a | (S)

26.  26 6.2:Chomsky Normal Form (1)  The Chomsky normal form places restrictions on the length and the composition of the right-hand side of a rule  Definition 6.4:  A CFG is in Chomsky normal form if each production rule has one of the following forms:  A → a  A → BC  S →   where B, C  NT  Read Theorem 6.6

27.  27 6.2:Chomsky Normal Form (2)  Algorithm Step 1  Make sure that the following are satisfied:  No  -productions (other than S →  )  No chain rules  No useless symbols

28.  28 6.2:Chomsky Normal Form (3)  Algorithm Step 2  Eliminate terminals from RHS of productions  For each production A → X 1 X 2 …X m  where X i  NT   If m  1, replace each terminal a  RHS of A  Add (if needed) C a → a for each a  , where each C a is new non-terminal.  In production A, replace terminal a with corresponding C a

29.  29 6.2:Chomsky Normal Form (4)  Algorithm Step 3  Eliminate productions with long RHS:  For each production:  A → B 1 B 2 …B m, m  2, where B i  NT  replace with productions  A → B 1 D 1  D 1 → B 2 D 2  …  D m-2 → B m-1 B m  where D 1 …D m-2 are new non-terminals.

30.  30 6.2:Chomsky Normal Form (5) 1. Original grammar (no chain rules, useless symbols, or  -productions): S  X a Y | Y b X  Y X a Y | a Y  S S | a X | b 2. Grammar after eliminating terminals from RHSs: S  X A Y | Y BA  a X  Y X A Y | aB  b Y  S S | A X | b 3. Grammar after eliminating long RHSs: S  X T | Y BT  A YA  a X  Y F | a F  X G B  b Y  S S | A X | b G  A Y Note: Could simplify by combining redundant variables T and G

31.  31 6.2:Chomsky Normal Form (6) 1. Original grammar (no chain rules, useless symbols, or  -productions): S  aXYZ | aX  aX | a Y  bcY | bcZ  cZ | c 2. Grammar after eliminating terminals from RHSs: S  AXYZ | aA  a X  AX | aB  b Y  BCY | BC C  c Z  CZ | c 3. Grammar after eliminating long RHSs: S  AF | aA  aF  XG X  AX | a B  b G  YZ Y  BH | BC C  cH  CY Z  CZ | c  See Example 6.8

32.  32 6.2: Greibach Normal Form (7)  A context-free grammar is in Greibach Normal Form if every production is of the form A → aX  where A  NT, X  NT *, and a  Σ  Examples:  G 1 = ({S, A}, {a, b}, S, {S → aSA | a, A → aA | b})  GNF  G 2 = ({S, A}, {a, b}, S, {S → AS | AAS, A → SA | aa})  not GNF  This grammar S  ABA  aA | bB | b B  b is not in GNF  This grammar S  aAB | bBB | bB A  aA | bB | bB  b is in GNF