Simplification of ContextFree Grammars

1
Simplification of Context-Free Grammars

• Some useful substitution rules.
• Removing useless productions.
• Removing ?-productions.
• Removing unit-productions.

2
Some Useful Substitution Rules

• G (V, T, S, P)
• A ? x1Bx2 ? P
• B ? y1 y2 ... yn ? P
• L(G) L(G)
• G (V, T, S, P)
• A ? x1y1x2 x1y2x2 ... x1ynx2 ? P

3
Example

• G (A, B, a, b, A, P)
• A ? a aaA abBc
• B ? abbA b
• G (V, T, S, P)
• A ? a aaA ababbAc abbc

4
Some Useful Substitution Rules

• G (V, T, S, P)
• A ? Ax1 Ax2 ... Axn ? P
• A ? y1 y2 ... ym ? P
• L(G) L(G)
• G (V?Z, T, S, P)
• A ? yi yiZ (i 1, m) ? P
• Z ? xi xiZ (i 1, n) ? P

5
Example

• G (A, B, a, b, A, P)
• A ? Aa aBc ?
• B ? Bb ba
• A ? aBc aBcZ Z ? A ? aBc aBcZ Z ?
• Z ? a aZ Z ? a aZ
• B ? Bb ba B ? ba baY
• Y ? b bY

6
Removing Useless Productions

• S ? aSb ? A
• A ? aA
• S ? A is redundant as A cannot be transformed
  into a terminal string.

7
Removing Useless Productions

• G (V, T, S, P)
• A ? V is useful iff there is w ? L(G) such that
  
• S ? xAy ? w
• A production is useless it it involves any
  useless variable.

8
Example

• G (S, A, B, a, b, S, P)
• S ? A
• A ? aA ?
• B ? bA

9
Example

• G (S, A, B, C, a, b, S, P)
• S ? aS A C S ? aS A S ? aS A
• A ? a A ? a A ? a
• B ? aa B ? aa
• C ? aCb

10
Example

• G (S, A, B, C, a, b, S, P)
• S ? aS A C S ? aS A S ? aS A
• A ? a A ? a A ? a
• B ? aa B ? aa
• C ? aCb dependency graph

11
Theorem

• Let G (V, T, S, P) be a context-free grammar.
• Then there exists an equivalent grammar G (V,
  T, S, P)
• that does not contain any useless variables or
  productions.

12
Theorem

• Let G (V, T, S, P) be a context-free grammar.
• Then there exists an equivalent grammar G (V,
  T, S, P)
• that does not contain any useless variables or
  productions.
• Proof ?

13
Theorem

• Proof
• Construct (V1, T, S, P1) such that V1 contains
  only variables A
• for which A ? w ? T.
• 1. Set V1 to ?.
• 2. Repeat until no more variables are added to
  V1
• For every A ? T for which P has a production
  of the form
• A ? x1x2... xn (xi ? T?V1)
• add A to V1.
• 3. Take P1 as all the productions in P with
  symbols in (V1? T).

14
Theorem

• Proof
• Draw the variable dependency graph for G1 and
  find all
• variables that cannot be reached from S.
• Remove those variables and the productions
  involving them.
• Eliminate any terminal that does not occur in a
  useful
• production.
• ? G (V, T, S, P)

15
Removing ?-Productions

• Any production of a context-free grammar of the
  form
• A ? ?
• is called a ?-production.
• Any variable A for which the derivation
• A ? ?
• is possible is called nullable.

16
Example

• S ? aS1b S ? aS1b ab
• S1 ? aS1b ? S1 ? aS1b ab

17
Theorem

• Let G (V, T, S, P) be a CFG such that ? ? L(G).
• Then there exists an equivalent grammar G having
  no
• ?-productions.

18
Theorem

• Proof
• Find the set VN of all nullable variables of G
• 1. For all productions A ? ?, put A into VN.
• 2. Repeat until no more variables are added to
  VN
• For all productions
• B ? A1A2... An (Ai ? VN)
• add B to VN.

19
Theorem

• Proof
• For each production in P of the form
• A ? x1x2... xm (m ? 1, xi ? V?T)
• put into P that production as well as all
  those generated by
• replacing null variables with ? in all
  possible combination.
• Exception if all xi are nullable, then A ? ?
  is not put into P.

20
Example

• S ? ABaC S ? ABaC BaC AaC ABa aC Aa
  Ba a
• A ? BC A ? B C BC
• B ? b ? B ? b
• C ? D ? C ? D
• D ? d D ? d
• VN A, B, C

21
Removing Unit-Productions

• Any production of a context-free grammar of the
  form
• A ? B
• is called a unit-production.

22
Theorem

• Let G (V, T, S, P) be a CFG without
  ?-productions.
• Then there exists an equivalent grammar G (V,
  T, S, P)
• that does not have any unit-productions.

23
Theorem

• Proof
• 1. Put into P all non-unit-productions of P.
• 2. Repeat until no more productions are added to
  P
• For every A and B ? V such that A ? B and B ?
  y1 y2 ... yn ? P
• add A ? y1 y2 ... yn to P.

24
Example

• S ? Aa B
• B ? A bb
• A ? a bc B

25
Example

• S ? Aa B S ? Aa
• B ? A bb A ? a bc
• A ? a bc B B ? bb
• S ? A S ? a bc bb
• S ? B A ? bb
• A ? B B ? a bc
• B ? A

26
Theorem

• Let L be a context-free language that does not
  contain ?.
• Then there exists a CFG that generates L and does
  not have
• any useless productions, ? -productions, or
  unit-productions.

27
Theorem

• Let L be a context-free language that does not
  contain ?.
• Then there exists a CFG that generates L and does
  not have
• any useless productions, ? -productions, or
  unit-productions.
• Proof
• 1. Remove ? -productions.
• 2. Remove unit-productions.
• 3. Remove useless-productions

28
Two Important Normal Forms

• Chomsky normal form.
• Greibach normal form.

29
Chomsky Normal Form

• A context-free grammar G (V, T, S, P) is in
  Chomsky
• normal form iff all productions are of the form
• A ? BC
• or
• A ? a
• where A, B, C ? V and a ? T.

30
Theorem

• Any context-free grammar G (V, T, S, P) such
  that ? ? L(G)
• has an equivalent grammar G (V, T, S, P) in
  Chomsky
• normal form.

31
Theorem

• Proof
• First, construct an equivalent grammar G1 (V1,
  T, S, P1).
• V1 V ? Ba a ? T P1 Ba ? a a ? T
• Remove all terminals from productions of length
  ? 1
• 1. Put all productions A ? a into P1.
• 2. Repeat until no more productions are added
  to P1
• For each production
• A ? x1x2... xn (n ? 2, xi ? T?V)
• add A ? C1C2... Cn to P1
• where Ci xi if xi ? V or Ci Ba if xi a

32
Theorem

• Proof
• Construct G (V, T, S, P) from G1 (V1, T,
  S, P1).
• V V1.
• Reduce the length of the right sides of the
  productions
• 1. Put all productions A ? a and A ? BC into
  P.
• 2. Repeat until no more productions are added
  to P
• For each production
• A ? C1C2... Cn (n ? 2)
• add A ? C1D1, D1 ? C2D2, ... , Dn-2 ? Cn-1Dn
  to P.

33
Example

• S ? ABa
• A ? aaB
• B ? aC

34
Greibach Normal Form

• A context-free grammar G (V, T, S, P) is in
  Greibach
• normal form iff all productions are of the form
• A ? ax
• where a ? T and x ? V.

35
Theorem

• Any context-free grammar G (V, T, S, P) such
  that ? ? L(G)
• has an equivalent grammar G (V, T, S, P) in
  Greibach
• normal form.

36
Theorem

• Proof
• Rewrite the grammar in Chomsky normal form.
• Relabel variables A1, A2, ..., An.
• Rewrite the grammar so that all productions have
  one of the following forms
• Ai ? Ajxj (j gt i)
• Zi ? Ajxj (j ? n, Zi introduced to eliminate
  left recursion)
• Ai ? axi (a ? T and xi ? V)
• Start from An ? axn to derive Greibach
  productions.

37
Example

• A2 ? A1A2 b
• A1 ? A2A2 a

38
Homework

• Exercises 3, 4, 5, 6, 7, 8, 17, 22 of Section
  6.1 - Linzs book.
• Exercises 2, 3, 4, 6, 9, 10, 11 of Section 6.2 -
  Linzs book.
• Presentations Section 6.3 and Section 7.4.