ICT role in 21st century education and it's challenges.
Chomsky & Greibach Normal Forms
1.
2. Presentation Outline
2
• Introduction
• Chomsky normal form
• Preliminary simplifications
• Final steps
• Greibach Normal Form
• Algorithm (Example)
• Summary
3. Introduction
3
Grammar: G = (V, T, P, S)
T = { a, b }T = { a, b }Terminals
V = A, B, CV = A, B, CVariables
SSStart Symbol
P = S → AP = S → AProduction
4. Introduction
4
Grammar example
S → aBSc
S → abc
Ba → aB
Bb → bb
L = { an
bn
cn
| n ≥ 1 }L = { an
bn
cn
| n ≥ 1 }
S aBSc aBabcc aaBbcc aabbcc⇒ ⇒ ⇒ ⇒
5. Introduction
5
Context free grammar
The head of any production contains only one
non-terminal symbol
S → P
P → aPb
P → ε
L = { an
bn
| n ≥ 0 }L = { an
bn
| n ≥ 0 }
6. Presentation Outline
6May 27, 2009
• Introduction
• Chomsky normal form
• Preliminary simplifications
• Final simplification
• Greibach Normal Form
• Algorithm (Example)
• Summary
7. Chomsky Normal Form
7
A → BC
A → α
A context free grammar is said to be in Chomsky
Normal Form if all productions are in the following
form:
• A, B and C are non terminal symbols
• α is a terminal symbol
8. Presentation Outline
8
• Introduction
• Chomsky normal form
• Preliminary simplifications
• Final steps
• Greibach Normal Form
• Algorithm (Example)
• Summary
9. Preliminary Simplifications
9
Eliminate Useless SymbolsEliminate Useless Symbols1
Eliminate ε productionsEliminate ε productions2
Eliminate unit productionsEliminate unit productions3
There are three preliminary simplifications
10. Preliminary Simplifications
10
Eliminate Useless Symbols
We need to determine if the symbol is useful by
identifying if a symbol is generating and is reachable
• X is generating if X ω for some terminal string ω.
• X is reachable if there is a derivation X αXβ
for some α and β
⇒
⇒
*
*
11. Preliminary Simplifications
11May 27, 2009
Example: Removing non-generating symbols
S → AB | a
A → b
S → AB | a
A → b Initial CFL grammar
S → AB | a
A → b
S → AB | a
A → b
Identify generating symbols
S → a
A → b
S → a
A → b
Remove non-generating
13. Preliminary Simplifications
13
The order is important.
S → AB | a
A → b
S → AB | a
A → b
Looking first for non-reachable symbols and then
for non-generating symbols can still leave some
useless symbols.
S → a
A → b
S → a
A → b
14. Preliminary Simplifications
14
Finding generating symbols
If there is a production A → α, and every symbol
of α is already known to be generating. Then A is
generating
S → AB | a
A → b
S → AB | a
A → b
We cannot use S → AB
because B has not been
established to be generating
15. Preliminary Simplifications
15
Eliminate Useless SymbolsEliminate Useless Symbols1
Eliminate ε productionsEliminate ε productions2
Eliminate unit productionsEliminate unit productions3
There are three preliminary simplifications
16. Preliminary Simplifications
16
Eliminate ε Productions
• In a grammar ε productions are convenient but
not essential
• If L has a CFG, then L – {ε} has a CFG
Nullable variable
A εA ε⇒*
17. Preliminary Simplifications
17
If A is a nullable variable
• Whenever A appears on the body of a production
A might or might not derive ε
S → ASA | aB
A → B | S
B → b | ε
Nullable: {A, B}
18. Preliminary Simplifications
18
• Create two version of the production, one with
the nullable variable and one without it
• Eliminate productions with ε bodies
S → ASA | aB
A → B | S
B → b | ε
S → ASA | aB | AS | SA | S | a
A → B | S
B → b
Eliminate ε Productions
19. Preliminary Simplifications
19
• Create two version of the production, one with
the nullable variable and one without it
• Eliminate productions with ε bodies
S → ASA | aB
A → B | S
B → b | ε
S → ASA | aB | AS | SA | S | a
A → B | S
B → b
Eliminate ε Productions
20. Preliminary Simplifications
20
• Create two version of the production, one with
the nullable variable and one without it
• Eliminate productions with ε bodies
S → ASA | aB
A → B | S
B → b | ε
S → ASA | aB | AS | SA | S | a
A → B | S
B → b
Eliminate ε Productions
21. Preliminary Simplifications
21
Eliminate Useless SymbolsEliminate Useless Symbols1
Eliminate ε productionsEliminate ε productions2
Eliminate unit productionsEliminate unit productions3
There are three preliminary simplifications
22. Preliminary Simplifications
22
Eliminate unit productions
A unit production is one of the form A → B where both
A and B are variables
A BA B⇒*
A → B, B → ω, then A → ω
Identify unit pairs
23. Eliminate unit productions
• Consider the CFG
S->0A | 1B|C
A-> 0S|00
B->1|A
C->01
remove unit productions.
S-> C
B-> A unit productions
S->0A|1B|01
A->0S|00
B->1|0S|00
May 27, 2009 23
