discrete math. and logic ii. normal forms - sfwr eng 2fa3se2fa3/notes/ln19new.pdf · 2014-03-27 ·...

Discrete Math. and Logic II. Normal FormsSFWR ENG 2FA3

Ryszard Janicki

Winter 2014

Acknowledgments: Material partially based on Automata and Computability by Dexter C. Kozen (Chapter 21).

Ryszard Janicki Discrete Math. and Logic II. Normal Forms 1 / 12

Normal Forms

Very often, it is helpful to assume that CFGs are in one oranother special restricted form

Two of the most useful such forms are

Chomsky normal form (CNF)

Greibach normal form (GNF)

De�nition

A CFG is in Chomsky normal form (CNF) if all productions are ofthe form

A −→ BC or A −→ a

where A,B,C ∈ N and a ∈ Σ.


Normal Forms

De�nition

A CFG is in Greibach normal form (GNF) if all productions are ofthe form

A −→ aB1B2 · · ·Bk

for some k ≥ 0, where A,B1, · · · ,Bk ∈ N and a ∈ Σ.

Note that k = 0 is allowed, giving productions of the formA −→ a

No grammar in Chomsky or Greibach form can generate thenull string (No production can generate ε which does notbelong to Σ)


Normal Forms

Example

Grammar for the set of nonnull strings of balanced parentheses

Most obvious grammar: S −→ [S ] | SS | εChomsky normal form

S −→ AB | AC | SS , C −→ SB,

A −→ [, B −→]

Greibach normal form

S −→ [B | [SB | [BS | [SBS , B −→]


Normal Forms

Theorem

For any CFG G, there is a CFG G ′ in Chomsky normal form and a

CFG G” in Greibach normal form such that

L(G”) = L(G ′) = L(G )− {ε}.

To prove this theorem, we must �rst show how to get rid ofall ε-productions A −→ ε and unit productions A −→ B


Normal Forms

These productions make it hard to determine whetherapplying a production makes any progress toward deriving astring of terminals

One might apply unit production n times, then useε-production n times to erase them =⇒ waste time!!!!

There might be useless productions!!

We cannot simply throw out the ε- and unit productions,because they may be needed to generate some strings in L(G )

We simply to replace them (with loosing only ε from thelanguage)


Getting Rid of ε- and Unit Productions

Lemma

For any CFG G = (N,Σ,P,S), there is a CFG G ′ with no ε- orunit productions such that L(G ′) = L(G )− {ε}.

Proof Idea.

Use the following rules to transform the productions of G to getrid of ε- and unit productions

If A −→ αBβ and B −→ ε is in the set of production, thenwe add A −→ αβ

If A −→ B and B −→ γ is in the set of production, then weadd A −→ γ

At the end we remove ε- and unit productions and thegrammer obtained generates L(G )− {ε}


Chomsky Normal Form (CNF)

Once we are rid of useless, ε- and unit productions, it is asimple matter to put the resulting grammar into Chomskynormal form

For each terminal a ∈ Σ, introduce a new nonterminal Aa andproduction Aa −→ a

After that, replace all occurrences of a on the right-hand sidesof old productions (except productions of the form B −→ a)with Aa

Then all productions are of the form

A −→ a or A −→ B1B2 · · ·Bk , k ≥ 2

where Bi are nonterminals.



For any production

A −→ B1B2 · · ·Bk

with k ≥ 3, introduce a new nonterminal C and replace thisproduction with the two productions

A −→ B1C and C −→ B2B3 · · ·B3

Keep doing this until all right-hand sides are of length at most2



Example

Derive a CNF grammar for the set {anbn | n ≥ 1}. We start withthe grammar

S −→ aSb | ε

Transformation:

1 Remove ε- and unit productions

S −→ aSb | ab

2 Adding nonterminal A,B and replace these productions with

S −→ ASB | AB, A −→ a, B −→ b



Example (Continued)

1 Finally, we add a nonterminal C and replace S −→ ASB with

S −→ AC C −→ SB

2 The �nal grammar (Under Chomsky Normal Form) isS −→ AB | AC , C −→ SB, A −→ a, B −→ b

Example (Home Work)

Derive a CNF grammar for the set of nonnull strings of balancedparentheses [ ]. Start with the grammar S −→ [S ] | SS | ε


Greibach Normal Form (GNF)

General transformation of any grammar (that does notgenerate ε) into appropriate Graibach Normal Form is rathercomplicated (Carefully read pages 144�147 from textbook).

The construction is rather easy if for any production X → Ax ,where x ∈ Σ∗, the productions starting from A have aterminal as the �rst symbol, i.e. A→ x implies x = ay wherea ∈ Sigma, y ∈ Σ∗.



The construction is based on the observation that anyproduction of the form A→ ax can easily be transformed intothe set of equivalent productions in Graibach Normal Form.

For example A→ aBbCa can be transformed into

A→ XaBXbCXa

Xa → a

Xa → a

where Xa,Xb are new variables/nonterminals.

If we have X → Ax and, for example, A→ ay | bz | ct, wecan replace X → Ax withX → ayx | bzx | ctx


discrete math. and logic ii. normal forms - sfwr eng 2fa3se2fa3/notes/ln19new.pdf · 2014-03-27 ·...

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