cnf&gnf forms

7/23/2019 Cnf&Gnf Forms

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Chomsky and Greibach Normal

FormsTeodor Rus

[email protected]

The University of Iowa, Department of Computer Science

Computation Theory p.1/2


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Simplifying a CFG

It is often convenient to simplify a CFG; One of the simplest and most useful simplified

forms of CFG is called the Chomsky normal

form;

Another normal form usually used in algebraicspecifications is Greibach normal form.

Note the difference between grammar cleaning and grammar simplification!



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Fact

Normal forms are useful when more advanced topics

in computation theory are approached, as we shall see

further.



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Definition

A context-free grammarGis in Chomsky normalform if every rule is of the form:

A BCA a

where a is a terminal, A, B,C are nonterminals, andB,Cmay not be the start variable (the axiom).



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Observation

The ruleS , whereSis the start variable, is notexcluded from a CFG in Chomsky normal form.



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Theorem 2.9

Any context-free language is generated by acontext-free grammar in Chomsky normal form.

Proof idea: Show that any CFGG can be converted into a CFGG in Chomsky

normal form;

Conversion procedure has several stages where the rules that violate

Chomsky normal form conditions are replaced with equivalent rules

that satisfy these conditions;

Order of transformations:(1) add a new start variable, (2) eliminate all

-rules, (3) eliminate unit-rules, (4) convert other rules;

Check that the obtained CFGG define the same language as the initial

CFGG.



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Proof

LetG= (N,,R,S )be the original CFG.

Step 1: add a new start symbolS0toN, and the ruleS0StoR

Note: this change guarantees that the start symbol of

G does not occur on therhsof any rule.



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Step 2: eliminate-rules

Repeat1. Eliminate the ruleA fromR whereAis not the start symbol;

2. For each occurrence ofAon therhs of a rule, add a new rule to R with

that occurrence ofA deleted;

Examples:(1) replaceB uAvbyB uAv|uv;

(2) replaceBuAvAwbyB uAvAw|uvAw|aAvw|uvw.

3. Replace the ruleB A, (if it is present) byBA|unless theruleB has been previously eliminated;

until allrules are eliminated.



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Step 3: remove unit rules

Repeat:1. Remove a unit ruleA BR;

2. For each ruleB uR, add the ruleA uto R, unless

B uwas a unit rule previously removed;

until all unit rules are eliminated.

Note: uis a string of variables and terminals.



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Convert all remaining rules

Repeat:1. Replace a ruleA u1u2 . . . uk,k 3, where eachui,1 ik, is

a variable or a terminal, by:

Au1A1,A1u2A2,. . .,Ak2uk1ukwhereA1,A2,. . .,Ak2 are new variables;

2. If k 2replace any terminaluiwith a new variableUiand add the rule

Uiui;

until no rules of the form Au1u2 . . . ukwith k3

remain.



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Removingrules

RemovingB:

S0 S

S ASA|aB|a

A B|S|

B b|

RemovingA :

S0 S

S ASA|aB|a|SA|AS|S

A B|S|

B b



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Removing unit rule

RemovingSS:

S0 S

S ASA|aB|a|SA|AS|S

A B|S

B b

RemovingS0S:

S0 S|ASA|aB|a|SA|AS

S ASA|aB|a|SA|AS

A B|S

B b



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More unit rulesRemovingAB:

S0 ASA|aB|a|SA|AS

S ASA|aB|a|SA|AS

A B|S|b

B b

RemovingA S:

S0 ASA|aB|a|SA|AS

S ASA|aB|a|SA|AS

A S|b|ASA|aB|a|SA|AS

B b



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Converting remaining rules

S0 AA1|U B|a|SA|AS

S AA1|U B|a|SA|AS

A b|AA1|U B|a|SA|AS

A1 SA

U a

B b



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Observation

The conversion procedure produces severalvariablesUialong with several rulesUia;

Since all these represent the same rule, we may

simplify the result using a single variableUand asingle ruleUa.



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Greibach Normal Form

A context-free grammarG= (V,,R,S )is inGreibach normal form if each rulerRhas theproperty: lhs(r)V,rhs(r) =a,aand

V.

Note:Greibach normal form provides a justification of

operator prefix-notation usually employed in algebra.



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Construction

1. Modify the rules inR so that ifAiAjRthenj > i2. Starting withA1 and proceeding toAm this is done as follows:

(a) Assume that productions have been modified so that for

1ik,AiAjRonly ifj > i;(b) IfAk Ajis a production withj < k, generate a new set of

productions substituting for theAj the rhs of eachAj production;

(c) Repeating (b) at mostk 1times we obtain rules of the form

Ak Ap,pk;

(d) Replace rulesAk Akby removing left-recursive rules.



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Facts

Letmbe the highest order nonterminal generatedduring Greibach normal form construction algorithm.

Note: if no left-recursive rules (Ai Ai, for some1im) are in the

grammar then we have:

1. The leftmost symbol on the rhs of each rule forAm must be a terminal,

sinceAm is the highest order variable.

2. The leftmost symbol on the rhs of any rule forAm1 must be either

Am or a terminal symbol. If it is Amwe can generate new rules

replacingAm with the rhs of the Am rules.

3. Repeating (2) forAm2, . . . , A2, A1 we obtain only rules whose rhsstart with terminal symbols.



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Left recursion

A CFG containing rules of the form

AA|

is called left-recursive inA.

The language generated by such rules is of the form

A

n

. If we replace the rulesAA|by

AB|,BB|

where Bis a new variable, then the language generated

by A is the same while no left-recursive A-rules are

used in the derivation.Computation Theory p.21/2


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Conversion algorithm

for (k=1; k beta

add A_k ---> beta alpha

remove A_k ---> A_j alpha

}

for (each rule A_k ---> A_k alpha)

{add rules B_k ---> alpha | alpha B_k

remove A_k ---> A_k alpha

}

for (each rule A_k ---> beta, beta does not begin with A_k)

add rule A_k ---> beta B_k

}

}



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More on Greibach NF

See Introduction to Automata Theory, Languages, andComputation, J.E, Hopcroft and J.D Ullman, Addison-

Wesley 1979, p. 9496.



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Example

Convert the CFGG= ({A1, A2, A3}, {a, b}, R , A1)

where

R={A1 A2A3, A2A3A1|b, A3A1A2|a}

into Greibach normal form.



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Solution

1. Step 1: ordering the rules: (OnlyA3 rules violate ordering conditions,hence onlyA3 rules need to be changed).

Following the procedure we replaceA3 rules by:

A3 A3A1A3A2|bA3A2|a

2. Eliminating left-recursion we get: A3 bA3A2B3|aB3|bA3A2|a,

B3A1A3A2|A1A3A2B3;

3. AllA3 rules start with a terminal. We use them to replaceA1 A2A3.

This introduces the rulesB3 A1A3A2|A1A3A2B3;

4. UseA1 production to make them start with a terminal.


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