cs314: formal languages and automata theory · pda can read and write into the stack by popping the...

Chapter 2: Context Free Grammar

CS314: FORMAL LANGUAGES

AND AUTOMATA THEORY L. NADA ALZABEN

Quick Note

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2.1 Context Free Grammar (CFG)

Useful in App’s that use recursion feature also it occurs in the specification and compilation of a programming language.

Also used in the study of human languages

Parsers in compilers extracts the meaning of a program before generating the compiled code. That is done when CFG is available.

The collection of languages associated with the CFG is called Context Free Language (CFL).

The machine that recognizes the CFL is called pushdown automata (PDA).

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2.1 Context Free Grammar (CFG)

E.g. G1: (Each part is called substitution rule or

production.)

Or “|” can be used to simplify the production.

Strings can be generated as follows:

A 0A1

A B

B #

Start variable terminal

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The sequence of substitutions to obtain a string is

called derivation. A derivation is:

Parse tree:

2.1 Context Free Grammar (CFG)

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E.g. G2:

2.1 Context Free Grammar (CFG)

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Formal definition of CFG

When we say Aw as rule in the grammar this

means uAv is uwv.

When we say u drives v, u v this means u=v

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E.g. G3:

E.g. G4: define if the string is generated by the G4

grammar

1. a + a x a 2. ( a + a) x a

2.1 Context Free Grammar (CFG)

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2.1 Context Free Grammar (CFG)

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A compiler translates the code to another form to

be executed. To do so the compiler extracts the

meaning of the code by the Parsing process as we

did in parse tree.

2.1 Context Free Grammar (CFG)

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Designing CFG

Designing CFG require creativity same as the finite

automata.

First method: is to break the CFL to smaller ones then

combine them with “|” operator.

E.g. design the CFG for

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Second method: if the language is regular we can construct the CFG by first giving the DFA of the language

Third method: if the CFG contains strings with two substrings that are linked which need to remember an unbounded amount of information about one of the substrings to verify that it corresponds to the other substrings.

Fourth method: for complex languages with strings of many structures that are recursively appearing as part of other or the same structure (mostly happens in the grammar that generate arithmetic expression. )

Designing CFG

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Ambiguity

When a grammar generates the same string in several

different ways (different parse trees) we say the string is

derived ambiguously in that grammar.

We call that grammar ambiguous grammar.

E.g. G5

test a + a x a

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TEST YOUR SELF

Use G2 to show the two different derivation of:

Submit on Thursday's lecture

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Some languages can be generated by ambiguous

grammars and also unambiguous grammar BUT for

those who only are generated by the ambiguous

grammar is called inherently ambiguous

Ambiguity

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CHOMSKY normal form

Simplifying the CFG is by using CHOMSKY normal form rules.

Any CFL is generated by a CFG in the CHOMSKY normal form.

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E.g. convert the G6 into CHOMSKY normal form:

CHOMSKY normal form

Step1: make new start

variable

S ASA | aB

A B|S

B b|ᵋ

𝑺𝟎 S

S ASA | aB

A B|S

B b|ᵋ

Step2: Remove ᵋ

𝑺𝟎 S

S ASA | aB|a

A B|S |ᵋ

B b|ᵋ

𝑺𝟎 S

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

A B|S |ᵋ

B b

Step3: Remove unit rule

𝑺𝟎 S| ASA | aB|a |AS|SA

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

A B|S

B b

𝑺𝟎 ASA | aB|a |AS|SA

S ASA | aB|a |AS|SA

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

B b Computer Science Department

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E.g. convert the G6 into CHOMSKY normal form:

CHOMSKY normal form

𝑺𝟎 ASA | aB|a |AS|SA

S ASA | aB|a |AS|SA

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

B b

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Step4: convert it to proper form

𝑺𝟎 AA1 | UB|a |AS|SA

S AA1 | UB|a |AS|SA

A b|AA1|UB|a |AS|SA

B b

A1 SA

U a

2.2 Pushdown Automata (PDA)

PDA is a generalization of NFA.

PDA structure

NFA part

(state diagram)

W string

Input tape (unbounded) Stack

(unbounded)

Read Head Read /write

Head

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PDA can read and write into the stack by popping the symbol in reading and pushing another symbol in writing.

The accept status is reached when the input tape and the stack are empty

Reading ᵋ from input tape or stack means ignoring the reading part.

2.2 Pushdown Automata (PDA)

Formal Definition of PDA

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Formal Definition of PDA

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E.g. the formal description for PDA machine

recognizing the language {0𝑛1𝑛 𝑛 ≥ 0 . Let

M1=( 𝑄, , 𝛤 , 𝛿, 𝑞1, 𝐹 where:

Q={q1,q2,q3,q4}

= 0,1 , 𝛤={0,$}

F={q1,q4} , 𝛿 is givin by the table:

2.2 Pushdown Automata (PDA)

input 0 1 ᵋ

Stack 0 $ ᵋ 0 $ ᵋ

0 $ ᵋ Q1 {(q2,$)}

Q2 {(q2,0)} {(q3,ᵋ)}

Q3 {(q3,ᵋ)} {(q4,ᵋ)}

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2.2 Pushdown Automata (PDA)

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E.g. M2 recognizes the language

{𝑎𝑖𝑏𝑗𝑐𝑘 𝑖, 𝑗, 𝑘 ≥ 0 𝑎𝑛𝑑 𝑖 = 𝑗 𝑜𝑟 𝑖 = 𝑘

2.2 Pushdown Automata (PDA)

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2.2 Pushdown Automata (PDA)

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