Lec4

Theory of Computation

Lecture 4: Context Free Grammar (CFG)

Cairo UniversityFCI

Dr. Hussien SharafComputer Science [email protected]

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p.434: Regular Languages and non-regular languages

Dr. Hussien M. Sharaf

Language Defined by

Corresponding Accepting Machine

Nondeter-minism=Determinism

Regular expression Finite automaton, transition graph

Yes

Context-free grammar

Pushdown automaton No

Type 0 grammar Turing machine, Post machine, Pushdown automaton

Yes

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CFG A context-free grammar is a notation for

defining context free languages.It is more powerful than finite automata or

RE’s, but still cannot define all possible languages.

Useful for nested structures, e.g., parentheses in programming languages.

Basic idea is to use “variables” to stand for sets of strings.

These variables are defined recursively, in terms of one another.


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CFG formal definitionC =(V, Σ, R, S)V: is a finite set of variables.Σ: symbols called terminals of the alphabet of

the language being defined.S V: a special start symbol.R: is a finite set of production rules of the

form A→ where A V, (V Σ)


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CFG -1Define the language { anbn | n > 1}.Terminals = {a, b}.Variables = {S}.Start symbol = S.Productions ={

S → ab,S → aSb }Summary

S → abS → aSb


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DerivationWe derive strings in the language of a CFG by

starting with the start symbol, and repeatedly replacing some variable A by the right side of one of its productions.

Derivation example for “aabb”Using S→ aSb

generates uncompleted string that still has a non- terminal S.

Then using S→ ab to replace the inner SGenerates “aabb”

S aSb aabb ……[Successful derivation of aabb]


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Balanced-parenthesesProd1 S → (S)Prod2 S → ()

Derive the string ((())).S → (S) …..[by prod1]

→ ((S)) …..[by prod1]→ ((())) …..[by prod2]


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PalindromeDescribe palindrome of a’s and b’s using CGF1] S → aS 2] S → bS3] S → b 4] S → a

Derive “baab” from the above grammar.S → bS [by 2]

→ baS [by 1]→ baaS [by 1]→ baab [by 3]


CFG – 2.1Describe anything (a+b)* using CGF1] S → Λ 2] S → Y 3] Y→ aY4] Y → bY 5] Y →a 6] Y→ b

Derive “aab” from the above grammar.S → aY [by 3]

Y → aaY [by 3]Y → aab [by 6]

Dr. Hussien M. Sharaf 9

CFG – 2.21] S → Λ 2] S → Y 3] Y→ aY4] Y → bY 5] Y →a 6] Y→ b

Derive “aa” from the above grammar.S → aY [by 3]

Y → aa [by 5]

Dr. Hussien M. Sharaf 10

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Remember CFG is about categorizing the grammar of a

language


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CFG – 3Describe anything (a+b)* using CGF1] S →aS 2] S → bS 3] S →ΛDerive “aab” from the above grammar.S → aS [by 1]

S → aaS [by 1]S → aabS [by 2]S → aabΛ [by 3] → aab


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CFG – 4Describe anything (a+b)*aa(a+b)* using CGF1] S →XaaX 2] X → aX 3] X → bX 4] X →ΛNote: rules 2, 3, 4 represents anything (a+b)*Derive “baabaab” from the above grammar.S → XaaX [by 1] X→ bXaaX [by 3]X→ bΛaaX [by 4] X → baabX [by 3]X → baabaX [by 2] X→ baabaaX [by 2]X→ baabaabX [by 3] X→ baabaabΛ[by 4]


Anything aa Anything

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CFG – 5Describe a language that has even number of a’s

and even number of b’s using CGF.i.e. aababbab

1] S →SS 2] S →BALANCED S3] S →S BALANCED 4] S →Λ5] S →UNBALANCED S UNBALANCED6] BALANCED → aa 7] BALANCED → bb8] UNBALANCED → ab9] UNBALANCED → ba


Even-Even grammar

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CFG – 5Derive “aababbab” from the above grammar.S → BALANCED S [by 1]BALANCED→ aa UNBALANCED S UNBALANCED

[by 5]UNBALANCED → aa ba S UNBALANCED [by 9]UNBALANCED → aa ba S ab [by 8]S → aa ba BALANCED S ab [by 3]UNBALANCED → aa ba bb S ab [by 7]S → aa ba bb ab [by 4]

→ aababbab


Even-Even grammar

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CFG – 6i.e. {Λ, ab, aaabbb,… }S → aSb|Λ Derive “aaaabbbb”S →aSb→aSb→aaSbb →aaaSbbb

→aaaaSbbbb →aaaabbbb


Balanced a-b grammar

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CFG – 7i.e. {Λ, ab, abbaabba,… }S → aSa| bSb| Λ Derive “abbaabba” “abba abba”S →aSa→abSba→abbSbba →abbaSabba→ abba abba→ abbaabba


Even-Plaindrome grammar

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CFG – 8i.e. {Λ, abb, ababa,… }S → aSa| bSb| a| b| Λ Derive “ababa” “ab a ba”S →aSa→abSba→abSba →abSba→ ababa

Note: anban can be represented by S → aSa| bBut anban bn+1 can not be represented by CFG !


Plaindrome grammar

i.e. {Λ, ab, abbaabba,… }S → aSa| bSb| Λ Derive abaaba

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Even-Plaindrome grammar

S

S

Λ

aa

S bb

S aa


CFG – 9

Deduce CFG of addition and parse the following expression 2+3+5

1] S→S+S|N2] N→1|2|3|4|5|6|7|8|9|0 N1|N2|N3|N4|N5|N6|N7|N8|N9|N0

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S

S+N

S+

N

5S

+

3

N

2

N

Can u makeanother parsingtree ?


CFG – 10

CFG – 11.1Deduce CFG of a addition/multiplication and

parse the following expression 2+3*5

1] S→S+S|S*S|N

2] N→1|2|3|4|5|6|7|8|9|0 N1|N2|N3|N4|N5|N6|N7|N8|N9|N0

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S

S*S

S*

N

5S+

3

N

2

N



CFG -11.2 without ambiguityDeduce CFG of a addition/multiplication and

parse the following expression 2*3+51] S→ Term|Term + S 2] Term → N|N * Term 3] N→1|2|3|4|5|6|7|8|9|0

N1|N2|N3|N4|N5|N6|N7|N8|N9|N0

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S

S+N

S+

N

5S

*

3

N

2

N



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Thank you


Lec4

Education

s balanced s

s s balanced

s s aa s bb s aa

s unbalanced s unbalanced

s xaax

inner s

s aabs

s termterm