Transcript
Theory of Computation
Lecture 4: Context Free Grammar (CFG)
Cairo UniversityFCI
Dr. Hussien SharafComputer Science Departmentdr.sharaf@from-masr.com
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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.
Dr. Hussien M. Sharaf
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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]
Dr. Hussien M. Sharaf
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]
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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]
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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]
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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
Dr. Hussien M. Sharaf
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
Dr. Hussien M. Sharaf
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
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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
Dr. Hussien M. Sharaf
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 !
Dr. Hussien M. Sharaf
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
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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 ?
Dr. Hussien M. Sharaf
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
Can u makeanother parsingtree ?
Dr. Hussien M. Sharaf
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
Can u makeanother parsingtree ?
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Thank you
Dr. Hussien M. Sharaf
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