Cs419 lec7 cfg
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This document introduces parsing and context-free grammars. It discusses how a parser uses a context-free grammar to determine if a program's syntax is correct by building a parse tree. It then provides examples of context-free grammars for simple languages involving parentheses, palindromes, and arithmetic expressions. The document also discusses ambiguity in grammars and ways to eliminate it, such as removing empty productions and avoiding left recursion.
![DERIVATION
We 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 nonterminal S.
Then using S→ ab to replace the inner S
Generates
“aabb”
S aSb aabb ……[Successful derivation of aabb]
Dr. Hussien M. Sharaf
7](https://image.slidesharecdn.com/cs419-20lec7-20-20cfg-140220150618-phpapp02/85/Cs419-lec7-cfg-7-320.jpg)
![CFG -1 : BALANCED-PARENTHESES
Prod1 S → (S)
Prod2 S → ()
Derive the string ((())).
S → (S)
…..[by prod1]
→ ((S))
…..[by prod1]
→ ((()))
…..[by prod2]
Dr. Hussien M. Sharaf
8](https://image.slidesharecdn.com/cs419-20lec7-20-20cfg-140220150618-phpapp02/85/Cs419-lec7-cfg-8-320.jpg)
![CFG -2 : PALINDROME
Describe palindrome of a’s and b’s using
CFG
1] S → aSa
2] S → bSb
3] S → Λ
Derive “baab” from the above grammar.
S → bSb
[by 2]
→ baSab
[by 1]
→ ba ab
[by 3]
Dr. Hussien M. Sharaf
9](https://image.slidesharecdn.com/cs419-20lec7-20-20cfg-140220150618-phpapp02/85/Cs419-lec7-cfg-9-320.jpg)
![CFG – 4
Describe anything (a+b)* using CGF
1] S → Λ
2] S → Y
3] Y→ aY
4] 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
11](https://image.slidesharecdn.com/cs419-20lec7-20-20cfg-140220150618-phpapp02/85/Cs419-lec7-cfg-11-320.jpg)
![CFG – 5
1] S → Λ
2] S → aS
3] S→ bS
Derive “aa” from the above grammar.
S → aS
[by 2]
→ aaS
[by 2]
→ aa
[by 1]
Dr. Hussien M. Sharaf
12](https://image.slidesharecdn.com/cs419-20lec7-20-20cfg-140220150618-phpapp02/85/Cs419-lec7-cfg-12-320.jpg)
![Ex 1
Deduce CFG of addition and parse the
following expression 2+3+5
1] 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
S
S+N
S
S
+
+
N
N
Can u make
another parsing
tree ?
5
N
3
2
Dr. Hussien M. Sharaf
17](https://image.slidesharecdn.com/cs419-20lec7-20-20cfg-140220150618-phpapp02/85/Cs419-lec7-cfg-17-320.jpg)
![Ex 2
Deduce 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|NN
S
S*S
S
S
+
*
N
N
Can u make
another parsing
tree ?
5
N
3
2
Dr. Hussien M. Sharaf
18](https://image.slidesharecdn.com/cs419-20lec7-20-20cfg-140220150618-phpapp02/85/Cs419-lec7-cfg-18-320.jpg)
![Ex 3 CFG without ambiguity
Deduce CFG of a addition/multiplication
and parse the following expression 2*3+5
1] S→ Term|Term + S
2] Term → N|N * Term
3] N→1|2|3|4|5|6|7|8|9|0
S
S+N
S
S
*
+
N
N
Can you make
another parsing
tree ?
5
N
3
2
Dr. Hussien M. Sharaf
19](https://image.slidesharecdn.com/cs419-20lec7-20-20cfg-140220150618-phpapp02/85/Cs419-lec7-cfg-19-320.jpg)
Cs419 lec7 cfg
- 1. WELCOME TO A JOURNEY TO CS419 Dr. Hussien Sharaf Computer Science Department dr.sharaf@from-masr.com
- 2. Dr. Hussien M. Sharaf PART ONE 2
- 3. PARSING A parser gets a stream of tokens from the scanner, and determines if the syntax (structure) of the program is correct according to the (context-free) grammar of the source language. Then, it produces a data structure, called a parse tree or an abstract syntax tree, which describes the syntactic structure of the program. Dr. Hussien M. Sharaf Stream of tokens parser Parse/syntax tree 3
- 4. 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 4
- 5. CFG FORMAL DEFINITION C =(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) Dr. Hussien M. Sharaf 5
- 6. CFG -1 Define the language { anbn | n > 1}. Terminals = {a, b}. Variables = {S}. Start symbol = S. Productions = S → ab S → aSb Summary S → ab S → aSb Dr. Hussien M. Sharaf 6
- 7. DERIVATION We 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 nonterminal S. Then using S→ ab to replace the inner S Generates “aabb” S aSb aabb ……[Successful derivation of aabb] Dr. Hussien M. Sharaf 7
- 8. CFG -1 : BALANCED-PARENTHESES Prod1 S → (S) Prod2 S → () Derive the string ((())). S → (S) …..[by prod1] → ((S)) …..[by prod1] → ((())) …..[by prod2] Dr. Hussien M. Sharaf 8
- 9. CFG -2 : PALINDROME Describe palindrome of a’s and b’s using CFG 1] S → aSa 2] S → bSb 3] S → Λ Derive “baab” from the above grammar. S → bSb [by 2] → baSab [by 1] → ba ab [by 3] Dr. Hussien M. Sharaf 9
- 10. CFG -3 : EVEN-PLAINDROME i.e. {Λ, ab, abbaabba,… } S → aSa| bSb| Λ Derive abaaba S a S a b S b a S a Λ Dr. Hussien M. Sharaf 10
- 11. CFG – 4 Describe anything (a+b)* using CGF 1] S → Λ 2] S → Y 3] Y→ aY 4] 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 11
- 12. CFG – 5 1] S → Λ 2] S → aS 3] S→ bS Derive “aa” from the above grammar. S → aS [by 2] → aaS [by 2] → aa [by 1] Dr. Hussien M. Sharaf 12
- 13. Dr. Hussien M. Sharaf PART TWO 13
- 14. Parsing CFG grammar is about categorizing the statements of a language. Parsing using CFG means categorizing a certain statements into categories defined in the CFG. Parsing can be expressed using a special type of graph called Trees where no cycles exist. A parse tree is the graph representation of a derivation. Programmatically; Parse tree can be represented as a dynamic data structure using a single root node. Dr. Hussien M. Sharaf 14
- 15. Parse tree (1)A vertex with a label which is a Non-terminal symbol is a parse tree. (2) If A → y1 y2 … yn is a rule in R, then the tree A y 1 y 2 ... y n is a parse tree. Dr. Hussien M. Sharaf 15
- 16. Ambiguity A grammar can generate the same string in different ways. Ambiguity occurs when a string has two or more leftmost derivations for the same CFG. There are ways to eliminate ambiguity such as using Chomsky Normal Form (CNF) which does n’t use Λ. Λ cause ambiguity. Dr. Hussien M. Sharaf 16
- 17. Ex 1 Deduce CFG of addition and parse the following expression 2+3+5 1] 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 S S+N S S + + N N Can u make another parsing tree ? 5 N 3 2 Dr. Hussien M. Sharaf 17
- 18. Ex 2 Deduce 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|NN S S*S S S + * N N Can u make another parsing tree ? 5 N 3 2 Dr. Hussien M. Sharaf 18
- 19. Ex 3 CFG without ambiguity Deduce CFG of a addition/multiplication and parse the following expression 2*3+5 1] S→ Term|Term + S 2] Term → N|N * Term 3] N→1|2|3|4|5|6|7|8|9|0 S S+N S S * + N N Can you make another parsing tree ? 5 N 3 2 Dr. Hussien M. Sharaf 19
- 20. Example 4 : AABB S A|AB A Λ| a | A b | A A B b|bc|Bc|bB Sample derivations: S AB AAB aAB aaB aabB aabb S AB AbB Aabb aabb A A B A b a B A b a Dr. Hussien M. Sharaf B A A a AAbb S S A Abb a b b 20
- 21. Ex 5 S A B A|AB Λ|a|Ab|AA b|bc|Bc|bB S A S B A A A b B a a S b A a A B A A b a b A A A a A A b A b a Dr. Hussien M. Sharaf 21
- 22. REMOVING AMBIGUITY Eliminate “useless” variables. Eliminate Λ-productions: A Λ. Avoid left recursion by replacing it with right-recursion. But if a language is ambiguous, it can’t be totally removed. We just need to the parsing to continue without entering an infinite loop. Dr. Hussien M. Sharaf 22
- 23. THANK YOU Dr. Hussien M. Sharaf 23