Backus Naur and Chomsky Normal Forms

NOTES
Title: Grammars for Compilers
Intro to BNF
Definition of BNF
Example : http://otal.umd.edu/drweb/c++tutorial/lessons/BNF.HTM
Variants of BNF: http://en.wikipedia.org/wiki/Backus–Naur_Form
Why BNF?
CNF Definition
Characteristics
BNF vs. CNF
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Backus Naur and Chomsky Normal Forms
Ashutosh Pandey
10/26/2009
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Backus-Naur Form
Created by John Warner Backus (1927-2007) to describe the grammar of ALGOL, the first, widely used high-level language.
The Backus–Naur Form or BNF grammars have significant similarities to Panini's Sanskrit grammar rules, and the notation is sometimes also referred to as Panini–Backus Form.
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Definition of BNF
A BNF specification consists of a set of production rules written as:-
<variable> ::= <expression>
where
<variable> is a non-terminal symbol
<expression> consists of one or more sequences of terminal and non-terminal symbol and each sequence is separated by the ‘ | ‘ symbol.
Terminal symbols cannot appear on the left side of a production
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Example of BNF
BNF Specification of the C++ grammar:-
<c++prog> ::= <sysdirectives> <decdefs> main () {<body>}
<sysdirectives> ::= ‘ ‘| #<includedir>
<includedir> ::= include '<'<filename>'>‘
.
.
.
<body> ::= | <statement> | <decdef> |<body> <statement> | <body> <decdef>
<statement> ::= <assignment>; | <inputstatement>; | <outputstatement>; | <condstatement>; | <iterativestatement>;
<assignment> ::= <id>++ | ++<id> | <id>-- | --<id> | <equalassign>
…
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Variants of BNF
Many variations of the BNF exist, for the sake of simplicity, and readability
Extended BNF (EBNF) is a popular variant, which uses regular expression repetition operators such as ‘+’ or ‘*’
Reduced BNF (RBNF) is used to encode rules of various routing protocols.
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Advantages of BNF
High human readability: BNF is informal in nature.
Ease in Parsing: simple LL parsers are sufficient.
Ease of adaptability: Can be customized for various application due to it’s informal nature
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Who uses BNF?
YACC
Definitions of markup languages such as XML (HTML does not have a grammar).
GOLD: A freeware parsing system.
Other parser generators for JAVA(Parser Objects), Python (Wormhole).
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Chomsky Normal Form
A grammar is in the Chomsky normal form if all it’s productions are of the form :-
S  AB
A  a
Where S is the start variable, AB are any two non – terminal symbol and a is a terminal symbol.
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Characteristics of CNF
Every parse tree is a binary tree with 2n – 1 nodes, where n is the length of the string to be parsed.
No Ambiguity.
Uses the CYK Algorithm whose time complexity is Θ(2n- 1), therefore efficient parsing is another characteristic
Every context-free grammar can be efficiently converted in to CNF.
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Uses of CNF
Required for some efficient parsing algorithms such as Cocke-Younger-Kasami (CYK) algorithm
Used for proofs in the field of languages and computability because of its linear order of growth.
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BNF vs. CNF
BNF
Easily understood by humans.
Useful for Parser Generators.
Uses LL, LR, LALR Parsing algorithms.
CNF
Often difficult to understand.
Useful for theorem proving.
Uses the CYK Parsing algorithm
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