context-free languages hinrich schütze cis, lmu, 2013-11-25 slides based on rpi csci 2400 thanks to...
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Context-Free Languages
Hinrich Schütze
CIS, LMU, 2013-11-25
Slides based on RPI CSCI 2400
Thanks to Costas Busch
Take-awayDefinition context-free grammarDefinition context-free languageDerivation, sentential form, sentenceDerivation treesAmbiguityContext-free grammars for natural
language
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TerminologyFor our purposes in this class:Context free grammar=Constituency grammar=Phrase structure grammar
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GrammarsGrammars express languages
Example: the English language
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A derivation of “the dog walks”:
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A derivation of “a cat runs”:
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Language of the grammar:
L = { “a cat runs”, “a cat walks”, “the cat runs”, “the cat walks”, “a dog runs”, “a dog walks”, “the dog runs”, “the dog walks” }
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Notation
Variable Terminal
Production Rules
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Another ExampleGrammar:
Derivation of sentence :
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Language?
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Grammar:
Derivation of sentence :
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Other derivations:
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Language of the grammar
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More Notation
Grammar
Set of variables
Set of terminal symbols
Start variable
Set of Production rules
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Example
Grammar :
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More NotationSentential Form: A sentence that contains variables and terminals
Example:
Sentential Forms sentence
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We write:
Instead of:
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In general we write:
If:
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By default:
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Example Grammar Derivations
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Grammar
Example
Derivations
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Another Grammar ExampleGrammar :
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Language?
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Grammar :
Derivations:
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More Derivations
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Language of a Grammar
For a grammar with start variable :
String of terminals
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ExampleFor grammar :
Since:
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A Convenient Notation
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Revisit first grammar A context-free grammar :
A derivation:
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A context-free grammar :
Another derivation:
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(((( ))))Describes parentheses:
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A context-free grammar :
Example
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Language?
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A context-free grammar :
Another derivation:
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A context-free grammar :
Example
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Language?
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A context-free grammar :
Two derivations:
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Interpretation?
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() ((( ))) (( ))
Describes matched parentheses:
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Definition: Context-Free Grammars
Grammar
Productions of the form:
String of variables and terminals
Variables Terminalsymbols
Startvariable
Variable
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Definition: Context-Free Languages
A language is context-free
if and only if
there is a context-free grammar with
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Derivation Order
Leftmost derivation:
Rightmost derivation:
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Language?
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Leftmost derivation:
Rightmost derivation:
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Language?
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Derivation Trees
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Derivation Tree
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yield
Derivation Tree
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Partial Derivation Trees
Partial derivation tree
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Partial derivation tree
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Partial derivation tree
sententialform
yield
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Same derivation tree
Sometimes, derivation order doesn’t matter
Leftmost:
Rightmost:
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Ambiguity
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leftmost derivation
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leftmost derivation
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Two derivation trees
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The grammar
is ambiguous:
string has two derivation trees
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string has two leftmost derivations
The grammar
is ambiguous:
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Definition:
A context-free grammar is ambiguous
if some string has:
two or more derivation trees
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In other words:
A context-free grammar is ambiguous
if some string has:
two or more leftmost derivations
(or rightmost)
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Why do we care about ambiguity?
take
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Correct result:
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• We want to remove ambiguity
• Ambiguity is bad for programming languages
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We fix the ambiguous grammar:
New non-ambiguous grammar:
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Unique derivation tree
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The grammar :
is non-ambiguous:
Every string hasa unique derivation tree
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Another Ambiguous Grammar
IF_STMT if EXPR then STMT
if EXPR then STMT else STMT
Ambiguity?
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If expr1 then if expr2 then stmt1 else stmt2
IF_STMT
expr1 then
elseif expr2 then
STMT
stmt1
if
IF_STMT
expr1 then else
if expr2 then
STMT stmt2if
stmt1
stmt2
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Inherent Ambiguity
Some context free languageshave only ambiguous grammars
Example:
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The string
has two derivation trees
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Ambiguity in natural language?
Take-awayDefinition context-free grammarDefinition context-free languageDerivation, sentential form, sentenceDerivation treesAmbiguityContext-free grammars for natural
language
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