Context-Free Languages Hinrich Schütze CIS, LMU, 2013-11-25 Slides based on RPI CSCI 2400 Thanks to Costas Busch.

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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