Languages and Finite Automata

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Formal Languages Context-Free Languages Hinrich
Schütze IMS, Uni Stuttgart, WS 2006/07 Slides
based on RPI CSCI 2400 Thanks to Costas Busch
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Regular Languages
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Context-Free Languages
Regular Languages
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Context-Free Languages
Context-Free Grammars
Pushdown Automata
stack
automaton
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Context-Free Grammars

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Grammars

• Grammars 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

Production Rules
Variable
Terminal
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Another Example

• Grammar
• 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 Notation

• Sentential 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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Example
Grammar
Derivations
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Another Grammar Example

• Grammar

Derivations
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Language?
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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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Example

• For grammar

Since
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A Convenient Notation

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Example

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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Example
A context-free grammar
A derivation
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Language?
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A context-free grammar
Another derivation
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(No Transcript)
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Example
A context-free grammar
A derivation
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Language?
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A context-free grammar
A derivation
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Interpretation?
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Describes matched parentheses
() ((( ))) (( ))
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Definition Context-Free Grammars
Grammar
Variables
Terminal symbols
Start variable
Productions of the form
String of variables and terminals
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

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Language?
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Language?
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Derivation Trees

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(No Transcript)
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Derivation Tree
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Derivation Tree
yield
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Partial Derivation Trees

Partial derivation tree
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Partial derivation tree
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sentential form

Partial derivation tree
yield
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Sometimes, derivation order doesnt matter
Leftmost

Rightmost
Same derivation tree
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Ambiguity

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leftmost derivation
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leftmost derivation
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(No Transcript)
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The grammar
is ambiguous
string
has two derivation trees
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The grammar
is ambiguous
string
has two leftmost derivations
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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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(No Transcript)
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Correct result
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• Ambiguity is bad for programming languages

• We want to remove ambiguity

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We fix the ambiguous grammar
New non-ambiguous grammar
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(No Transcript)
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Unique derivation tree
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The grammar
is non-ambiguous
Every string has a unique
derivation tree
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Another Ambiguous Grammar
IF_STMT
if EXPR then STMT
if EXPR then STMT else STMT
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If expr1 then if expr2 then stmt1 else stmt2
IF_STMT
if
expr1
then
STMT
else
if
expr2
then
stmt1
stmt2
IF_STMT
if
expr1
then
else
STMT
stmt2
if
expr2
then
stmt1
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Inherent Ambiguity

• Some context free languages
• have only ambiguous grammars

Example
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The string
has two derivation trees
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Ambiguity in natural language?