Lecture 6: Context-Free Languages

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Lecture 6 Context-Free Languages

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Content

• Language Recognizer vs. Language Generator
• Context-Free Grammars
• Regular Languages vs. Context-Free Languages
• Pushdown Automata
• Pushdown Automata vs. Context-Free Grammars
• Properties of Context-Free Languages
• Context-Sensitive Grammars

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Lecture 6 Context-Free Languages

• Language Recognizer vs.
• Language Generator

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Recognizer vs. Generator
Parsing
Machines
Grammas
Generation
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Example Regular Expressions
Is the expression power of r. e. strong enough?
A rule (grammar) to generate strings ?
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Regular Expressions As Grammars
S, M, A, B
? nonterminals
Production Rules
S
? start symbol
a, b
? terminals
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Regular Expressions As Grammars
Example
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Lecture 6 Context-Free Languages

• Context-Free Grammars

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Definition
A context-free grammar G is a quadruple
(V, ?, P, S)
where
V finite set of nonterminals
V ? ? ?
? finite set of terminals
P finite set of rules, i.e., subset of V? (V?
?)
S start symbol, S?V
In the form of A?? with A?V and ??(V? ?)
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Notation Conventions

• The capital letters A, B, C, D, E, and S denote
  nonterminals S is the start symbol.
• The lower-case letters a, b, c, d, e, digits and
  boldface strings are terminals.
• The capital letters X, Y, and Z that may be
  either terminals or variables.
• The lower-case letters u, v, w, x, y, and z
  denote strings of terminals.
• The lower-case Greek letters ?, ?, and ? denote
  strings of variables and terminals.

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Notation Conventions
a derivation of G of ?m from ?1.
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Context-Free Languages
The language generated by G (V, ?, P, S),
denoted as L(G), is
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Example
L(G)?
G (V, ?, P, E)
E Expression T Term F Factor
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Example
L(G)?
R1
R2
R3
R4
R5
R6
R7
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Example
L(G)?
x1
x1(x2x1x1)
x1(x2(x1x1))
x1x2x1x2
?L(G)?
x1(
xx2
x2
2x2
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Example
L(G)?
G (V, ?, P, S)
VS ?a, b
L(G)anbnn?0
S ? aSb
? aaSbb
? a3Sb3
? ? ? ? ? anSbn
? anbn
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Example
L(G)?
G (V, ?, P, S)
Is the expression power of context-free grammar
stronger than regular expression?
VS ?a, b
L(G)anbnn?0
S ? aSb
? aaSbb
? a3Sb3
? ? ? ? ? anSbn
? anbn
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Lecture 6 Context-Free Languages

• Regular Languages
• vs.
• Context-Free Languages

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Definition ? Regular Grammar
A CFG G (V, ?, P, S) is regular iff
P ? V ? ?(V??).
tailing nonterminal
RHS
nonterminal
leading terminal(s)
LHS
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Example
Consider G (V, ?, P, S) with
G is regular.
VS, A, B ?a, b
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Regular Grammars vs. Regular Languages
Consider G (V, ?, P, S) with
G is regular.
VS, A, B ?a, b
? babaS
?
S ? bA
? ababS
S ? aB
?
Regular Expression
L(G)(abab baba)
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Theorem
Let L be a language.
L is regular iff L is generated by a regular
grammar.
accepted by a DFA
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Theorem
L is regular iff L is generated by a regular
grammar.
Pf)
?
Given a DFA M (K, ?, ?, s, F), we will
construct a regular grammar, say, G st. L(M)
L(G).
Consider G (K, ?, P, S) st.
From the correspondence btw. M and G, it is
easily seen that
Hence,
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Theorem
L is regular iff L is generated by a regular
grammar.
Pf)
?
Given a regular grammar G (V, ?, P, S), we will
construct an NFA, say, M st. L(G) L(M).
Consider M (K, ?, ?, s, F) st.
From the correspondence btw. M and G, it is
easily seen that
Hence,
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Example
Consider G (V, ?, P, S) with
G is regular.
VS, A, B ?a, b
f
B
A
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The Chomsky Hierarchy
Chomsky Hierarchy Languages Grammars Automaton
Type 0 Recursively enumerable unrestricted Turing Machine
Recursive unrestricted Decider
Type 1 Context-Sensitive Context-Sensitive Linear-Bounded Automaton
Type 2 Context-Free Context-Free Push-Down Automaton
Type 3 Regular Regular NFA or DFA
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The Chomsky Hierarchy
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Lecture 6 Context-Free Languages

• Pushdown Automata

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Basic Concept
How to accept language LwwR w???
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Definition
A pushdown automata is a sextuple
(K, ?, ?, ?, s, F)
where
K a finite set of states
? a finite set of input symbols
? a finite set of stack symbols
? the transition function K ???? ? K ??
s ?K the initial state
F ?K the set of final states
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Definition ? More Precise Version
A PDA is possibly nondeterministic.
A pushdown automata is a sextuple
(K, ?, ?, ?, s, F)
where
K a finite set of states
? a finite set of input symbols
? a finite set of stack symbols
? the transition function K ???? ? 2K ??
s ?K the initial state
F ?K the set of final states
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The Transition Function of PDA
K ???? ? K ??
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The Transition Function of PDA
?K ???? ? K ??
?(p, v, ?)? (q, ?)
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Stack Operations
Case 3 ? ?, ? ? ?
Case 1 ? ? ?, ? ? ?
replacement
push
Case 2 ? ? ?, ? ?
Case 4 ? ?, ? ?
pop
none
?(p, v, ?)? (q, ?)
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Yields in One Step
if ?(p, v, ?)? (q, ?)
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Yields
in some (including zero) steps
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Languages Accepted by PDAs
M (K, ?, ?, ?, s, F)
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Example
A deterministic PDA
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Example
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Example
How to design a PDA to accept L wwR?
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Example
?
-
s
abbbba
s
a
1
bbbba
s
ba
2
bbba
s
bba
2
bba
bba
3
f
bba
f
ba
5
ba
f
a
5
a
?
f
?
4
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Example
A nondeterministic PDA
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Lecture 6 Context-Free Languages

• Pushdown Automata
• vs.
• Context-Free Grammars

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Leftmost and Rightmost Derivations

• Given a context-free grammar there may be several
  ways of rewriting its nonterminals to arrive at
  precisely the same string.
• Two systematic ways of derivations are
• Leftmost derivation
• Rightmost derivation

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Example
Note that there are other derivations that are
neither leftmost nor rightmost.
VS, A, B, ?a, b, c
Consider G (V, ?, P, S) with
? abcbabb
S ? aABb
? abAbBb
? abcbBb
? abcbaBb
S ? aABb
? aAaBb
? aAabb
? abAbabb
? abcbabb
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Lemma
For any context-free grammar G (V, ?, P, S) and
any string w? ?,
See textbook for the proof
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Theorem
The class of languages accepted by pushdown
automata is exactly the class of context-free
languages.
Pf)
? Pushdown Automaton
Context-Free Grammar
G (V, ?, P, S)
M ?
? Context-Free Grammar
Pushdown Automaton
M (K, ?, ?, ?, s, F)
G ?
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Theorem
The class of languages accepted by pushdown
automata is exactly the class of context-free
languages.
Pf)
Context-Free Grammar
? Pushdown Automaton
? M (p, q, ?, V? ?, ?, p, q)
G (V, ?, P, S)
M ?
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Theorem
The class of languages accepted by pushdown
automata is exactly the class of context-free
languages.
Pf)
Context-Free Grammar
? Pushdown Automaton
? M (p, q, ?, V, ?, p, q)
G (V, ?, P, S)
M ?
One can show that L(M)L(G).
(see text)
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Theorem
The class of languages accepted by pushdown
automata is exactly the class of context-free
languages.
Pf)
Pushdown Automaton
? Context-Free Grammar
M (K, ?, ?, ?, s, F)
G ?
see text
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Lecture 6 Context-Free Languages

• Properties of
• Context-Free Languages

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Review ? Properties of Regular Languages

• The regular languages are closed under
• union
• concatenation
• Keene star
• complementation
• intersection.

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Properties of Context-Free Languages

• The context-free languages are closed under
• union
• concatenation
• Keene star
• complementation
• intersection.

?
?
?
?
?
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Properties of Context-Free Languages

• The context-free languages are closed under
• union
• concatenation
• Keene star.

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Properties of Context-Free Languages

• The context-free languages are closed under
• union
• concatenation
• Keene star.

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Properties of Context-Free Languages

• The context-free languages are closed under
• union
• concatenation
• Keene star.

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Review The Pumping Theoremfor Regular Languages
Let L be an infinite regular set. Then there are
strings u,v, and w st. vgt0 and uvnw?L for all n
? 0.
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What CFGs Can? What CFGs cannot?
Which of the following languages are context-free?
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The Pumping Theoremfor Context-Free Languages
See textbook for the proof.
Let G be context-free gramma. Then there is a
number K, depending on G, such that any w ? L(G)
with wgtK can be rewritten as w uvxyz in such
a way that either v or y is nonempty and uvnxynz
? L(G) for all n ? 0.
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Theorem
Let G be context-free gramma. Then there is a
number K, depending on G, such that any w ? L(G)
with wgtK can be rewritten as w uvxyz in such
a way that either v or y is nonempty and uvnxynz
? L(G) for all n ? 0.
is not context-free.
Pf)
Let
It is impossible for us to rewrite w as w uvxyz
such that vygt0 and uvnxynz ? L(G) for all n ? 0.
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Theorem
The context-free languages is not closed under
intersection or complementation.
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Theorem
The context-free languages is not closed under
intersection or complementation.
Pf)
Facts
context-free
? not context-free
? CFLs are not closed under intersection.
Fact

• If CFLs are closed under complementation,
• they would be also closed under intersection.

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Theorem Algorithmic Properties
See textbook for the proof.
There are algorithms for answering the following
questions about context-free grammars

1. Given a grammar G, and a string w, is w?L(G)?
2. Is L(G)??

How?
What is compiler-compiler?
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The Chomsky Hierarchy
Chomsky Hierarchy Languages Grammars Automaton
Type 0 Recursively enumerable unrestricted Turing Machine
Recursive unrestricted Decider
Type 1 Context-Sensitive Context-Sensitive Linear-Bounded Automaton
Type 2 Context-Free Context-Free Push-Down Automaton
Type 3 Regular Regular NFA or DFA
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The Chomsky Hierarchy
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Lecture 6 Context-Free Languages

• Context-Sensitive Grammars

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Definition ?Context-Free Grammar (Review)
A context-free grammar G is a quadruple
(V, ?, P, S)
where
V finite set of nonterminals
V ? ? ?
? finite set of terminals
P finite set of rules in the form of
A?? with A?V and ??(V? ?)
S start symbol, S?V
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Definition ?Context-Sensitive Grammar (I)
A context-free grammar G is a quadruple
A context-sensitive grammar G is a quadruple
(V, ?, P, S)
where
V finite set of nonterminals
V ? ? ?
? finite set of terminals
? ? ?
P finite set of rules in the form of
A?? with A?V and ??(V? ?)
with ?, ??(V? ?) and ? ? ?.
S start symbol, S?V
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Definition ?Context-Sensitive Grammar (II)
A context-free grammar G is a quadruple
A context-sensitive grammar G is a quadruple
(V, ?, P, S)
where
V finite set of nonterminals
V ? ? ?
? finite set of terminals
? ? ?
?A? ? ???
P finite set of rules in the form of
A?? with A?V and ??(V? ?)
with ?, ??(V? ?) and ? ? ?.
with ?, ??(V? ?), A?V and ??(V? ?).
S start symbol, S?V
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Definition ?Context-Sensitive Grammar (II)
A language L is said to be context-sensitive if
there exists a context-sensitive grammar G, such
that
L L(G)
or
L L(G)??
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Review
is not context-free.
Is it context-sensitive?
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Example
Consider G (V, ?, P, S) with
VS, A, B ?a, b, c
L(G)?
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Example
Consider G (V, ?, P, S) with
VS, A, B ?a, b, c
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Example
Consider G (V, ?, P, S) with
VS, A, B ?a, b, c
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The Chomsky Hierarchy
Chomsky Hierarchy Languages Grammars Automaton
Type 0 Recursively enumerable unrestricted Turing Machine
Recursive unrestricted Decider
Type 1 Context-Sensitive Context-Sensitive Linear-Bounded Automaton
Type 2 Context-Free Context-Free Push-Down Automaton
Type 3 Regular Regular NFA or DFA
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The Chomsky Hierarchy
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Exercises

• Construct pushdown automata for the following
  languages over the alphabet a, b
• w w has an equal number of as and bs
• w the length of w is odd and its middle
  symbol is a b
• Prove that the following languages are not
  context free
• aibjck 0 lt i lt j lt k
• wwRw w ? a,b
• Construct context-free grammars that define the
  following languages over a, b
• ambn m ? n
• aibjck i j k