TDDD65
Introduction to the Theory of Computation
Lecture 3
Gustav Nordh
Department of Computer and Information Science
gustav.nordh@liu.se
2012-09-05
Outline
Context-free Grammars
Ambiguity
Pumping Lemma
Pushdown Automata
Summary of Context-free Languages
Context-free Languages (CFL)
What can be computed with restricted access to unlimited
memory?
Context-free Languages (CFL)
What can be computed with restricted access to unlimited
memory?
input tape
0 0 1 1
control
stack
..
.
···
Context-free Languages (CFL)
What can be computed with restricted access to unlimited
memory?
input tape
0 0 1 1
0
stack
..
.
control
···
Context-free Languages (CFL)
What can be computed with restricted access to unlimited
memory?
input tape
0 0 1 1
0
0
stack
..
.
control
···
Context-free Languages (CFL)
What can be computed with restricted access to unlimited
memory?
input tape
0 0 1 1
0
0
stack
···
control
..
.
A push-down automaton (PDA) is a NFA with a stack
Context-free Languages (CFL)
Recall: DFAs correspond to regular expressions
Push-down automatas (PDAs) correspond to Context-free
Grammars (CFGs)
Context-free Languages (CFL)
Noam Chomsky (1928 -)
Context-free Grammars: Motivation
Describing (parts of) natural languages
Describing the syntax of programming languages
Example of a Context-free Grammars (CFG)
Example
S→A
S→B
A → 0A1
A→ε
B → aBa
B → bBb
B→a
B→b
B→ε
Example of a Context-free Grammars (CFG)
Example
S→A
S→B
A → 0A1
A→ε
B → aBa
B → bBb
B→a
B→b
B→ε
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
Example of a Context-free Grammars (CFG)
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
A string is in the language of the grammar if it can be generated
by:
Example of a Context-free Grammars (CFG)
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
A string is in the language of the grammar if it can be generated
by:
1
Writing down the start variable
Example of a Context-free Grammars (CFG)
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
A string is in the language of the grammar if it can be generated
by:
1
Writing down the start variable
2
Replacing a variable that is written down by the right hand
side of a rule starting with that variable
Example of a Context-free Grammars (CFG)
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
A string is in the language of the grammar if it can be generated
by:
1
Writing down the start variable
2
Replacing a variable that is written down by the right hand
side of a rule starting with that variable
3
Repeating Step 2 until no variable remains
Example of a Context-free Grammars (CFG)
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
A string is in the language of the grammar if it can be generated
by:
1
Writing down the start variable
2
Replacing a variable that is written down by the right hand
side of a rule starting with that variable
3
Repeating Step 2 until no variable remains
S⇒B
Example of a Context-free Grammars (CFG)
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
A string is in the language of the grammar if it can be generated
by:
1
Writing down the start variable
2
Replacing a variable that is written down by the right hand
side of a rule starting with that variable
3
Repeating Step 2 until no variable remains
S ⇒ B ⇒ aBa
Example of a Context-free Grammars (CFG)
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
A string is in the language of the grammar if it can be generated
by:
1
Writing down the start variable
2
Replacing a variable that is written down by the right hand
side of a rule starting with that variable
3
Repeating Step 2 until no variable remains
S ⇒ B ⇒ aBa ⇒ abBba
Example of a Context-free Grammars (CFG)
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
A string is in the language of the grammar if it can be generated
by:
1
Writing down the start variable
2
Replacing a variable that is written down by the right hand
side of a rule starting with that variable
3
Repeating Step 2 until no variable remains
S ⇒ B ⇒ aBa ⇒ abBba ⇒ abba
Definition of Context-free Grammar (CFG)
Definition
A context-free grammar (CFG) is a 4-tuple (V , Σ, R, S) where
V is a finite set of variables
Σ is a finite set of terminals
R is a finite set of rules
S ∈ V is the start variable
The language of a CFG
If u, v , and w are strings of variables and terminals, and A → w
is a rule of the grammar we say that uAv yields uwv , written
uAv ⇒ uwv .
The language of a CFG
If u, v , and w are strings of variables and terminals, and A → w
is a rule of the grammar we say that uAv yields uwv , written
uAv ⇒ uwv .
∗
u derives v written u ⇒ v if u ⇒ u1 ⇒ u2 ⇒ · · · ⇒ uk ⇒ v
The language of a CFG
If u, v , and w are strings of variables and terminals, and A → w
is a rule of the grammar we say that uAv yields uwv , written
uAv ⇒ uwv .
∗
u derives v written u ⇒ v if u ⇒ u1 ⇒ u2 ⇒ · · · ⇒ uk ⇒ v
Definition
The language of a CFG G = (V , Σ, R, S) is
∗
{w ∈ Σ∗ | S ⇒ w}
written L(G).
The language of a CFG
If u, v , and w are strings of variables and terminals, and A → w
is a rule of the grammar we say that uAv yields uwv , written
uAv ⇒ uwv .
∗
u derives v written u ⇒ v if u ⇒ u1 ⇒ u2 ⇒ · · · ⇒ uk ⇒ v
Definition
The language of a CFG G = (V , Σ, R, S) is
∗
{w ∈ Σ∗ | S ⇒ w}
written L(G).
Definition
A language that is generated by some context-free grammar is
called a context-free language
The language of a CFG
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
The language of a CFG
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
Example
The language generated by
A → 0A1 | ε
is LA = {0n 1n | n ≥ 0}
The language of a CFG
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
Example
The language generated by
A → 0A1 | ε
is LA = {0n 1n | n ≥ 0}
Example
The language generated by
B → aBa | bBb | a | b | ε
is LB = {s ∈ {a, b}∗ | s is a palindrome}
CFG, Ambiguity
Definition
A derivation of a string w in a grammar G is a leftmost
derivation if at every step the leftmost remaining variable is the
one being replaced
CFG, Ambiguity
Definition
A derivation of a string w in a grammar G is a leftmost
derivation if at every step the leftmost remaining variable is the
one being replaced
Definition
A string s is derived ambiguously in a CFG G if it has two
different leftmost derivations. A CFG G is ambiguous if it
generates some string ambiguously.
CFG, Ambiguity
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
CFG, Ambiguity
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
Is this grammar ambiguous?
CFG, Ambiguity
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
Yes!
S⇒B⇒ε
S⇒A⇒ε
CFG, Ambiguity
Example
S→A|B
A → 0A1 | ε
B → aBa | bBb | a | b | ε
Equivalent and unambiguous:
Example
S→A|B|ε
A → 0A1 | 01
B → aBa | bBb | a | b | aa | bb
Pushdown Automata
input tape
0 0 1 1
control
stack
..
.
···
Pushdown Automata
input tape
0 0 1 1
0
stack
..
.
control
···
Pushdown Automata
input tape
0 0 1 1
0
0
stack
..
.
control
···
CFL, pushdown automata, definition
Definition
A pushdown automaton (PDA) is a 6-tuple (Q, Σ, Γ, δ, q0 , F )
where
Q is the finite set of states
Σ is the input alphabet
Γ is the stack alphabet
δ : Q × (Σ ∪ {ε}) × (Γ ∪ {ε}) → P(Q × (Γ ∪ {ε})) is the
transition function
q0 ∈ Q is the start state
F ⊆ Q is the set of accept states
CFL, pushdown automata
Pushdown automata are nondeterministic!
CFL, pushdown automata
Pushdown automata are nondeterministic!
Theorem
There are languages recognized by PDAs that are not
recognized by any deterministic PDA. For example the
language {ww R | w ∈ {0, 1}∗ }.
CFL, pushdown automata
Example
Describe a pushdown automaton recognizing the language
{0n 1n | n ≥ 0}
CFL, pushdown automata
Example
Describe a pushdown automaton recognizing the language
{0n 1n | n ≥ 0}
1
Start pushing the 0’s read on the stack.
2
When the first 1 appears, start popping a 0 from the stack
for each 1 that is read.
3
Should a 0 appear as input in this stage, then reject the
string.
4
If the input is finished and 0’s remains on the stack, or if
the stack is emptied before the input is finished, then reject
the string.
5
Otherwise, accept the string.
CFL, pushdown automata
Theorem
A language is context-free if and only if some pushdown
automaton recognizes it
nonCFLs
L = {0n 1n 2n | n ≥ 0} is not a CFL
Why?
nonCFLs
L = {0n 1n 2n | n ≥ 0} is not a CFL
Why?
Imagine a PDA that recognize L
When reading a string the PDA needs to keep track of the
number of 0’s so that it can check that the same number of
1’s and 2’s follow
The number of 0’s is unbounded so the PDA needs to use
its stack for this
To check that the same number of 1’s follow, the PDA
needs to empty its stack
Now, the PDA has no way of checking that the same
number of 2’s follow
Pumping Lemma for CFLs
Lemma
If L is a CFL, then there exists a positive integer p (the pumping
length) such that every string s ∈ L, |s| ≥ p, can be partitioned
into five pieces, s = uvxyz, such that the following conditions
hold:
|vy | > 0,
|vxy | ≤ p, and
for each i ≥ 0, uv i xy i z ∈ L
Summary of Context-free Languages
The context-free languages are the languages generated
by context-free grammars
A context-free grammar is ambiguous if the same string
can be derived using two different left-most derivations
A language is context-free iff it is recognized by a PDA
There are simple languages that are not context-free