A Finite Automaton

Transcription

A Finite Automaton

A Finite Automaton
011011001
read
unread
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University.
– p.1
A Pushdown Automaton
0110110
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a
b
b
a
pop
push
– p.2
can push symbols onto the stack
– p.3
can pop them (read them back) later
– p.3
stack can grow unboundedly
– p.3
stack can grow unboundedly
yet at any moment stack has finite size.
– p.3
An Example
Recall that the language {0n 1n | n ≥ 0} is not regular.
Consider the following PDA:
read input symbols
– p.4
An Example
read input symbols
for each 0, push it on the stack
– p.4
An Example
read input symbols
as soon as a 1 is seen, pop a 0 for each 1 read
– p.4
An Example
read input symbols
accept if stack is empty when last symbol read
– p.4
An Example
read input symbols
reject if
– p.4
An Example
read input symbols
reject if
stack is non-empty when last symbol read
– p.4
An Example
read input symbols
reject if
stack is empty but input symbol(s) still exist,
– p.4
An Example
read input symbols
reject if
stack is empty but input symbol(s) still exist,
0 is read after 1.
– p.4
On PDA vs. Finite Automata
Nondeterminism
– p.5
Nondeterminism
PDA may be deterministic or
non-deterministic.
– p.5
Nondeterminism
non-deterministic.
Unlike finite automata, non-determinism adds
power.
– p.5
Nondeterminism
non-deterministic.
power.
There are some languages accepted only by
non-deterministic PDAs.
– p.5
Nondeterminism
non-deterministic.
power.
There are some languages accepted only by
non-deterministic PDAs.
Transition function δ looks different than DFA or
NFA cases, reflecting stack functionality.
– p.5
The Transition Function
Denote input alphabet by Σ and stack alphabet by Γ.
the domain of the transition function δ is
– p.6
current state: Q
– p.6
current state: Q
next input symbol, if any: Σε (=Σ ∪ {ε})
– p.6
current state: Q
stack symbol popped, if any: Γε
– p.6
current state: Q
and its range is
– p.6
current state: Q
and its range is
new state: Q
– p.6
current state: Q
and its range is
new state: Q
stack symbol pushed, if any: Γε
– p.6
current state: Q
and its range is
new state: Q
non-determinism: P(· · · )
– p.6
current state: Q
and its range is
new state: Q
non-determinism: P(· · · )
δ : Q × Σε × Γε → P(Q × Γε )
– p.6
Formal Definitions
A pushdown automaton (PDA) is a 6-tuple
(Q, Σ, Γ, δ, q0 , F ), where
Q is a finite set called the states,
– p.7
Formal Definitions
(Q, Σ, Γ, δ, q0 , F ), where
Σ is a finite set called the input alphabet,
– p.7
Formal Definitions
(Q, Σ, Γ, δ, q0 , F ), where
Γ is a finite set called the stack alphabet,
– p.7
Formal Definitions
(Q, Σ, Γ, δ, q0 , F ), where
δ : Q × Σε × Γε → P(Q × Γε ) is the
transition function,
– p.7
Formal Definitions
(Q, Σ, Γ, δ, q0 , F ), where
q0 ∈ Q is the start state, and
– p.7
Formal Definitions
(Q, Σ, Γ, δ, q0 , F ), where
q0 ∈ Q is the start state, and
F ⊆ Q is the set of accept states.
– p.7
Conventions
Question: When is the stack empty?
– p.8
Conventions
start by pushing $ onto stack
– p.8
Conventions
when you see it again, stack is empty.
– p.8
Conventions
Question: When is input string exhausted?
– p.8
Conventions
doesn’t matter – PDA need not know this
explicitly
– p.8
Conventions
doesn’t matter – PDA need not know this
explicitly
accepting state accepts only if inputs
exhausted!
– p.8
Notation
Transition a, b → c means
– p.9
Notation
read a from input
– p.9
Notation
read a from input
pop b from stack
– p.9
Notation
read a from input
pop b from stack
push c onto stack
– p.9
Notation
read a from input
pop b from stack
push c onto stack
Meaning of ε transitions:
– p.9
Notation
read a from input
pop b from stack
push c onto stack
if a = ε, don’t read inputs
– p.9
Notation
read a from input
pop b from stack
push c onto stack
if b = ε, don’t pop any symbols
– p.9
Notation
read a from input
pop b from stack
push c onto stack
if b = ε, don’t pop any symbols
if c = ε, don’t push any symbols
– p.9
Example
q1
ε,ε $
q2
1,0
q4
ε,$ ε
q3
0,ε $
ε
1,0
ε
– p.10
Example
q1
ε,ε $
q2
1,0
q4
ε,$ ε
q3
0,ε $
ε
1,0
ε
What does this PDA accept?
– p.10
Example
q1
ε,ε $
q2
1,0
q4
ε,$ ε
q3
0,ε $
ε
1,0
ε
What does this PDA accept?
{0n 1n |n ≥ 1}.
– p.10
Another Example
A PDA that accepts
i j k
a b c |i, j, k > 0 and i = j or i = k
Informally:
read and push a’s
– p.11
Another Example
A PDA that accepts
i j k
Informally:
read and push a’s
either pop and match with b’s
– p.11
Another Example
A PDA that accepts
i j k
Informally:
read and push a’s
or else pop and match with c’s
– p.11
Another Example
A PDA that accepts
i j k
Informally:
read and push a’s
non-deterministic choice!
– p.11
Another Example
A PDA that accepts
i j k
Informally:
read and push a’s
non-deterministic choice!
Note: non-determinism essential here!
Unlike finite automata, non-determinism
does add power
– p.11
Another Example
c,ε ε
ε,$ ε
q3
q4
b,a ε
ε,ε $
ε,ε
ε
q1
ε
ε
ε
q5 ε,ε
q2 ε,ε
q0 ε,ε
q7
a,ε a
b,ε ε
c,a ε
This PDA accepts
i j k
– p.12
Yet Another Example
A palindrome is a string w satisfying w = w R.
“Madam I’m Adam”
– p.13
Yet Another Example
“Dennis and Edna sinned”
– p.13
Yet Another Example
“Red rum, sir, is murder”
– p.13
Yet Another Example
“Able was I ere I saw Elba”
– p.13
Yet Another Example
“In girum imus nocte et consumimur igni”
– p.13
Yet Another Example
“νιψoν ανoµηµατ α µη µoναν oψιν”
– p.13
Yet Another Example
“νιψoν ανoµηµατ α µη µoναν oψιν”
Palindromes also appear in nature. For example as
DNA sites (strings over {A, C, T, G}) being cut
by restriction enzymes.
– p.13
Yet Another Example
q1
ε,ε $
q2
0,ε 0
1,ε 1
ε,ε ε
q4
ε,$ ε
q3
0,0 ε
1,1 ε
This PDA accepts binary palindromes of even length.
– p.14
Equivalence Theorem
Theorem: A language is context free if and only if
some pushdown automata accepts it.
– p.15
Equivalence Theorem
Theorem: A language is context free if and only if
some pushdown automata accepts it.
This time, the proofs of both the “if” part and the “only
if” part are interesting.
– p.15
If Part
Theorem: If a language is context free, then some
pushdown automaton accepts it.
Let A be a context-free language.
– p.16
If Part
By definition, A has a context-free grammar G
generating it.
– p.16
If Part
generating it.
On input w, the PDA P should figure out if there
is a derivation of w using G.
– p.16
If Part
generating it.
Question: How does P figure out which substitution
to make?
– p.16
If Part
generating it.
Question: How does P figure out which substitution
to make?
Answer: It guesses.
– p.16
CFL Implies PDA
Informally:
P pushes start variable S on stack
– p.17
CFL Implies PDA
Informally:
keeps making substitutions, storing intermediate
strings
– p.17
CFL Implies PDA
Informally:
strings
when only terminals remain . . .
– p.17
CFL Implies PDA
Informally:
strings
when only terminals remain . . .
tests whether derived string equals input
– p.17
CFL Implies PDA
Where do we keep the intermediate string?
Can’t put it all on the stack
– p.18
CFL Implies PDA
Keep on stack only symbols after first variable
– p.18
CFL Implies PDA
Terminal symbols before first variable matched
immediately to input string symbols.
– p.18
CFL Implies PDA
Terminal symbols before first variable matched
immediately to input string symbols.
011001
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A
1
A
0
$
pop
push
intermediate string: 01A1A0
– p.18
CFL Implies PDA
Informal description:
push S$ on stack
– p.19
CFL Implies PDA
push S$ on stack
if top of stack is a variable A,
non-deterministically select rule and substitute.
– p.19
CFL Implies PDA
push S$ on stack
if top of stack is terminal a read next input and
compare. If they differ, reject on this branch of
the nondeterminism.
– p.19
CFL Implies PDA
push S$ on stack
if top of stack is terminal a read next input and
compare. If they differ, reject on this branch of
the nondeterminism.
if top of stack is $, enter accept state (here we
accept only if input has all been read, as accept
state will have no exits labeled with input
symbols!).
– p.19
CFL Implies PDA
Need shorthand to push entire string onto stack (in
this example the pushed string is w = xyz).
(r, w) ∈ δ(q, a, s)
Easy to do by introducing intermediate states.
a,s
q
q
a,s
r
z
ε,ε
xyz
y
r
ε,ε
x
– p.20
CFL Implies PDA
States of P are
start state, qs
– p.21
CFL Implies PDA
States of P are
start state, qs
accept state, qa
– p.21
CFL Implies PDA
States of P are
start state, qs
accept state, qa
loop state, q`
– p.21
CFL Implies PDA
States of P are
start state, qs
accept state, qa
loop state, q`
E states, shorthand for pushing entire strings
– p.21
Transition Function
Initialize stack – δ(qs , ε, ε) = {q` , S$}
– p.22
Transition Function
Top of stack is variable –
δ(q` , ε, A) = {(q` , w)| where A → w is a rule }
– p.22
Transition Function
Top of stack is terminal – δ(q` , a, a) = {(q` , ε)}
– p.22
Transition Function
Top of stack is terminal – δ(q` , a, a) = {(q` , ε)}
End of Stack – δ(q` , ε, $) = {(qa , ε)}
– p.22
Transition Function
qs
ε,ε S$
ε,A w
a,a ε
ql
ε,$ ε
qa
– p.23
Example
S → aT b|b
T → T a|ε
Initialization:
qs
ε,ε S$
ql
– p.24
Example
S → aT b|b
T → T a|ε
Rules for S
ε,ε T
qs
ε,ε S$
ε,S
b
ε,S
b
ε,ε a
ql
qa
– p.25
Example
S → aT b|b
T → T a|ε
Rules for T
ε,ε T
qs
ε,ε S$
ε,S
ε,T
b
ε
ε,S
b
ε,T
a
ε,ε a
ql
ε,ε a
qa
– p.26
Example
S → aT b|b
T → T a|ε
Rules for terminals
ε,ε T
qs
ε,ε S$
ε,S b
ε,T ε
aa ε
bb ε
ε,S
b
ε,T
a
ε,ε a
ql
ε,ε a
qa
– p.27
Example
S → aT b|b
T → T a|ε
Termination:
ε,ε T
qs
ε,ε S$
ε,S
ε,T
aa
bb
b
ε
ε
ε
ε,$
ε,S
b
ε,T
a
ε,ε a
ql
ε
ε,ε a
qa
– p.28

A Finite Automaton

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