FIT2014 Theory of Computation Lecture 8 Kleene’s Theorem. I. Regexp -3mu NFA -3mu FA
Monash University
Faculty of Information Technology
FIT2014 Theory of Computation
Lecture 8
Kleene’s Theorem. I.
Regexp −→ NFA −→ FA
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Overview
I Questions
I Kleene’s Theorem
I Convert Regular Expressions to NFA
I Convert NFA to FA
I Next lecture:
Convert FA to Regular Expression
Kleene (1909–1994)
https://mathshistory.st-andrews.
ac.uk/Biographies/Kleene/
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https://mathshistory.st-andrews.ac.uk/Biographies/Kleene/
https://mathshistory.st-andrews.ac.uk/Biographies/Kleene/
Questions
I Can every language which is represented by a regular expression
be described by a finite automaton?
I Can every language which is described by a finite automaton
be represented by a regular expression?
I Can every language be represented by a regular expression or a finite automaton?
{ all languages } representable by
regexp
?
recognised by
FA
?
?
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Kleene’s Theorem
Theorem.
Any language which can be defined by
I Regular Expressions
I Finite Automata
I Nondeterministic Finite Automata (NFA)
I Generalized Nondeterministic Finite Automata (GNFA)
can be defined by any of the other methods.
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Kleene’s Theorem
Regular Expression NFA
Finite AutomatonGNFA
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Converting Regular Expression to NFA
Start with:
Regular Expression
Apply the following rules until all edges are labelled with a letter or ε:
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Converting Regular Expression to NFA
∅ (R)
R
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Converting Regular Expression to NFA
RS
R S
R ∪ S
R
S
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Converting Regular Expression to NFA
R∗
ε ε
R
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Converting Regular Expression to NFA
The two ε transitions
are necessary,
in general.
Here, you can’t
match PR∗Q
or SR∗T ,
in general.
So the R loop
cannot be at
left node or
right node.
P
Q
S
T
P
Q
S
T
R∗
ε ε
R
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Converting Regular Expression to NFA. Example: a((ab)∗ ∪ (ba))b∗
a((ab)∗ ∪ (ba))b∗
a (ab)
∗ ∪ (ba) b∗
a
(ab)∗
ba
b∗
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a
(ab)∗
b a
b∗
a
ε ε
ab
b a
b∗
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a
ε ε
ab
b a
ε ε
b
a
ε ε
b a
ε ε
ba b
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Converting a NFA to a FA
Complexity?
How reversible is this construction?
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Kleene’s Theorem
Regular Expression NFA
Finite AutomatonGNFA
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Converting a NFA to a FA
In a FA:
I Any string w traces a unique path, starting from the Start State and ending at
some unique state, which we’ll call endState(w).
I The string w is accepted if endState(w) is a Final State, otherwise it is rejected.
I endState(ε) = Start State.
In a NFA:
I Any string w traces a set of paths, starting from the Start State and ending at
some set of states, which we’ll call endStates(w).
I The set might have zero, one or more members.
I The string w is accepted if endStates(w) contains a Final State, otherwise it is
rejected.
I endStates(ε) = { Start State } if there are no ε transitions.
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Converting a NFA to a FA
1 2 3
a,b
b a,b
endStates(ab) = {1, 2}
endStates(aba) = {1, 3}
In general, if w is a string and x is a single letter, then
endStates(wx) = {q : for some state p ∈ endStates(w), there is a transition p x−→ q}
. . . provided there are no empty transitions.
This suggests part of a method for constructing endStates(w) for all strings w .
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Converting a NFA to a FA
Idea:
sets of states in the NFA −→ states in the FA.
Informally (and assuming no empty transitions, for the time being):
Start with the one-element set { Start State }.
I This is endStates(ε).
I It’s the set of NFA-states we can possibly be in at the very start.
Construct endStates(a), the set of all states we could then get to by reading a single a.
Construct endStates(b), the set of all states we could then get to by reading a single b.
For each set of states, X , that we construct:
I find the set of states we can get to from X , by reading a single a.
I find the set of states we can get to from X , by reading a single b.
Keep doing this, until we no longer get any new sets of states.
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Converting a NFA to a FA
1 2 3
a,b
b a,b
state a b
Start {1}
state a b
Start {1} {1} {1,2}
{1,2}
state a b
Start {1} {1} {1,2}
{1,2} {1,3} {1,2,3}
{1,3}
{1,2,3}
· · ·
state a b
Start {1} {1} {1,2}
{1,2} {1,3} {1,2,3}
{1,3} {1} {1,2}
{1,2,3}
· · · · · ·
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Converting a NFA to a FA
1 2 3
a,b
b a,b
· · ·
state a b
Start {1} {1} {1,2}
{1,2} {1,3} {1,2,3}
{1,3} {1} {1,2}
{1,2,3} {1,3} {1,2,3}
state a b
Start {1} {1} {1,2}
{1,2} {1,3} {1,2,3}
Final {1,3} {1} {1,2}
Final {1,2,3} {1,3} {1,2,3}
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Converting a NFA to a FA
1 2 3
a,b
b a,b
state a b
Start {1} {1} {1,2}
{1,2} {1,3} {1,2,3}
Final {1,3} {1} {1,2}
Final {1,2,3} {1,3} {1,2,3}
{1}
{1,2}
{1,3}
{1,2,3}
a
b
a
b
a
b
a
b
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Algorithm: Conversion of NFA without empty transitions to FA
Input: a NFA
NextSetOfStatesOfNFA := { Start State of NFA }.
Create new incomplete row in FA table, for Start State called NextSetOfStatesOfNFA.
while the FA table still has at least one incomplete row do
CurrentStateInFA := the state for the first incomplete row of the FA.
for each letter x in the alphabet do
NextSetOfStatesOfNFA :=
{q : for some NFA-state p in CurrentStateInFA, ∃ transition p x−→ q}
Write NextSetOfStatesOfNFA in table entry for row CurrentStateInFA, column x .
if NextSetOfStatesOfNFA is new then
Create new incomplete row in table, using set NextSetOfStatesOfNFA as state.
Any FA state which (as a set) contains an NFA Final State is labelled Final.
Output: the FA 22 / 28
Converting a NFA to a FA
Now suppose that the NFA might have empty transitions, q1
ε−→ q2.
These allow change of state without reading any letter of the input string.
Every time we include a new state q in NextSetOfStatesOfNFA, we also need to
include any state we can reach from it along empty transitions.
Look at all paths from q that just use ε transitions . . .
q
ε−→ q1
ε−→ q2
ε−→ · · · · · · ε−→ qi
. . . and include all states on such paths.
Modify earlier algorithm, for constructing the sets of NFA states, to take account of
empty transitions.
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Algorithm: Conversion of NFA to FA
Input: a NFA
NextSetOfStatesOfNFA := { Start State of NFA }.
for each q ∈ NextSetOfStatesOfNFA do
Add, to NextSetOfStatesOfNFA, all states reachable from q along ε-transitions.
Create new incomplete row in FA table, for Start State called NextSetOfStatesOfNFA.
while the FA table still has at least one incomplete row do
CurrentStateInFA := the state for the first incomplete row of the FA.
for each letter x in the alphabet do
NextSetOfStatesOfNFA :=
{q : for some NFA-state p in CurrentStateInFA, ∃ transition p x−→ q}
for each q ∈ NextSetOfStatesOfNFA do
Add, to NextSetOfStatesOfNFA, all states reachable from q along ε-transitions.
Write NextSetOfStatesOfNFA in table entry for row CurrentStateInFA, column x .
if NextSetOfStatesOfNFA is new then
Create new incomplete row in table, using set NextSetOfStatesOfNFA as state.
Any FA state which (as a set) contains an NFA Final State is labelled Final.
Output: the FA 24 / 28
Converting a NFA to a FA
1 2 3
a
ε
b
ε
state a b
Start {1,2,3}
state a b
Start {1,2,3} {1,2,3} {2,3}
{2,3}
state a b
Start {1,2,3} {1,2,3} {2,3}
{2,3} ∅ {2,3}
∅
state a b
Start {1,2,3} {1,2,3} {2,3}
{2,3} ∅ {2,3}
∅ ∅ ∅
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Converting a NFA to a FA
1 2 3
a
ε
b
ε
state a b
Start {1,2,3} {1,2,3} {2,3}
{2,3} ∅ {2,3}
∅ ∅ ∅ {1,2,3} {2,3} ∅
a
b
b
a
a,b
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Converting a NFA to a FA
Complexity?
I Think about how many states the FA may have,
as a function of the number of states in the NFA.
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Revision
Today:
I Understand Kleene’s Theorem
I Be able to convert Regular Expression −→ NFA
I Be able to convert NFA −→ Finite Automaton
Next lecture:
I Be able to convert FA −→ Regular Expression
Reading:
Sipser, Ch 1, especially pp. 54–58, 66–69.
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