Finite Automata Question: What is a computer? real computers too complex for any theory Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.1
Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Finite AutomataQuestion: What is a computer?
real computers too complex for any theory
need manageable mathematical abstraction
idealized models: accurate in some ways, but notin all details
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.1
Finite AutomataQuestion: What is a computer?
real computers too complex for any theory
need manageable mathematical abstraction
idealized models: accurate in some ways, but notin all details
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.1
Finite AutomataQuestion: What is a computer?
real computers too complex for any theory
need manageable mathematical abstraction
idealized models: accurate in some ways, but notin all details
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.1
Finite Automata – Important ideas
formal definition of finite automata
deterministic vs. non-deterministic finiteautomata
regular languages
operations on regular languages
regular expressions
pumping lemma
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
Finite Automata – Important ideas
formal definition of finite automata
deterministic vs. non-deterministic finiteautomata
regular languages
operations on regular languages
regular expressions
pumping lemma
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
Finite Automata – Important ideas
formal definition of finite automata
deterministic vs. non-deterministic finiteautomata
regular languages
operations on regular languages
regular expressions
pumping lemma
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
Finite Automata – Important ideas
formal definition of finite automata
deterministic vs. non-deterministic finiteautomata
regular languages
operations on regular languages
regular expressions
pumping lemma
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
Finite Automata – Important ideas
formal definition of finite automata
deterministic vs. non-deterministic finiteautomata
regular languages
operations on regular languages
regular expressions
pumping lemma
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
Finite Automata – Important ideas
formal definition of finite automata
deterministic vs. non-deterministic finiteautomata
regular languages
operations on regular languages
regular expressions
pumping lemma
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.2
Example: An Automatic Door
frontpad
rearpad
door
open when person approaches
hold open until person clears
don’t open when someone standing behind door
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.3
Example: An Automatic Door
frontpad
rearpad
door
open when person approaches
hold open until person clears
don’t open when someone standing behind door
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.3
Example: An Automatic Door
frontpad
rearpad
door
open when person approaches
hold open until person clears
don’t open when someone standing behind door
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.3
The Automatic Door as DFA
closed open
FRONT
NEITHER
FRONTREARBOTH
REARBOTH
NEITHER
States:
OPENCLOSED
Sensor:FRONT: someone on rear padREAR: someone on rear padBOTH: someone on both padsNEITHER no one on either pad.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
The Automatic Door as DFA
closed open
FRONT
NEITHER
FRONTREARBOTH
REARBOTH
NEITHER
States:OPEN
CLOSED
Sensor:FRONT: someone on rear padREAR: someone on rear padBOTH: someone on both padsNEITHER no one on either pad.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
The Automatic Door as DFA
closed open
FRONT
NEITHER
FRONTREARBOTH
REARBOTH
NEITHER
States:OPENCLOSED
Sensor:FRONT: someone on rear padREAR: someone on rear padBOTH: someone on both padsNEITHER no one on either pad.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
The Automatic Door as DFA
closed open
FRONT
NEITHER
FRONTREARBOTH
REARBOTH
NEITHER
States:OPENCLOSED
Sensor:
FRONT: someone on rear padREAR: someone on rear padBOTH: someone on both padsNEITHER no one on either pad.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
The Automatic Door as DFA
closed open
FRONT
NEITHER
FRONTREARBOTH
REARBOTH
NEITHER
States:OPENCLOSED
Sensor:FRONT: someone on rear pad
REAR: someone on rear padBOTH: someone on both padsNEITHER no one on either pad.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
The Automatic Door as DFA
closed open
FRONT
NEITHER
FRONTREARBOTH
REARBOTH
NEITHER
States:OPENCLOSED
Sensor:FRONT: someone on rear padREAR: someone on rear pad
BOTH: someone on both padsNEITHER no one on either pad.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
The Automatic Door as DFA
closed open
FRONT
NEITHER
FRONTREARBOTH
REARBOTH
NEITHER
States:OPENCLOSED
Sensor:FRONT: someone on rear padREAR: someone on rear padBOTH: someone on both pads
NEITHER no one on either pad.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
The Automatic Door as DFA
closed open
FRONT
NEITHER
FRONTREARBOTH
REARBOTH
NEITHER
States:OPENCLOSED
Sensor:FRONT: someone on rear padREAR: someone on rear padBOTH: someone on both padsNEITHER no one on either pad.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.4
The Automatic Door as DFA
DFA is Deterministic Finite Automata
closed open
FRONT
NEITHER
FRONTREARBOTH
REARBOTH
NEITHER
neither front rear bothclosed closed open closed closedopen closed open open open
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.5
The Automatic Door as DFA
DFA is Deterministic Finite Automata
closed open
FRONT
NEITHER
FRONTREARBOTH
REARBOTH
NEITHER
neither front rear bothclosed closed open closed closedopen closed open open open
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.5
The Automatic Door as DFA
DFA is Deterministic Finite Automata
closed open
FRONT
NEITHER
FRONTREARBOTH
REARBOTH
NEITHER
neither front rear bothclosed closed open closed closedopen closed open open open
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.5
DFA: Informal Definition
q1
0
q2 3q
11 0
0,1
The machine � :
states: � ���
��� , and ��� .
start state: (arrow from “outside”).
accept state: (double circle).
state transitions: arrows.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.6
DFA: Informal Definition
q1
0
q2 3q
11 0
0,1
The machine � :
states: � ���
��� , and ��� .
start state: � � (arrow from “outside”).
accept state: (double circle).
state transitions: arrows.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.6
DFA: Informal Definition
q1
0
q2 3q
11 0
0,1
The machine � :
states: � ���
��� , and ��� .
start state: � � (arrow from “outside”).
accept state: �� (double circle).
state transitions: arrows.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.6
DFA: Informal Definition
q1
0
q2 3q
11 0
0,1
The machine � :
states: � ���
��� , and ��� .
start state: � � (arrow from “outside”).
accept state: �� (double circle).
state transitions: arrows.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.6
DFA: Informal Definition (cont.)
q1
0
q2 3q
11 0
0,1
On an input string
DFA begins in start stateafter reading each symbol, DFA makesstate transition with matching label.
After reading last symbol, DFA produces output:accept if DFA is an accepting state.reject otherwise.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
DFA: Informal Definition (cont.)
q1
0
q2 3q
11 0
0,1
On an input stringDFA begins in start state � �
after reading each symbol, DFA makesstate transition with matching label.
After reading last symbol, DFA produces output:accept if DFA is an accepting state.reject otherwise.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
DFA: Informal Definition (cont.)
q1
0
q2 3q
11 0
0,1
On an input stringDFA begins in start state � �
after reading each symbol, DFA makesstate transition with matching label.
After reading last symbol, DFA produces output:accept if DFA is an accepting state.reject otherwise.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
DFA: Informal Definition (cont.)
q1
0
q2 3q
11 0
0,1
On an input stringDFA begins in start state � �
after reading each symbol, DFA makesstate transition with matching label.
After reading last symbol, DFA produces output:
accept if DFA is an accepting state.reject otherwise.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
DFA: Informal Definition (cont.)
q1
0
q2 3q
11 0
0,1
On an input stringDFA begins in start state � �
after reading each symbol, DFA makesstate transition with matching label.
After reading last symbol, DFA produces output:accept if DFA is an accepting state.
reject otherwise.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
DFA: Informal Definition (cont.)
q1
0
q2 3q
11 0
0,1
On an input stringDFA begins in start state � �
after reading each symbol, DFA makesstate transition with matching label.
After reading last symbol, DFA produces output:accept if DFA is an accepting state.reject otherwise.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.7
Informal Definition - Example
q1
0
q2 3q
11 0
0,1
What happens on input strings
� � � �
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.8
Informal Definition - Example
q1
0
q2 3q
11 0
0,1
What happens on input strings
� � � �
� � � �
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.8
Informal Definition - Example
q1
0
q2 3q
11 0
0,1
What happens on input strings
� � � �
� � � �
� � � � �
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.8
Informal Definition
q1
0
q2 3q
11 0
0,1
This DFA accepts
all input strings that end with a 1
all input strings that contain at least one 1, andend with an even number of 0’s
no other strings
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.9
Languages and Alphabets
An alphabet is a finite set of letters.
� ����
��
��� � � �
� �
– the English alphabet.
� ��� �
� � � � �
� �
– the Greek alphabet.
� � ��
� �
– the binary alphabet.
� � ��
��� � � �
� �
– the digital alphabet.
The collection of all strings over is denoted by .
For the binary alphabet, , , , ,
are all members of .
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.10
Languages and Alphabets
An alphabet is a finite set of letters.
� ����
��
��� � � �
� �
– the English alphabet.
� ��� �
� � � � �
� �
– the Greek alphabet.
� � ��
� �
– the binary alphabet.
� � ��
��� � � �
� �
– the digital alphabet.
The collection of all strings over is denoted by
�
.
For the binary alphabet, , , , ,
are all members of .
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.10
Languages and Alphabets
An alphabet is a finite set of letters.
� ����
��
��� � � �
� �
– the English alphabet.
� ��� �
� � � � �
� �
– the Greek alphabet.
� � ��
� �
– the binary alphabet.
� � ��
��� � � �
� �
– the digital alphabet.
The collection of all strings over is denoted by
�
.
For the binary alphabet, �,
�
,
�
,
� � � � � � � � �
,
� � � � � � � � � �
are all members of
�
.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.10
Languages and Examples
A language over is a subset
� �
. For example
Modern English.
Ancient Greek.
All prime numbers, writen using digits.
has at most seventeen 0’s
has an equal number of 0’s and 1’s
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
Languages and Examples
A language over is a subset
� �
. For example
Modern English.
Ancient Greek.
All prime numbers, writen using digits.
has at most seventeen 0’s
has an equal number of 0’s and 1’s
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
Languages and Examples
A language over is a subset
� �
. For example
Modern English.
Ancient Greek.
All prime numbers, writen using digits.
has at most seventeen 0’s
has an equal number of 0’s and 1’s
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
Languages and Examples
A language over is a subset
� �
. For example
Modern English.
Ancient Greek.
All prime numbers, writen using digits.
� � � � � has at most seventeen 0’s
�
has an equal number of 0’s and 1’s
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
Languages and Examples
A language over is a subset
� �
. For example
Modern English.
Ancient Greek.
All prime numbers, writen using digits.
� � � � � has at most seventeen 0’s
�
� � � � � � �� � �
has an equal number of 0’s and 1’s
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
Languages and Examples
A language over is a subset
� �
. For example
Modern English.
Ancient Greek.
All prime numbers, writen using digits.
� � � � � has at most seventeen 0’s
�
� � � � � � �� � �
� � � � � has an equal number of 0’s and 1’s
�
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.11
Languages and DFA
Definition:
� � �
, the language of a DFA , is theset of strings
�
that accepts,
� � � � �.
Note that
may accept many strings, but
accepts only one language
What language does accept if it accepts no strings?
A language is called regular if some deterministic finite
automaton accepts it.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.12
Languages and DFA
Definition:
� � �
, the language of a DFA , is theset of strings
�
that accepts,
� � � � �.
Note that
may accept many strings, but
accepts only one language
What language does accept if it accepts no strings?
A language is called regular if some deterministic finite
automaton accepts it.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.12
Languages and DFA
Definition:
� � �
, the language of a DFA , is theset of strings
�
that accepts,
� � � � �.
Note that
may accept many strings, but
accepts only one language
What language does accept if it accepts no strings?
A language is called regular if some deterministic finite
automaton accepts it.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.12
Languages and DFA
Definition:
� � �
, the language of a DFA , is theset of strings
�
that accepts,
� � � � �.
Note that
may accept many strings, but
accepts only one language
What language does accept if it accepts no strings?
A language is called regular if some deterministic finite
automaton accepts it.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.12
Formal DefinitionsA deterministic finite automaton (DFA) is a 5-tuple
�
� �
��
��� �
�
, where
is a finite set called the states,
is a finite set called the alphabet,
��� � is the transition function,
�� � is the start state, and
is the set of accept states.
Slides modified by Benny Chor, based on original slides by Maurice Herlihy, Brown University. – p.13
Related Documents
