Lecture Three:Finite Automata
Finite Automata, Lecture 3, slide 1
Amjad Ali
Finite Automata, Lecture 3, slide 2
DFA: Deterministic Finite Automaton An informal definition (formal version later):
A diagram with a finite number of states represented by circles
An arrow points to one of the states, the unique start state
Double circles mark any number of the states as accepting states
For every state, for every symbol in , there is exactly one arrow labeled with that symbol going to another state (or back to the same state)
Finite Automata, Lecture 3, slide 3
DFAs Define Languages
Given any string over , a DFA can read the string and follow its state-to-state transitions
At the end of the string, if it is in an accepting state, we say it accepts the string
Otherwise it rejects The language defined by a DFA is the
set of strings in * that it accepts
Finite Automata, Lecture 3, slide 4
The 5-Tuple
Q is the set of states Drawn as circles in the diagram We often refer to individual states as qi The definition requires at least one: q0, the start state
is the alphabet (that is, a finite set of symbols)
A DFA M is a 5-tuple M = (Q, , , q0, F), :where Q is the finite set of states is the alphabet (that is, a finite set of symbols) (Q Q) is the transition functionq0 Q is the start stateF Q is the set of accepting states
Finite Automata, Lecture 3, slide 5
The 5-Tuple
F is the set of all those in Q that are accepting states Drawn as double circles in the diagram
is the transition function A function (q,a) that takes the current state q and next
input symbol a, and returns the next state Represents the same information as the arrows in the
diagram
A DFA M is a 5-tuple M = (Q, , , q0, F), where:Q is the finite set of states is the alphabet (that is, a finite set of symbols) (Q Q) is the transition functionq0 Q is the start stateF Q is the set of accepting states
Finite Automata, Lecture 3, slide 6
Example: This DFA defines {xa | x {a,b}*} Formally, M = (Q, , , q0, F), where
Q = {q0,q1} = {a,b} F = {q1} Start sate q0 (q0,a) = q1, (q0,b) = q0, (q1,a) = q1, (q1,b) = q0
We could just say M = ({q0,q1}, {a,b}, , q0, {q1})
q0 q1
b
a
a
b
q0q1
q1 q0q1 q0
a b
Finite Automata, Lecture 3, slide 7
A regular language is one that is L(M) for some DFA M.
For any DFA M = (Q, , , q0, F), L(M) denotes the language accepted by M, which is L(M) = {x * | *(q0, x) F}.
Regular Languages
To show that a language is regular, give a DFA for it. We'll see additional ways later
To show that a language is not regular is much harder; we'll see how later
Finite Automata, Lecture 3, slide 8
Example #1:
Q = {q0, q1, q2}Σ = {a, b, c}Start state is q0
F = {q2}a b c
q0 q0 q0 q1
q1 q1 q1 q2
q2 q2 q2 q2
Since δ is a function, at each step M has exactly one option.
It follows that for a given string, there is exactly one computation.
q1q0 q2
a
b
a
b
c c
a/b/c
Finite Automata, Lecture 3, slide 9
Example #2:
1 0 0 1 1 q0 q0 q1 q0 q0 q0
One state is final/accepting, all others are rejecting. The above DFA accepts those strings that contain an
even number of 0’s
q0q1
0
0
1
1
Finite Automata, Lecture 3, slide 10
Example #3:
a c c c baccepted
q0 q0 q1 q2 q2 q2
a a crejected
q0 q0 q0 q1
Accepts those strings that contain at least two c’s
q1q0q2
a
b
a
b
c c
a/b/c
Finite Automata, Lecture 3, slide 11
Example #4:
Give a DFA M such that:
L(M) = {x | x is a string of 0’s and 1’s and |x| >= 2}
q1q0q2
0/1
0/1
0/1
Finite Automata, Lecture 3, slide 12
Example #5:
Give a DFA M such that:
L(M) = {x | x is a string of (zero or more) a’s and b’s such that x does not contain the substring aa}
q2q0
a
a/b
aq1
b
b
Finite Automata, Lecture 3, slide 13
Example #6:
Give a DFA M such that:
L(M) = {x | x is a string of a’s, b’s and c’s such that x
contains the substring aba}
q2q0
a
a/b
bq1
b a
b
q3
a
Finite Automata, Lecture 3, slide 14
Example #7:
Give a DFA M such that:
L(M) = {x | x is a string of a’s and b’s such that x contains both aa and bb}
q0
b
q7
q5q4 q6b
b
b
a
q2q1 q3a
a
a
b
a/bb
a
a
a b
Finite Automata, Lecture 3, slide 15
Example #8:
Let Σ = {0, 1}. Give DFAs for {}, {ε}, Σ*, and Σ+.
For {}: For {ε}:
For Σ*: For Σ+:
0/1
q0
0/1
q0
q1q0
0/1
0/1
0/1q0 q1
0/1
Finite Automata, Lecture 3, slide 16
Definitions for DFAs Let M = (Q, Σ, δ,q0,F) be a DFA and let w be in Σ*. Then
w is accepted by M iff δ^(q0,w) = p for some state p in F.
Let M = (Q, Σ, δ,q0,F) be a DFA. Then the language accepted by M is the set:
L(M) = {w | w is in Σ* and δ^(q0,w) is in F}
Another equivalent definition:L(M) = {w | w is in Σ* and w is accepted by
M} Let L be a language. Then L is a regular language iff
there exists a DFA M such that L = L(M). Let M1 = (Q1, Σ1, δ1, q0, F1) and M2 = (Q2, Σ2, δ2, p0, F2) be
DFAs. Then M1 and M2 are equivalent iff L(M1) = L(M2).
Finite Automata, Lecture 3, slide 17
Notes:
A DFA M = (Q, Σ, δ,q0,F) partitions the set Σ* into two sets: L(M) andΣ* - L(M).
If L = L(M) then L is a subset of L(M) and L(M) is a subset of L.
Similarly, if L(M1) = L(M2) then L(M1) is a subset of L(M2) and L(M2) is a subset of L(M1).
Some languages are regular, others are not. For example, if
L1 = {x | x is a string of 0's and 1's containing an even number of 1's} and
L2 = {x | x = 0n1n for some n >= 0} then L1 is regular but L2 is not.
Finite Automata, Lecture 3, slide 18
Questions: How do we determine whether or not a
given language is regular? How could a program “simulate” a DFA?
Finite Automata, Lecture 3, slide 19
The * Function The function gives 1-symbol moves We'll define * so it gives whole-string results (by
applying zero or more moves) A recursive definition:
*(q,) = q For all w in Σ* and a in Σ
*(q,wa) = (*(q,w),a) That is:
For the empty string, no moves For any string wa (w is any string and a is any final
symbol) first make the moves on w, then one final move on a
Finite Automata, Lecture 3, slide 20
M Accepts x *(q,w) is the state of M ends up in,
starting from state q and reading all of string w
So *(q0,w) tells us whether M accepts w :
A string w * is accepted by a DFA M = (Q, , , q0, F) if and only if *(q0, w) F.
Finite Automata, Lecture 3, slide 21
Example #1:
What is δ*(q0, 011)? Informally, it is the state entered by M after processing 011 having started in state q0.
Formally: δ*(q0, 011) = δ (δ*(q0,01), 1) by rule #2
= δ (δ ( δ*(q0,0), 1), 1) by rule #2
= δ (δ (δ (δ*(q0, λ), 0), 1), 1) by rule #2= δ (δ (δ(q0,0), 1), 1) by rule #1= δ (δ (q1, 1), 1) by definition
of δ= δ (q1, 1) by definition
of δ= q1 by definition
of δ
Is 011 accepted? No, since δ*(q0, 011) = q1 is not a final state.
q0q1
0
0
1
1
Finite Automata, Lecture 3, slide 22
Example #2:
What is δ(q0, 011)? Informally, it is the state entered by M after processing 011 having started in state q0.
Formally: δ(q0, 011) = δ (δ*(q0,01), 1) by rule #2
= δ (δ (δ*(q0,0), 1), 1) by rule #2
= δ (δ(δ(δ*(q0 , λ), 0),1), 1) by rule #2= δ (δ(δ(q0 , 0), 1),1) by definition
of δ= δ (δ(q1 , 1),1) by definition
of δ= δ (q1 , 1) by definition
of δ = q1 Is 011 accepted? No, since δ(q0, 011) = q1 is not a final
state.
q1q0 q2
1 1
00
1
0
Finite Automata, Lecture 3, slide 23
Example #3:
What is δ(q0, 100)? δ(q0, 100) = δ (δ*(q0,10), 0) by rule #2
= δ (δ (δ*(q0,1), 0), 0) by rule #2
= δ (δ(δ(δ*(q0 , λ), 1),0), 0) by rule #2= δ (δ(δ(q0 , 1), 0),0) by definition
of δ= δ (δ(q0 , 0),0) by definition
of δ= δ (q1 , 0) by definition
of δ = q2
Is 100 accepted? Yes, since δ(q0, 100) = q2 is a final state.
0
q1q0 q2
1 1
0
1
0
Finite Automata, Lecture 3, slide 24
Formal Definition of ComputationLet M = (Q, , , q0, F) and w = w1w2w3w4…wn be a string where each wi ∑. Then M accepts w if a sequence of states r0, r1, …, rn in Q exists with three conditions:
1. r0 = q0
2. (ri, wi+1) = ri+1 for i = 0, …, n-13. rn F
Formal Definition of DFAA DFA M is a 5-tuple M = (Q, , , q0, F) :
where Q is the finite set of states is the alphabet (that is, a finite set of symbols) (Q Q) is the transition functionq0 Q is the start stateF Q is the set of accepting states
Finite Automata, Lecture 3, slide 25
Example #1:
Q = {q0, q1, q2}Σ = {a, b}Start state is q0
F = {q2}
δ : a b q0 q1 q0
q1 q2 q0
q2 q2 q2
q1q0 q2
b
b
a a
a, b
Finite Automata, Lecture 3, slide 26
w = a b a b a b a a b w = w1 w2 w3 w4 w5 w6 w7 w8 w9 , |w| = 9 , n = 9 r0 = q0
(ri , wi+1) = ri+1 for i = 0, …, 8
a b a b a b a a b q0 q1 q0 q1 q0 q1 q0 q1 q2 q2
r0 r1 r2 r3 r4 r5 r6 r7 r8 r9
1. (r0 , w1) = r1 2. (r1 , w2) = r2 3. (r2 , w3) = r3 4. (r3 , w4) = r4
(q0 , a) = q1 (q1 , b) = q0 (q0 , a) = q1 (q1 , b) = q0
5. (r4 , w5) = r5 6. (r5 , w6) = r6 7. (r6 , w7) = r7 8. (r7 , w8) = r8
(q0 , a) = q1 (q1 , b) = q0 (q0 , a) = q1 (q1 , a)= q2
9. (r8 , w9) = r9
(q2 , b) = q2 , As r9=q2 and q2 F, then String is accepted
q1q0 q2
b
b
a a
a, b
DFA Applications
Finite Automata, Lecture 3, slide 27
We have seen how DFAs can be used to define formal languages. In addition to this formal use, DFAs have practical applications. DFA-based pieces of code lie at the heart of many commonly used computer programs.
Finite Automata, Lecture 3, slide 28
DFA Applications Programming language processing
Scanning phase: dividing source file into "tokens" (keywords, identifiers, constants, etc.), skipping whitespace and comments
Command language processing Typed command languages often
require the same kind of treatment Text pattern matching
Unix tools like awk, egrep, and sed, mail systems like ProcMail, database systems like MySQL, and many othersFinite Automata, Lecture 3, slide 29
More DFA Applications Signal processing
Speech processing and other signal processing systems use finite state models to transform the incoming signal
Controllers for finite-state systems Hardware and software A wide range of applications, from
industrial processes to video gamesFinite Automata, Lecture 3, slide 30