Finite Automata

What can be computed with finite memory?

state

0 1 1 0 · · ·input tape

Finite Automata

What can be computed with finite memory?

state

0 1 1 0 · · ·input tape

Finite Automata

What can be computed with finite memory?

state

0 1 1 0 · · ·input tape

A “Wild” Finite Automaton

The input is the numbers pressed by theuser

The correct code is 1234

Actually the door is opened if{0, . . . ,9}∗1234{0, . . . ,9}∗ is entered

The machine needs to remember if theinput given so far contains thesubsequence 1234

Finite Automata: Informal Definition

q1 q2

1

0

0 1

The machine M:

States: q1 and q2

Start state: q1 (arrow from nowhere)

Accept state: q2 (double circle)

State transitions: arrows

Finite Automata: Informal Definition

q1 q2

1

0

0 1

The machine M:On input string s = s1s2 · · · sn, M operates as follows:

Begins in start state q1 and reads the string s from left torightWhen reading symbol si it follows the transition labeled si

from the current stateAfter reading sn, the last symbol in the string, it

accepts s if it is in an accept staterejects s if it is not in an accept state

Finite Automata: Informal Definition

q1 q2

1

0

0 1

The machine M:On input:011

Finite Automata: Informal Definition

q1 q2

1

0

0 1

The machine M:On input:011 ACCEPT

Finite Automata: Informal Definition

q1 q2

1

0

0 1

The machine M:On input:011 ACCEPT10

Finite Automata: Informal Definition

q1 q2

1

0

0 1

The machine M:On input:011 ACCEPT10 REJECT

Finite Automata: Informal Definition

q1 q2

1

0

0 1

The machine M:On input:011 ACCEPT10 REJECT110

Finite Automata: Informal Definition

q1 q2

1

0

0 1

The machine M:On input:011 ACCEPT10 REJECT110 REJECT

A “Wild” Finite Automaton

The input is the numbers pressed by theuser

The correct code is 1234

Actually the door is opened if{0, . . . ,9}∗1234{0, . . . ,9}∗ is entered

The machine needs to remember if theinput given so far contains thesubsequence 1234

A “Wild” Finite Automaton

The input is the numbers pressed by theuser

The correct code is 1234

Actually the door is opened if{0, . . . ,9}∗1234{0, . . . ,9}∗ is entered

The machine needs to remember if theinput given so far contains thesubsequence 1234

qs q1 q12 q123 q12341 2 3 4

0,2, . . . ,9

1

0,3, . . . ,9

0, . . . ,90,2,4, . . . ,9

0,2,3,5, . . . ,9

1

1

Representation of finite automata

State diagram

q1 q2

1

0

0 1

Transition table0 1

→ q1 q1 q2

F q2 q1 q2

Definition of DFAs

Definition

A deterministic finite automaton (DFA) is a 5-tuple(Q,Σ, δ,q0,F ) where

Q is a finite set called the states

Σ is a finite set called the alphabet

δ : Q × Σ → Q is the transition function

q0 ∈ Q is the start state

F ⊆ Q is the set of accept states

The language recognized by a DFA

Definition

Let M = (Q,Σ, δ,q0,F ) be a DFA and s = s1s2 · · · sn a stringover Σ. M accepts s if there is a sequence of states r0, r1, . . . rn

from Q such that

r0 = q0,

δ(ri , si+1) = ri+1 (i = 0, . . . ,n − 1), and

rn ∈ F

The language recognized by a DFA

Definition

Let M = (Q,Σ, δ,q0,F ) be a DFA and s = s1s2 · · · sn a stringover Σ. M accepts s if there is a sequence of states r0, r1, . . . rn

from Q such that

r0 = q0,

δ(ri , si+1) = ri+1 (i = 0, . . . ,n − 1), and

rn ∈ F

Definition

M recognizes language L if L = {s | M accepts s}

L(M) denotes the language recognized by M

Regular Languages

DefinitionA language is a regularlanguage if some DFArecognizes it

Nondeterminism

The machines we have seen so far have beendeterministic. The next state follows uniquely from thecurrent state and the input symbol

In a nondeterministic machine several possible next statesmay follow from the current state and input symbol. Thesepossibilities can be thought of as being explored in parallel

Understanding the power of nondeterminism is a centraltopic in the theory of computation (and this course)

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

An NFA accepts a string s if, after reading the last symbolof s, at least one of its active states is an accept state

An NFA rejects a string s if, after reading the last symbol ofs, none if its active states is an accept state

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

0101

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

0101

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

0101

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

0101

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

0101

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

0101

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

0101

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

0101

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

0101

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

0101 REJECT

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

110

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

110

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

110

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

110

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

110

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

110

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

110

Example of a nondeterminstic finite automaton (NFA)

p q r1 0,1

0,1

110 ACCEPT

Example of a NFA with ε transitions

p q r1 0,1

0,1

q1 q2

1

0

0 1

Example of a NFA with ε transitions

p q r1 0,1

0,1

q1 q2

s

ε

ε

1

0

0 1

Example of a NFA with ε transitions

p q r1 0,1

0,1

q1 q2

s

ε

ε

1

0

0 1

01

Example of a NFA with ε transitions

p q r1 0,1

0,1

q1 q2

s

ε

ε

1

0

0 1

01

Example of a NFA with ε transitions

p q r1 0,1

0,1

q1 q2

s

ε

ε

1

0

0 1

01

Example of a NFA with ε transitions

p q r1 0,1

0,1

q1 q2

s

ε

ε

1

0

0 1

01 ACCEPT

Definition of NFAs

The power set of Q, written P(Q), is the set of all subsets of Q

Example

If A = {1,2} then P(A) = {∅, {1}, {2}, {1,2}}

Definition of NFAs

Definition

A nondeterministic finite automaton (NFA) is a 5-tuple(Q,Σ, δ,q0,F ) where

Q is a finite set called the states

Σ is a finite set called the alphabet

δ : Q × (Σ ∪ {ε}) → P(Q) is the transition function

q0 ∈ Q is the start state

F ⊆ Q is the set of accept states

The language recognized by a NFA

Definition

Let N = (Q,Σ, δ,q0,F ) be a NFA and s = s1s2 · · · sm a stringover Σ. N accepts s if there is a sequence of states r0, r1, . . . rn

from Q and s can be written as s = s1, s2 . . . , sn where eachsi ∈ Σ ∪ {ε} such that

r0 = q0,

ri+1 ∈ δ(ri , si+1) (i = 0, . . . ,n − 1), and

rn ∈ F

Equivalence of DFAs and NFAs

Definition

Two machines are equivalent if they recognize the samelanguage

Equivalence of DFAs and NFAs

TheoremEvery NFA has an equivalent DFA

Equivalence of DFAs and NFAs

TheoremEvery NFA has an equivalent DFA

Proof idea.

Given a NFA we need to construct a DFA that simulate the NFA

The DFA need to keep track of the set of active states ofthe NFA at each step

If k is the number of states of the NFA, then the DFA mightneed up to 2k states (one for each subset of states of theNFA)

So, the states of the DFA should be P(Q) where Q is thestates of the NFA

Equivalence of DFAs and NFAs

TheoremEvery NFA has an equivalent DFA

Notation: Let E(R) be the set of states that can be reachedfrom R using 0 or more ε transitions.

Equivalence of DFAs and NFAs

Theorem

Every NFA has an equivalent DFA

Notation: Let E(R) be the set of states that can be reachedfrom R using 0 or more ε transitions.

Proof.

The subset construction: Given a NFA N = (Q,Σ, δ,q0,F ) weconstruct an equivalent DFA M = (Q′,Σ, δ′,q′

0,F′) where

Q′ = P(Q)

q′

0 = E({q0})

F ′ = {R ∈ Q′ | R ∩ F 6= ∅}

Equivalence of DFAs and NFAs

TheoremEvery NFA has an equivalent DFA

Notation: Let E(R) be the set of states that can be reachedfrom R using 0 or more ε transitions.

Proof.

The subset construction: Given a NFA N = (Q,Σ, δ,q0,F ) weconstruct an equivalent DFA M = (Q′,Σ, δ′,q′

0,F′) where

For R ∈ Q′ and a ∈ Σ,δ′(R,a) = {q ∈ Q | q ∈ E(δ(r ,a)) for some r ∈ R}

δ′(R,a) is the set of all states that can be reached (in theNFA) by first following a transition labeled a from a state inR and then following 0 or more ε transitions

Equivalence of DFAs and NFAs: Example

p q r1 0,1

0,1

q1 q2

s

ε

ε

1

0

0 1

0 1→ {s,q1,p} {q1,p} {q2,p,q}

Equivalence of DFAs and NFAs: Example

p q r1 0,1

0,1

q1 q2

s

ε

ε

1

0

0 1

0 1→ {s,q1,p} {q1,p} {q2,p,q}{q1,p} {q1,p} {q2,p,q}

Equivalence of DFAs and NFAs: Example

p q r1 0,1

0,1

q1 q2

s

ε

ε

1

0

0 1

0 1→ {s,q1,p} {q1,p} {q2,p,q}{q1,p} {q1,p} {q2,p,q}F {q2,p,q} {q1,p, r} {q2,p,q, r}

Equivalence of DFAs and NFAs: Example

p q r1 0,1

0,1

q1 q2

s

ε

ε

1

0

0 1

0 1→ {s,q1,p} {q1,p} {q2,p,q}{q1,p} {q1,p} {q2,p,q}F {q2,p,q} {q1,p, r} {q2,p,q, r}F {q1,p, r} {q1,p} {q2,p,q}

Equivalence of DFAs and NFAs: Example

p q r1 0,1

0,1

q1 q2

s

ε

ε

1

0

0 1

0 1→ {s,q1,p} {q1,p} {q2,p,q}{q1,p} {q1,p} {q2,p,q}F {q2,p,q} {q1,p, r} {q2,p,q, r}F {q1,p, r} {q1,p} {q2,p,q}F {q2,p,q, r} {q1,p, r} {q2,p,q, r}

Equivalence of DFAs and NFAs: Example

p q r1 0,1

0,1

q1 q2

s

ε

ε

1

0

0 1

{q1,p} {q2,p,q, r}0

{q2,p,q} {q1,p, r}

{s,q1,q}

0

1 1

1

0

0

1

0

1

Equivalence of DFAs and NFAs

TheoremEvery NFA has an equivalent DFA

Corollary

Every language recognized by a NFA can be recognized by aDFA