Nondeterministic Finite Automata

Formal Language, chapter 5, slide 1 Copyright © 2007 by Adam Webber
Chapter Five:
Nondeterministic Finite Automata
AL-ofairi Adel Ahmed

A DFA has exactly one transition from every state on every
symbol in the alphabet. By relaxing this requirement we get
a related but more flexible kind of automaton: the
nondeterministic finite automaton or NFA.
NFAs are a bit harder to think about than DFAs, because
they do not appear to define simple computational processes.
They may seem at first to be unnatural, like puzzles invented
by professors for the torment of students. But have patience!
NFAs and other kinds of nondeterministic automata arise
naturally in many ways, as you will see later in this book,
and they too have a variety of practical applications.

Outline
• 5.1 Relaxing a Requirement
• 5.2 Spontaneous Transitions
• 5.3 Nondeterminism
• 5.4 The 5-Tuple for an NFA
• 5.5 The Language Accepted by an NFA

Not A DFA
• Does not have exactly one transition from
every state on every symbol:
– Two transitions from q0 on a
– No transition from q0 (on either a or b)
• Though not a DFA, this can be taken as
defining a language, in a slightly different way
q1
a,b
q0
a

Possible Sequences of Moves
• We'll consider all possible sequences of moves the machine
might make for a given string
• For example, on the string aa there are three:
– From q0 to q0 to q0, rejecting
– From q0 to q0 to q1, accepting
– From q0 to q1, getting stuck on the last a
• Our convention for this new kind of machine: a string is in L(M) if
there is at least one accepting sequence
q1
a,b
q0
a

Nondeterministic Finite
Automaton (NFA)
• L(M) = the set of strings that have at least one accepting
sequence
• In the example above, L(M) = {xa | x ∈ {a,b}*}
• A DFA is a special case of an NFA:
– An NFA that happens to be deterministic: there is exactly one
transition from every state on every symbol
– So there is exactly one possible sequence for every string
• NFA is not necessarily deterministic
q1
a,b
q0
a

NFA Advantage
• An NFA for a language can be smaller and easier to construct
than a DFA
• Strings whose next-to-last symbol is 1:
DFA:
NFA:
0
0
0
1
1
1
1
0
0,1
0,1
1

Outline

Spontaneous Transitions
• An NFA can make a state transition
spontaneously, without consuming an input
symbol
• Shown as an arrow labeled with ε
• For example, {a}* ∪ {b}*:
q0
aq1
q2 b

ε-Transitions To Accepting States
• An ε-transition can be made at any time
• For example, there are three sequences on the empty string
– No moves, ending in q0, rejecting
– From q0 to q1, accepting
– From q0 to q2, accepting
• Any state with an ε-transition to an accepting state ends up
working like an accepting state too
q0
aq1
q2 b

ε-transitions For NFA Combining
∀ε-transitions are useful for combining smaller
automata into larger ones
• This machine is combines a machine for {a}*
and a machine for {b}*
• It uses an ε-transition at the start to achieve
the union of the two languages
q0
aq1
q2 b

Incorrect Union
A = {an
| n is odd}
B = {bn
| n is odd}
A ∪ B ?
No: this NFA accepts aab
a
a
b
b
a
a
b
b

Correct Union
A = {an
| n is odd}
B = {bn
| n is odd}
A ∪ B
a
a
b
b
a
a
b
b

Incorrect Concatenation
A = {an
| n is odd}
B = {bn
| n is odd}
{xy | x ∈ A and y ∈ B} ?
No: this NFA accepts abbaab
a
a
b
b
a
a
b
b

Correct Concatenation
A = {an
| n is odd}
B = {bn
| n is odd}
{xy | x ∈ A and y ∈ B}
a
a
b
b
a
a
b
b

Outline

DFAs and NFAs
• DFAs and NFAs both define languages
• DFAs do it by giving a simple computational procedure for
deciding language membership:
– Start in the start state
– Make one transition on each symbol in the string
– See if the final state is accepting
• NFAs do it without such a clear-cut procedure:
– Search all legal sequences of transitions on the input string?
– How? In what order?

Nondeterminism
• The essence of nondeterminism:
– For a given input there can be more than one legal
sequence of steps
– The input is in the language if at least one of the legal
sequences says so
• We can achieve the same result by deterministically
searching the legal sequences, but…
• ...this nondeterminism does not directly correspond to
anything in physical computer systems
• In spite of that, NFAs have many practical
applications

Outline

Powerset
• If S is a set, the powerset of S is the set of all subsets of S:
P(S) = {R | R ⊆ S}
• This always includes the empty set and S itself
• For example,
P({1,2,3}) = {{}, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}

The 5-Tuple
• The only change from a DFA is the transition function δ
∀δ takes two inputs:
– A state from Q (the current state)
– A symbol from Σ∪{ε} (the next input, or ε for an ε-transition)
∀δ produces one output:
– A subset of Q (the set of possible next states)
An NFA M is a 5-tuple M = (Q, Σ, δ, q0, F), where:
Q is the finite set of states
Σ is the alphabet (that is, a finite set of symbols)
δ ∈ (Q × (Σ∪{ε}) → P(Q)) is the transition function
q0 ∈ Q is the start state
F ⊆ Q is the set of accepting states

Example:
• Formally, M = (Q, Σ, δ, q0, F), where
– Q = {q0,q1,q2}
Σ = {a,b} (we assume: it must contain at least a and b)
– F = {q2}
δ(q0,a) = {q0,q1}, δ(q0,b) = {q0}, δ(q0,ε) = {q2},
δ(q1,a) = {}, δ(q1,b) = {q2}, δ(q1,ε) = {}
δ(q2,a) = {}, δ(q2,b) = {}, δ(q2,ε) = {}
• The language defined is {a,b}*
q0 q1
a
q2
b
a,b

Outline

The δ* Function
• The δ function gives 1-symbol moves
• We'll define δ* so it gives whole-string results
(by applying zero or more δ moves)
• For DFAs, we used this recursive definition
δ*(q,ε) = q
δ*(q,xa) = δ(δ*(q,x),a)
• The intuition is the similar for NFAs, but the
ε-transitions add some technical hair

NFA IDs
• An instantaneous description (ID) is a
description of a point in an NFA's execution
• It is a pair (q,x) where
– q ∈ Q is the current state
– x ∈ Σ* is the unread part of the input
• Initially, an NFA processing a string x has the
ID (q0,x)
• An accepting sequence of moves ends in an
ID (f,ε) for some accepting state f ∈ F

The One-Move Relation On IDs
• We write
if I is an ID and J is an ID that could follow
from I after one move of the NFA
• That is, for any string x ∈ Σ* and any ω ∈ Σ
or ω = ε,
I a J
q,ωx( )a r ,x()if and only if r ∈δq,w( )

The Zero-Or-More-Move Relation
• We write
if there is a sequence of zero or more moves
that starts with I and ends with J:
• Because it allows zero moves, it is a reflexive
relation: for all IDs I,
I a∗
J
I a L a J
I a∗
I

The δ* Function
• Now we can define the δ* function for NFAs:
• Intuitively, δ*(q,x) is the set of all states the
NFA might be in after starting in state q and
reading x
• Our definition allows ε-transitions, including
those made before the first symbol of x is
read, and those made after the last
δ∗
q,x( )=r q,x( )a∗
r,ε( ){ }

M Accepts x
• Now δ*(q,x) is the set of states M may end in,
starting from state q and reading all of string x
• So δ*(q0,x) tells us whether M accepts x:
A string x ∈ Σ* is accepted by an NFA M = (Q, Σ, δ, q0, F)
if and only if δ*(q0, x) contains at least one element of F.

For any NFA M = (Q, Σ, δ, q0, F), L(M) denotes
the language accepted by M, which is
L(M) = {x ∈ Σ* | δ*(q0, x) ∩ F ≠ {}}.
The Language An NFA Defines

Nondeterministic Finite Automata

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