Lecture Four: Nondeterministic Finite Automata

Lecture Four: Nondeterministic Finite

Automata

NFA, Lecture 4, slide 1

Amjad Ali

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,

and they too have a variety of practical applications.

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

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

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

Definitions for NFAs

Let N = (Q, Σ, δ, q0, F) be an NFA and let w be in Σ*. Then w is

accepted by N iff δ({q0}, w) contains at least one state in F.

Let N = (Q, Σ, δ, q0, F) be an NFA. Then the language accepted

by N is the set:

L(N) = {w | w is in Σ* and δ({q0},w) contains at least one state in F}

Another equivalent definition:

L(N) = {w | w is in Σ* and w is accepted by N}NFA, Lecture 4, slide 6

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:

Spontaneous Transitions An NFA can make a state transition

spontaneously, without consuming an input symbol

Shown as an arrow labeled with For example, {a}* {b}*:

-Transitions To Accepting States

An -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 -transition to an accepting state ends up working like an accepting state too

-transitions For NFA Combining

-transitions are useful for combining smaller automata into larger ones

This machine combines a machine for {a}* and a machine for {b}*

It uses an -transition at the start to achieve the union of the two languages

Incorrect Union

A = {an | n is odd}

B = {bn | n is odd}

A B ?No: this NFA accepts aab

Correct UnionA = {an | n is odd}

B = {bn | n is odd}

Incorrect Concatenation

A = {an | n is odd}

B = {bn | n is odd}

{xy | x A and y B} ?No: this NFA accepts abbaab

Correct Concatenation

A = {an | n is odd}

B = {bn | n is odd}

{xy | x A and y B}

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

Formal Definition of Nondeterministic Finite Automata

An NFA is a five-tuple:

N = (Q, Σ, δ, q0, F)

Q A finite set of statesΣ A finite input alphabetq0 The initial/starting state, q0 is in QF A set of final/accepting states, which is a subset of Qδ A transition function, which is a total function from Q x Σ to 2Q

δ: (Q x Σ) → P(Q) -P(Q) is the power set of Q, the set of all subsets of Q

δ(q,s) -The set of all states p such that there is a transition labeled s from q to p

δ(q,s) is a function from Q x S to P(Q) (but not to Q)NFA, Lecture 4, slide 16

Example : Formal Definition of NFA given below: (pair of 0’s or pair of 1’s)

Q = {q0, q1, q2 , q3 , q4}

Σ = {0, 1} Start state is q0

F = {q2, q4}

δ: 0 1q0

{q0, q3} {q0, q1}

{} {q2}

{q2} {q2}

{q4} {}

{q4} {q4}

0 0q3q4

Example:Possible Sequences of 001

Determining if a given NFA accepts a given string (001) can be done algorithmically:

q0 q0 q0 q0

q3 q3 (stuck) q1

accepted Each level will have at most n states

0 0q3q4

Formal Definition of Computation (NFA)

The * Function: What we currently have: δ : (Q x Σ) → P(Q) What we want (why?): δ : (P(Q) x Σ*) → P(Q)

δ*(R, ε) = Rδ*(R, xa) = δ (δ*(R, x), a) - for any subset R of Q, for any w=xa in Σ*, a in Σ, and subset R of Q.

andδ(R, a) = δ(q, a) for all subsets R of Q, and symbols a in Σ

Example:

What is δ*(q0, 011) ? Is 011 Accepted?

δ*({q0}, 011) = δ(δ*({q0 }, 01), 1)

= δ(δ(δ*({q0 }, 0), 1), 1)

= δ(δ(δ(δ*({q0 }, ε), 0), 1), 1)

= δ(δ(δ({q0 }, 0), 1), 1)

= δ(δ({q1, q2, q3 }, 1), 1)

= δ(δ({q1}, 1) U δ({q2}, 1) U δ({q3}, 1), 1)

= δ(({q2 ,q3} U {q3} U {}), 1)

= δ({q2 ,q3}, 1)= δ({q2}, 1) U δ({q3} , 1)= {q3} U {} = {q3}

Is 011 accepted? Yes, since δ*({q0}, 011) = {q3 } is a final state.

q00 1q1

Example: Possible Sequences of 011

q0 q1 q2 q3 (Accepted)

q2 q3 (Stuck)

q3(Stuck)

q3 (Stuck)

q00 1q1

Definitions for NFAs

Let N = (Q, Σ, δ,q0,F) be an NFA and let w be in Σ*. Then w is accepted by N iff δ*({q0}, w) contains at least one state in F.

Let N = (Q, Σ, δ,q0,F) be an NFA. Then the language accepted by N is the set:

L(N) = {w | w is in Σ* and δ*({q0},w) contains at least one state in F}

Another equivalent definition:

L(N) = {w | w is in Σ* and w is accepted by N}

For any NFA N = (Q, , , q0, F), L(N) denotes the language accepted by N which is

L(N) = {x * | *(q0, x) F {}}.

The Language An NFA Defines

ε-Non-deterministic Finite Automate

(NFA-ε)

Formal Definition of NFAs with ε Moves

An NFA-ε is a five-tuple:N = (Q, Σ, δ, q0, F)

Q A finite set of statesΣ A finite input alphabetq0 The initial/starting state, q0 is in QF A set of final/accepting states, which is a subset of Qδ A transition function, which is a total function from Q x Σ U {ε} to P(Q)

δ: (Q x (Σ U {ε})) → P(Q)

A String w in Σ* is accepted by NFA iff there exists a path in NFA from q0 to a state in F labeled by w and zero or more ε transitions.

Sometimes referred to as an NFA-ε other times, simply as an NFA.

Formally: Q = {q0, q1, q2 , q3 }

Σ = {0, 1} Start state is q0

F = {q2, q4}

δ: δ: (Q x (Σ U {ε})) → P(Q) 0 1 εq0

{q0} { } {q1}

{q1, q2} {q0, q3} {q2}

{q2} {q2} { }

{ } { } { }

Example:

Formal Definition of Computation (NFA- ε)

The * Function: What we currently have: δ : (Q x (Σ U {ε})) → P(Q) What we want (why?): δ : (P(Q) x Σ*) → P(Q)

δ*(R, ε) = ε-closure(R) - for any subset R of Q

δ *(R, xa) = ε-closure(δ(δ*(R,x), a)) - for any w=xa in Σ*, a in Σ, and subset R of Q

andδ(R, a) = δ(q, a) for all subsets R of Q, and symbols a in Σ

Example: Computation of ε-closure

Define ε-closure(q) to denote the set of all states reachable from q by zero or more ε transitions.

ε-closure(q0) = {q0, q1, q2}ε-closure(q1) = {q1, q2} ε-closure(q2) = {q2}ε-closure(q3) = {q3}

ε-closure(q) can be extended to sets of states by defining:ε-closure(P) = ε-closure(q)

ε-closure({q1, q2}) = ε-closure(q1) U ε-closure(q2) = {q1, q2} U {q2}= {q1, q2}ε-closure({q0, q3}) = ε-closure(q0) U ε-closure(q3) = {q0, q1, q2} U {q3}={q0,

q1, q2, q3}

What is δ*(q0, 010) ? Is 010 Accepted?

δ*({q0}, 010) = E(δ(δ*({q0 }, 01), 0))

= E(δ(E(δ(δ*({q0 }, 0), 1)), 0))

= E(δ(E(δ(E(δ(δ*({q0 }, ε), 0)), 1)), 0))

= E(δ(E(δ(E(δ(E({q0 }), 0)), 1)), 0))

= E(δ(E(δ(E(δ({q0, q1, q2}, 0)), 1)), 0))

= E(δ(E(δ(E(δ({q0}, 0) U δ({q1}, 0) U δ({q2}, 0)), 1)), 0))

= E(δ(E(δ(E({q0} U {q2} U {q2}), 1)), 0))

= E(δ(E(δ(E({q0, q2})), 1)), 0))

= E(δ(E(δ( (E({q0 }) U E({q2})) , 1)), 0))

= E(δ(E(δ( ({q0, q1, q2 } U {q2} ) , 1)), 0))

= E(δ(E(δ( {q0, q1, q2 } , 1)), 0))

Example:

= E(δ(E(δ( {q0, q1, q2 } , 1)), 0))

= E(δ(E((δ({q0}, 1) U δ({q1}, 1) U δ({q2},1)), 0))

= E(δ(E( {} U {q0, q3} U {q2} ), 0))

= E(δ(E( {q0, q2 , q3} ), 0))

= E(δ( (E({q0}) U E({q2}) U E({q3})) , 0))

= E(δ( ({q0 , q1 , q2} U {q2} U {q3} ) , 0))

= E(δ( {q0 , q1 , q2 , q3} , 0))

= E( ( δ({q0}, 0) U δ({q1}, 0) U δ({q2}, 0) U δ({q3}, 0)) )

= E( ( {q0} U {q1, q2} U {q2} U {} ) )

= E( {q0 , q1, q2 } )

= E({q0}) U E({q1}) U E({q2})

= {q0 , q1 , q2} U {q1 , q2}) U {q2}

= {q0 , q1 , q2}

(Is 010 accepted? Yes, since {q2 } is a final state )

Definitions for NFA-ε Machines

Let N = (Q, Σ, δ,q0,F) be an NFA-ε and let w be in Σ*. Then w is accepted by N iff δ*({q0}, w) contains at least one state in F.

Let N = (Q, Σ, δ,q0,F) be an NFA-ε. Then the language accepted by N is the set:

L(N) = {w | w is in Σ* and δ*({q0},w) contains at least one state in F}

Another equivalent definition:L(N) = {w | w is in Σ* and w is accepted by N}

Equivalence of NFAs and NFA-εs

Do NFAs and NFA-ε machines accept the same class of languages?

Is there a language L that is accepted by a NFA, but not by any NFA-ε?

Is there a language L that is accepted by an NFA-ε, but not by any DFA?

Observation: Every NFA is an NFA-ε.

Therefore, if L is a regular language then there exists an NFA-ε N such that L = L(N).

It follows that NFA-ε machines accept all regular languages.

But do NFA-ε machines accept more?NFA, Lecture 4, slide 32

Lemma 1: Let M be an NFA. Then there exists a NFA-ε M’ such that L(M) = L(M’).

Proof: Every NFA is an NFA-ε. Hence, if we let M’ = M, then it follows that L(M’) = L(M).

Lemma 2: Let M be an NFA-ε. Then there exists a NFA M’ such that L(M) = L(M’).

Proof: (sketch)Let M = (Q, Σ, δ,q0,F) be an NFA-ε.

Define an NFA M’ = (Q, Σ, δ’,q0,F’) as:

F’ = F U {q0} if ε-closure(q0) contains at least one state from FF’ = F otherwise

δ’(q, a) = δ^(q, a) - for all q in Q and a in Σ

Notes: δ’: (Q x Σ) –> 2Q is a function M’ has the same state set, the same alphabet, and the same start

state as M M’ has no ε transitions

Example: Equivalence of NFAs and NFA-ε s

Step #1: Same state set as M q0 is the starting state

Step #2:

F’ = F U {q0} if ε-closure(q0) contains at least one state from FF’ = F otherwise

q0 becomes a final state

Step #3:Q = {q0, q1, q2, q3}

δ’(q0, 0) = δ^ (q0, 0)

Step #3Q = {q0, q1, q2, q3}

δ’(q0, 1) = δ^ (q0, 1)

Step #4:Q = {q0, q1, q2, q3}

δ’(q1, 0) = δ^ (q1, 0)

Step #4:Q = {q0, q1, q2, q3}

δ’(q1, 1) = δ^ (q1, 1)

Step #5:Q = {q0, q1, q2, q3}

δ’(q2, 0) = δ^ (q2, 0)

Step #5:Q = {q0, q1, q2, q3}

δ’(q2, 1) = δ^ (q2, 1)

Step #6:Q = {q0, q1, q2, q3}

δ’(q3, 0) = δ^ (q3, 0)

Step #6:Q = {q0, q1, q2, q3}

δ’(q3, 1) = δ^ (q3, 1)

Step #7: Done

Equivalence of DFAs and NFAs

Do DFAs and NFAs accept the same class of languages? Is there a language L that is accepted by a DFA, but not by

any NFA? Is there a language L that is accepted by an NFA, but not

by any DFA? Observation: Every DFA is an NFA.

Therefore, if L is a regular language then there exists an NFA M such that L = L(M).

It follows that NFAs accept all regular languages.

But do NFAs accept more?

Lemma 1: Let M be an DFA. Then there exists a NFA M’ such that L(M) = L(M’).

Proof: Every DFA is an NFA. Hence, if we let M’ = M, then it follows that L(M’) = L(M).

Lemma 2: Let M be an NFA. Then there exists a DFA M’ such that L(M) = L(M’).

Proof: (sketch)Let M = (Q, Σ, δ,q0,F) be an NFA, and define a DFA M’ = (Q’, Σ,

δ’,q’0,F’) as:

1. Q’ = P(Q) - Every state of M’ is a set of states of M

2. q’0 = E({q0})

3. δ’ (R, a) = {q є Q | q є E(δ (r, a)) for some r є R}or δ’ (R, a) = E ( δ (r, a) ) - R є Q’ and a є ∑. If R is a state of

M’, it is also a set of states of M. When M’ reads symbol a in state R, it shows where a takes each state in R.

4. F’ = {R є Q’ | R contains an accept state of M}

Here Q = {1, 2, 3} - Q is set of NFA StatesAs Q’ = P(Q) 1. Q’ = { Ф,{1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} }

2. q’0 = E({q0}) –q0 is the start state of NFA and q0’ is start state of

DFA q’

0 = E({1})

q0’= { 1, 3 } - start state of DFA

3. δ’ (R, a) = E ( δ (r, a) )

Example: Equivalence of DFAs and NFAs

δ’ (R, a) = E ( δ (r, a) )

(i) δ’ ({}, a) = E ( δ({}, a) ) = E ( {} ) = { }

(ii) δ’ ({}, b) = E ( δ({}, b) ) = E ( {} ) = {}

(i) δ’ ({1}, a) = E ( δ({1}, a) ) = E ( {} ) = { }

(ii) δ’ ({1}, b) = E ( δ({1}, b) ) = E ( {2} ) = {2}

δ’ (R, a) = E ( δ (r, a) )

(i) δ’ ({2}, a) = E ( δ({2}, a) ) = E ( {2, 3} ) = E ( {2} ) U E ( {3} ) = {2} U {3} = {2, 3}

(ii) δ’ ({2}, b) = E ( δ({2}, b) ) = E ( {3} ) = {3}

δ’ (R, a) = E ( δ (r, a) )

(i) δ’ ({3}, a) = E ( δ({3}, a) ) = E ( {1} ) = E ( {1} ) = {1,3}

(ii) δ’ ({3}, b) = E ( δ({3}, b) ) = E ( {} ) = {}

δ’ (R, a) = E ( δ (r, a) )

(i) δ’ ({1,2}, a) = E ( δ({1}, a) U δ({2}, a) ) = E ( {} U {2,3} ) = E({2,3}) = E({2}) U E({3}) = {2} U {3} = {2,3}

(ii) δ’ ({1,2}, b) = E ( δ({1}, b) U δ({2}, b) ) = E ( {2} U {3} ) = E({2,3}) = E({2}) U E({3}) = {2} U {3} = {2,3}

3. δ’ (R, a) = E ( δ (r, a) )

(i) δ’ ({1, 3}, a) = E ( δ({1}, a) U δ({3}, a) ) = E ( { } U {1} ) = E ( {1} ) = {1, 3}

(ii) δ’ ({1, 3}, b) = E ( δ({1}, b) U δ({3}, b) ) = E ( {2} U {} ) = E ( {2} ) = {2}

δ’ (R, a) = E ( δ (r, a) ) (i) δ’ ({2,3}, a) = E ( δ({2}, a) U δ({3}, a) )

= E ( {2,3} U {1} ) = E({1,2,3}) = E({1}) U E({2}) U E({3}) = {1,3} U {2} U {3} = {1,2,3}

(ii) δ’ ({2,3}, b) = E ( δ({2}, b) U δ({3}, b) ) = E ( {3} U {} ) = E({3}) = {3}

δ’ (R, a) = E ( δ (r, a) ) (i) δ’ ({1,2,3}, a) = E ( δ({1}, a) U δ({2}, a) U δ({3}, a) )

= E ( { } U {2,3} U {1} ) = E ({1,2,3}) = E({1}) U E({2}) U E({3}) = {1,3} U {2} U {3} = {1,2,3}

(ii) δ’ ({1,2,3}, b) = E ( δ({1}, b) U δ({2}, b) U δ({3}, b) ) = E ( {2} U {3} U { } ) = E({2,3}) = E({2}) U E({3}) = {2} U {3} ={2,3}

Ф {2} {1,2}{1}

{1,3} {2,3} {1,2,3}

Ф {2}

{1,3} {2,3} {1,2,3}

Minimized DFA:Removing unreachable states from start state

q’0 = E({1})

q0’= { 1, 3 } - start state of DFA

δ’ (R, a) = E ( δ (r, a) )

(i) δ’ ({1, 3}, a) = E ( δ({1}, a) U δ({3}, a) ) = E ( { } U {1} ) = E ( {1} ) = {1, 3}

(ii) δ’ ({1, 3}, b) = E ( δ({1}, b) U δ({3}, b) ) = E ( {2} U {} ) = E ( {2} ) = {2}

Alternate Method

δ’ (R, a) = E ( δ (r, a) )

(i) δ’ ({2}, a) = E ( δ({2}, a) ) = E ( {2, 3} ) = E ( {2} ) U E ( {3} ) = {2} U {3} = {2, 3}

(ii) δ’ ({2}, b) = E ( δ({2}, b) ) = E ( {3} ) = {3}

Alternate Method

δ’ (R, a) = E ( δ (r, a) ) (i) δ’ ({2,3}, a) = E ( δ({2}, a) U δ({3}, a) )

= E ( {2,3} U {1} ) = E({1,2,3}) = E({1}) U E({2}) U E({3}) = {1,3} U {2} U {3} = {1,2,3}

(ii) δ’ ({2,3}, b) = E ( δ({2}, b) U δ({3}, b) ) = E ( {3} U {} ) = E({3}) = {3}

Alternate Method

δ’ (R, a) = E ( δ (r, a) )

(i) δ’ ({3}, a) = E ( δ({3}, a) ) = E ( {1} ) = E ( {1} ) = {1,3}

(ii) δ’ ({3}, b) = E ( δ({3}, b) ) = E ( {} ) = {}

Alternate Method

δ’ (R, a) = E ( δ (r, a) ) (i) δ’ ({1,2,3}, a) = E ( δ({1}, a) U δ({2}, a) U δ({3}, a) )

= E ( { } U {2,3} U {1} ) = E ({1,2,3}) = E({1}) U E({2}) U E({3}) = {1,3} U {2} U {3} = {1,2,3}

(ii) δ’ ({1,2,3}, b) = E ( δ({1}, b) U δ({2}, b) U δ({3}, b) ) = E ( {2} U {3} U { } ) = E({2,3}) = E({2}) U E({3}) = {2} U {3} ={2,3}

Alternate Method

δ’ (R, a) = E ( δ (r, a) )

(i) δ’ ({}, a) = E ( δ({}, a) ) = E ( {} ) = { }

(ii) δ’ ({}, b) = E ( δ({}, b) ) = E ( {} ) = {}

Alternate Method

Ф {2}

{1,3} {2,3} {1,2,3}

Alternate Method

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?

Lecture Four: Nondeterministic Finite Automata

Documents

Transcript of Lecture Four: Nondeterministic Finite Automata

Computability, Complexity, and Languages: Fundamentals of ......Complexity, and Languages Fundamentals of Theoretical Computer Science ... Finite Automata 237 2. Nondeterministic Finite

Nondeterministic Finite Automata - Universitas Atma Jaya ...fti.uajm.ac.id/ajar/Automata Dan Teori Bahasa Formal/4fa3.pdfSurprisingly, for any NFA there is a DFA that accepts the same

Antichain-based Inclusion Checking on Finite ...vojnar/Vienna-15/03-antichains.pdf · Antichain-based Inclusion Checking on Finite Nondeterministic Word and Tree Automata Tomáš

Automata - IOE Notes · Solution NONDETERMINISTIC FINITE AUTOMATA ... Page 6 of 30 Examples ... Chapter 2: Finite Automata 1 2 1 2 ( ...

Forward Bisimulations for Nondeterministic Symbolic Finite ... · Forward Bisimulations for Nondeterministic Symbolic Finite Automata Loris D’Antoni1 and Margus Veanes2 1 University

On the Power of Finite Automata with both Nondeterministic ...avi/PUBLICATIONS/MYPAPERS/CONDON/JOURNAL/final.pdf · On the Power of Finite Automata with both Nondeterministic and

Automata with Modulo Counters and Nondeterministic Counter ...

Unconventional Finite Automata and Algorithms · Two-Way Finite Automata Ultrametric Finite Automata Counting With Automata Two-Way Frequency Automata Ultrametric Query Algorithms

Modal Interfaces: Unifying Interface Automata and Modal ......Modal Specifications Key Concepts • Modal Automata: similar to Nondeterministic Finite Automata, but with must and may

NonDeterministic Finite Automata...Perbedaan DFA dan NFA •DFA (Deterministic Finite Automata) FA di dalam menerima input mempunyai tepat satu busur keluar. •NFA (Non Deterministic

Happy Birthday Michael !!. Probabilistic & Nondeterministic Finite Automata Avi Wigderson Institute for Advanced Study Very old (1996) joint work with.

Finite Automata - Gunadarmarisma.staff.gunadarma.ac.id/Downloads/files/58923/finite-automata.… · Finite Automata Finite Automata • Two types – both describe what are called

Nondeterministic Finite Automata

Nondeterministic Finite Automata in Hardware - the Case …tjt7a/docs/levenshtein_automata.pdf · Nondeterministic Finite Automata in Hardware - the Case of the Levenshtein Automaton

Nondeterministic Finite Automata · 2020. 11. 16. · Is it legal, i.e. a \proper" DFA? -S0-a a 6 S 1- b b 6 S 2 c- c 6 S 3 A. It makes sense, but it is nondeterministic: A nondeterministic

Simulation Reduction of Finite Nondeterministic Word and ...vojnar/Vienna-15/02-simulations.pdfSimulation Reduction of Finite Nondeterministic Word and Tree Automata Tomáš Vojnar

Implementation: Finite Automata - KSUfac.ksu.edu.sa/sites/default/files/lexical_analysis...Deterministic and Nondeterministic Automata •Deterministic Finite Automata (DFA) –One

1 Basics of automata theory Nondeterministic Finite Automata (NFA) Nondeterministic Finite Automata on infinite words (NFW)

Nondeterministic Finite Automat

1 Non Deterministic Automata. 2 Alphabet = Nondeterministic Finite Accepter (NFA)