Lecture 2 Finite Automata 2-0 An Example - - PDF document

✬ ✫ ✩ ✪ The University of Melbourne

• Dept. of Computer Science and Software Eng.

433–330 Theory of Computation Harald Søndergaard

Lecture 2

Finite Automata

2-0

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An Example Automaton

Imagine a vending machine selling tea or coffee for $2. It accepts 1- and 2-dollar coins. If we let ‘1’ (‘2’) stand for the event that a 1-dollar (2-dollar) coin enters the coin slot, and C (T) stand for the push of button ‘C’ (‘T’) and subsequent delivery of a cup of coffee (tea), then the following automaton describes the acceptable event sequences:

• 2
• 1
• 1,2
• C,T
• 1,2
• That’s “acceptable” from a greedy owner’s point
• f view, for example, 2T11C22C is accepted, but

111C1T is not.

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Example 2

Here is an automaton for recognising unsigned number literals in some programming language:

• 1

d

• 2

d

• .
• E
• 3

d

• 4

d

• E
• 5

+,−

• d
• 6

d

• 7

d

• d is an abbreviation for 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

(that is, the digits).

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Formal Definition

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

• Q is a finite set of states,
• Σ is a finite alphabet,
• δ : Q × Σ → Q is the transition function,
• q0 ∈ Q is the start state, and
• F ⊆ Q are the accept states.

Here δ is a total function.

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Example 3

The automaton M1:

• q1
• 1
• q2

1

• q3

0,1

• can be described precisely as

M1 = ({q1, q2, q3}, {0, 1}, δ, q1, {q2}), where δ is given by:

1 q1 q1 q2 q2 q3 q2 q3 q2 q2

The language recognised by M1, L(M1) =        w

• w contains at least one 1,

and an even number of 0s follow the last 1       

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Example 4

• q1

a

• b
• q2

b

• a
• s

a

• b
• r1

b

• a
• r2

a

• b
• Which language is recognised by this machine?

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Exercise

Consider the alphabet Σ = {0, 1}. We can interpret strings in Σ∗ as numbers in binary representation. Construct an automaton over Σ to recognise exactly those numbers that are multiples of 5. Hint: Consider five states:

• q1
• q3
• q0
• q2
• q4

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Acceptance, Recognition

We can say precisely what it means for an automaton to accept a string. Let M = (Q, Σ, δ, q0, F) and let w = v1v2 · · · vn be a string from Σ∗. M accepts w iff there is a sequence of states r0, r1, . . . , rn, with each ri ∈ Q, such that

• 1. r0 = q0
• 2. δ(ri, vi+1) = ri+1 for i = 0, . . . , n − 1
• 3. rn ∈ F

M recognises language A iff A = {w | M accepts w}

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Regular Languages

A language is regular iff there is a finite automaton that recognises it. We shall soon see that there are languages which are not regular.

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Regular Operations

Recall that for our purposes, a language is simply a set of strings. Let A and B be languages. The regular operations are:

• Union:

A ∪ B

• Concatenation:

A ◦ B = {xy | x ∈ A, y ∈ B}

• Kleene star:

A∗ = {x1x2 · · · xk | k ≥ 0, each xi ∈ A} Note that the empty string, ǫ, is always in A∗.

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Regular Operations (cont.)

Let A = {aa, abba} and B = {a, ba, bba, bbba, . . .}. A ∪ B = {a, aa, abba, ba, bba, bbba, . . .} A ◦ B = {aaa, abbaa, aaba, abbaba, aabba, abbabba, . . .} A∗ = {ǫ, aa, abba, aaaa, aaabba, abbaaa, abbaabba, aaaaaa, aaaaabba, aaabbaaa, aaabbaabba, . . .} The regular languages are closed under the regular operations. It will be easier to show this after we have considered non-deterministic automata.

2-10

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Nondeterminism

The type of machine we have seen so far is called a deterministic finite automaton, or DFA. We now turn to non-deterministic finite automata, or NFAs. Here is an NFA for recognising the language   w

• w ∈ {0, 1}∗ has length 3 or more,

and the third last symbol in w is 1   

• q1

0,1

• 1
• q2

0,1

• q3

0,1

• q4

Note: No transitions from q4, and two possible transitions when we meet a 1 in state q1.

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Nondeterminism (cont.)

The NFA is much easier to design and understand than a DFA for the same language:

• 000
• 1
• 100
• 1
• 010
• 1
• 110
• 1
• 001
• 1
• 101
• 1
• 011
• 1
• 111
• 1
• This is the simplest DFA that will do the job!

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Epsilon Transitions

NFAs may also be allowed to move from one state to another without consuming input. Such a transition is called an ǫ transition. Amongst other things this is useful for constructing machines to recognise unions A ∪ B

• f languages:

machine for A

• ǫ
• ǫ
• machine for B

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Formal Definition

For any alphabet Σ let Σǫ denote Σ ∪ {ǫ}. An NFA is a 5-tuple (Q, Σ, δ, q0, F), where

• Q is a finite set of states,
• Σ is a finite alphabet,
• δ : Q × Σǫ → P(Q) is the transition function,
• q0 ∈ Q is the start state, and
• F ⊆ Q are the accept states.

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Acceptance, Recognition

The definition of what it means for an NFA N to accept a string says that it has to be possible to make the necessary transitions. Let N = (Q, Σ, δ, q0, F) be an NFA and let w = v1v2 · · · vn where each vi is a member of Σǫ. N accepts w iff there is a sequence of states r0, r1, . . . , rn, with each ri ∈ Q, such that

• 1. r0 = q0
• 2. ri+1 ∈ δ(ri, vi+1) for i = 0, . . . , n − 1
• 3. rn ∈ F

N recognises language A iff A = {w | N accepts w}

2-15