Example 1.11 I

Example 1.11 I

Fig 1.12

Example 1.11 II

s

q₁

q₂

r₁

r₂

a b

a

b b

a

b a

a b

Example 1.11 III

L(M) =?

a . . . a, b . . . b

where “. . .” can be any string of a and b First we check that any string accepted by the machine must be

a . . . a, b . . . b Second we check that any

a . . . a, b . . . b can be recognized by the machine

Example 1.11 IV

This machine handles strings with the same character in the beginning and in the end

Example 1.13 I

Figure 1.14

Example 1.13 II

q₀

q₁

q₂ 2,⟨reset

⟩ 1

0, ⟨reset⟩

0

1 2

0 2

1, ⟨reset⟩

Example 1.13 III

Σ = {⟨reset⟩, 0, 1, 2}

L(M) = . . . ⟨reset⟩ . . . ⟨reset⟩ . . .

= {sum of the last segment mod 3 = 0}

Example:

10⟨reset⟩22⟨reset⟩012

Example 1.13 IV

An example of running a string

q₀ −→ q¹ ₁ −→ q⁰ ₁ −−−→ q^⟨reset⟩ ₀ −→ q² ₂ −→ q² ₁

⟨reset⟩

−−−→ q₀ −→ q⁰ ₀ −→ q¹ ₁ −→ q² ₀ Accepted

Each node stores the sum of the current segment mod 3

Formal Definition of Computation I

M accepts w = w₁· · · w_n if ∃ states r₀· · · r_n such that

1 r₀ = q₀

2 δ(r_i, w_{i +1}) = r_{i +1}, i = 0, . . . , n − 1

3 r_n ∈ F

Definition: a language is regular if recognized by some automata

This is a very important definition

Examples described earlier are regular languages We say some automata, so it’s possible to have several automata for the same language

Formal Definition of Computation II

As long as there is one, then the language is regular

Designing Automata I

Given a language, how do we construct a machine to recognize it?

Basically we need to get a state diagram (where the number of states is finite)

Earlier we had the opposite: a machine is given and we check the corresponding language

Example: an automaton recognizing {0, 1} strings with an odd # of 1’s

Fig 1.20

Designing Automata II

q_e q_o

1

0 0

1 Sample strings

01

q_e −→ q⁰ _e −→ q¹ _o 010101

q −→ q⁰ −→ q¹ −→ q⁰ −→ q¹ −→ q⁰ −→ q¹

Designing Automata III

Two ways to think about the design

After the first 1, we go to qo. Subsequently, every 1, . . . , 1 pair is cancelled out by

q_o −→ q¹ _e → · · · → q_e −→ q¹ _o qe, qo respectively remember whether the number of 1’s so far is even or odd

Example 1.21

Language: strings containing 001 Fig 1.22

Designing Automata IV

q q₀ q₀₀ q₀₀₁

1

0

1

0

0

1

0, 1 q₀, q₀₀ indicate that before the current input

character, we have 0 and 00, respectively