Regular Operations I
Regular operations can be used to study whether languages are regular or not
That is, these operations can help us to check if for a given language, whether there are finite automata to recognize it or not
We mainly consider three operations.
Assume A, B are given languages union
A ∪ B
Regular Operations II
concatenation
A ◦ B = {xy | x ∈ A, y ∈ B}
star:
A∗ = {x1· · · xk | k ≥ 0, xi ∈ A}
Regular Operations III
If k = 0, what do we mean x1· · · xk? We define
ϵ : empty string in this situation
Thus
ϵ ∈ A∗
Regular Operations IV
Example
Σ = {a, . . . , z}
A = {good , bad } B = {boy , girl }
A ◦ B = {goodboy , . . .}
A∗ : {ϵ, good , bad , goodgood , . . .}
We say an operation R is closed if the following property holds
if x ∈ A, y ∈ A, then xRy ∈ A
Regular Operations V
Example: N = {1, 2, . . .} is closed under multiplication
Th 1.25: regular languages are closed under the union operation
A1, A2 are regular languages
⇒A1 ∪ A2 is regular Proof
Regular Operations VI
Assume we are given two automata M1 = (Q1, Σ, δ1, q1, F1) M2 = (Q2, Σ, δ2, q2, F2)
Question: you want to think about why we can consider the same Σ
Idea: we construct a parallel machine to run two machines simultaneously
Regular Operations VII
Definition of our new machine M = (Q, Σ, δ, q0, F )
Q = {(r1, r2) | r1 ∈ Q1, r2 ∈ Q2} δ((r1, r2), a) = (δ1(r1, a), δ2(r2, a)) q0 = (q1, q2)
F = {(r1, r2) | r1 ∈ F1 or r2 ∈ F2} Example: combining
{w | w has an odd # 1’s}∪
{w | w has an odd # 0’s}
Regular Operations VIII
qe qo
1
0 0
1
se so
0
1 1
0
Regular Operations IX
qe, se
qe, so
qo, se
qo, so 0
0
1 1 1
1
0 0
Is this proof rigorously enough?
Regular Operations X
A formal proof should be done by induction. But we don’t provide it here
Th 1.26: closed under concatenation
If A, B are regular, then A ◦ B is regular But the proof is not easy
It’s unclear where to break the input
To easily do the proof, we introduce a new technique called nondeterminism
