sliek môn máy tự động chương 4 properties of regular languages

Chapter 4: Properties of Regular LanguagesOctober 11, 2009... Theorem 4.3 - Linz ’s bookTheorem The family of regular languages is closed under homomorphism.. IfLis regular, then so ishL

Chapter 4: Properties of Regular Languages

October 11, 2009

Objectives

Theorem 4.1 - Linz ’s book

Theorem

concatenation, complement, and star-closure

IfL1 andL2 are regular, then so areL1∩ L2,L1∪ L2 ,L1L2 ,L1,L∗1

Proof of Theorem 4.1 - Linz ’s book

By theorem 3.2 - Linz ’s book:

L1 = L(r1) and L2= L(r2)

L(r1+ r2) = L(r1) ∪ L(r2)

L(r1.r2) = L(r1)L(r2)

L(r1∗) = (L(r1))∗

Proof of Theorem 4.1 - Linz ’s book

M = (Q,P, δ, q0, F ) accepts L1

M = (Q,P, δ, q0, Q − F ) accepts L1

Proof of Theorem 4.1 - Linz ’s book

Example 4.1

L2 = {anb | n ≥ 0}

a

a

b

bb

Theorem 4.2 - Linz ’s book

Theorem

IfLis regular, then so isLR

Proof: ???

Definition

h(L) = {h(w ) : w ∈ L}

Definition

symbol of r

Theorem 4.3 - Linz ’s book

Theorem

The family of regular languages is closed under homomorphism

IfLis regular, then so ish(L)

Proof of Theorem 4.3 - Linz ’s book

Let L(r) = L for some regular expression r

h(L(r)) = L(h(r))

Right Quotient

Definition

quotientof L1 with L2 is defined as:

L1/L2= {x | xy ∈ L1∧ y ∈ L2}

Example 4.5

L1/L2 = {anbm| n ≥ 1, m ≥ 0}

Theorem 4.4 - Linz ’s book

Theorem

The family of regular languages is closed under right quotient

IfL1 andL2 are regular, then so isL1/L2

Example 4.6

L1 = L(a∗baa∗)

L2 = L(ab∗)

L1/L2 =?

Questions about RL

Problem (Theorem 4.5 - Linz ’s book)

Problem (Theorem 4.6 - Linz ’s book)

Is there an algorithm to determine whether or not a regularlanguage isempty,finite, or infinite?

Problem (Theorem 4.7 - Linz ’s book)

Problem (Theorem 4.8 - Linz ’s book)

Is there an algorithm to determine whether or not a language is

regular?

Theorem 4.8 - Linz ’s book

Theorem (Pumping Lemma)

positive integer m such that any w ∈ L with |w | ≥ m can be

Exercises: 2, 4, 6, 8, 9, 11, 18, 22 of Section 4.1 - Linzs book

Exercises: 1, 2, 3, 5, 9 of Section 4.2 - Linzs book

Exercises: 3, 4, 5, 6, 8, 10, 12 of Section 4.3 - Linzs book