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Chapter 4: Properties of Regular Languages

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Tiêu đề Properties of Regular Languages
Tác giả Quan Thanh Tho
Trường học Hochiminh City University of Technology
Chuyên ngành Computer Science
Thể loại Bài báo
Thành phố Ho Chi Minh City
Định dạng
Số trang 33
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Chapter 4: Properties of Regular Languages Quan Thanh Tho qttho@cse.hcmut.edu.vn Theorem 4.1 If L 1 and L 2 are regular, then so are L 1 ∩L 2 , L 1 ∪L 2 , L 1 L 2 , L 1 , L 1 * . (The family of regular languages is closed under intersection, union, concatenation, complement, and star-closure.) Proof • L 1 = L(r 1 ) L 2 = L(r 2 ) L(r 1 + r 2 ) = L(r 1 )∪L(r 2 ) L(r 1 . r 2 ) = L(r 1 )L(r 2 ) L(r 1 * ) = (L(r 1 )) * Proof (cont’d) • M = (Q, Σ, δ, q 0 , F) accepts L 1 . M = (Q, Σ, δ, q 0 , Q – F) accepts L 1 . Proof (cont’d)  M1 = (Q, Σ, δ1, q0, F1) accepts L1.  M2 = (P, Σ, δ2, p0, F2) accepts L2. q 0 q f a 1 a n p 0 p f a 1 a n δ 2 (p j , a) = p l δ 1 (q i , a) = q k δ 1 ((q i , p j ), a) = (q k , p l ) Example 4.1 L 1 = {ab n | n ≥ 0} L 2 = {a n b | n ≥ 0} L 1 ∩L 2 = {ab} Example 4.2  Find intersection dfa of L 1 ={a 2n b m : n, m ≥ 0} and L 2 ={a 3n b 2m : n, m ≥ 0} p 1 p 0 p 2 p 3 a a a b b b L 2 p 4 q 1 p 1 q 0 p 0 q 2 p 4 q 1 p 2 q 2 p 3 q 0 p 2 q 0 p 1 q 1 p 0 a a a a a a b b b L 1 ∩ L 2 q 1 q 0 q 2 aa b L 1 b Theorem 2 The family of regular languages is closed under reversal: If L is regular, then so is L R . Proof ? Suppose Σ and Γ are alphabets. h: Σ → Γ* is called a homomorphism Homomorphism [...]... (cont’d) If r is a regular expression on Σ, then the regular expression h(r) is obtained by applying the homomorphism to each Σ symbol of r Example 4.4 Σ = {a, b} Γ = {b, c, d} h(a) = dbcc h(b) = bdc r = (a + b*)(aa)* h(r) = (dbcc + (bdc)*)(dbccdbcc)* Theorem 4.3 The family of regular languages is closed under homomorphism: If L is regular, then so is h(L) Proof Let L(r) = L for some regular expression... representation of a regular language is one of the followings: – – – Finite automaton Regular expression Regular grammar Questions about RL 1 Given a regular language L on Σ and any w ∈ Σ*, is there an algorithm to determine whether or not w ∈ L? Yes Questions about RL 2 Is there an algorithm to determine whether or not a regular language is empty, finite, or infinite? Yes Questions about RL 3 Given two regular. .. y ∈ L2 Example 4.5 (cont’d) L1 = {anbm | n ≥ 1, m ≥ 0}∪{ba} L2 = {bm | m ≥ 1} b q1 b q2 δ*(q0, x) = qi δ*(qi, y) ∈ F and y ∈ L2 Theorem 4.4 The family of regular languages is closed under right quotient: If L1 and L2 are regular, then so is L1/L2 Proof • M = (Q, Σ, δ, q0, F) accepts L1 M^ = (P, Σ, δ, q0, F^) accepts L1/L2 If y ∈ L2 and δ*(qi, y) ∈ F ⇒ add qi to F^ Example 4.6 L1 = L(a*baa*) L2 = L(ab*)... language or not? Further Reading   Pumping lemma L = {anbn : n ≥ 0} revisited with pumping lemma Homework • Exercises: 2, 4, 6, 8, 9, 11, 18, 22 of Section 4.1 - Linz’s book • Exercises: 1, 2, 3, 5, 9 of Section 4.2 Linz’s book • Exercises: 3, 4, 5, 6, 8, 10, 12 of Section 4.3 - Linz’s book ... empty, finite, or infinite? Yes Questions about RL 3 Given two regular languages L1 and L2, is there an algorithm to determine whether or not L1 = L2? Yes Pigeonhole principle  When n objects are placed in m boxes, provided n > m, there is at least one box storing more than one objects  Example: L = {anbn : n ≥ 0} ; L is a regular language or not? Further Reading   Pumping lemma L = {anbn : n... languages is closed under homomorphism: If L is regular, then so is h(L) Proof Let L(r) = L for some regular expression r Prove: h(L(r)) = L(h(r)) Right Quotient Let L1 and L2 be languages on the same alphabet Then the right quotient of L1 with L2 is defined as: L1/L2 = {x | xy ∈ L1 and y ∈ L2} Example 4.5 L1 = {anbm | n ≥ 1, m ≥ 0}∪{ba} L2 = {bm | m ≥ 1} L1/L2 = {anbm | n ≥ 1, m ≥ 0} Example 4.5 (cont’d)

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