Recitation Course on 1022 jinnjy Outline l Myhill

Outline l Myhill Nerode Thm and application l Minimization of DFA and application l

Application of MN Thm l Myhill Nerode Thm and application l Minimization of DFA

Application of MN Thm l Eg: Let L={0 n 1 n|n≥ 0}. ¡ ¡

Application of MN Thm l Eg: S={0 n|n≥ 0} : infinite set, and L={ww|w∈{0,

DFA Minimization Key: 把equivalent state合併 l Two states p, q are indistinguishable if l

DFA Minimization l Usage of table-filling algorithm: ¡ ¡ Decide whether two regular languages

Example of DFA minimization l e. g. L(A)={w| w has at least 2 a’s}

Example of DFA minimization l Basis: (1, 3), (2, 3), (3, 4) are distinguishable

Example of DFA minimization l Induction: d(1, a)=2, d(2, a)=3, d(4, a)=2. (1, 2)

Example of DFA minimization l (1, 4) is indistinguishable. We have the equivalent DFA.

Summary of Chapter 1 DFA, NFA, regular expression l Pumping Lemma and Myhill-Nerode Thm

重點提示 l l l DFA, NFA, regular expression的關係轉換 Pumping Lemma和Myhill Nerode Thm的使用利用closure properties

重點提示 l 證明一個language ¡ l 是regular的方法找出它的DFA, NFA, regular expression 證明一個language 不是regular的方法 Pumping lemma

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Recitation Course on 10/22 jinnjy

Outline l Myhill Nerode Thm and application l Minimization of DFA and application l Context-Free Grammar

Myhill-Nerode Theorem

Application of MN Thm l Myhill Nerode Thm and application l Minimization of DFA and application l Context-Free Grammar

Application of MN Thm l Eg: Let L={0 n 1 n|n≥ 0}. ¡ ¡ The intuitive reason L is not regular : we must remember how many 0’s we have seen S={0 n|n≥ 0} : infinite set 0 i , 0 j ∈S, i≠j 0 i 1 i∈L but 0 j 1 i∉ L. ⇒ 1 i distinguishes 0 i and 0 j ⇒ The relation IL has infinitely many distinct equivalence classes and that L is not regular.

Application of MN Thm l Eg: S={0 n|n≥ 0} : infinite set, and L={ww|w∈{0, 1}*} is a language. z=1 n 0 n 1 n is a string. ⇒ 0 nz ∈ L , 0 mz∉ L ⇒ z distinguishes 0 n and 0 m

DFA Minimization Key: 把equivalent state合併 l Two states p, q are indistinguishable if l

DFA Minimization

DFA Minimization l Usage of table-filling algorithm: ¡ ¡ Decide whether two regular languages are equal Obtain the unique minimum state DFA from any given DFA

L(A)=L(B) ? ? q. A q. B A q. A A B C q. B B

Example of DFA minimization l e. g. L(A)={w| w has at least 2 a’s} a, b b 1 a b 2 a 4 b a 3

Example of DFA minimization l Basis: (1, 3), (2, 3), (3, 4) are distinguishable pairs. a, b b 1 a b 2 a 4 b a 3

Example of DFA minimization l Induction: d(1, a)=2, d(2, a)=3, d(4, a)=2. (1, 2) and (2, 4) are distinguishable pairs. a, b b 1 a b 2 a 4 b a 3

Example of DFA minimization l (1, 4) is indistinguishable. We have the equivalent DFA. b 1, 4 a a, b b 2 a 3

Summary of Chapter 1 DFA, NFA, regular expression l Pumping Lemma and Myhill-Nerode Thm l Closure Properties of regular languages l Decision Properties of regular languages l

重點提示 l l l DFA, NFA, regular expression的關係轉換 Pumping Lemma和Myhill Nerode Thm的使用利用closure properties 及pumping lemma證明某些language不是regular Closure properties 及Decision Properties Table-filling Algorithm的用途

重點提示 l 證明一個language ¡ l 是regular的方法找出它的DFA, NFA, regular expression 證明一個language 不是regular的方法 Pumping lemma ¡ Myhill-Nerode Thm ¡ l 在證明一個language是regular時千萬不要提到 pumping lemma, pumping lemma 一個language是regular 不能拿來證