2017/08/31 Chapter 1.2 & 1.3 in Sipser Ø Announcement:

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1 Regular Expressions Human-aware and Robo.cs Operations 2017/08/31 Chapter 1.2 & 1.3 in Sipser Ø Announcement: q q q q Many thanks to students who have responded so far! There is still time to respond to the poll. (Today s slides are adjusted accordingly.) Piazza registration: Slides for this lecture are here: REO.pdf HW2 on NFA will be released soon; HW1 due tonight 11:59PM u Happy Labor Day! 1

2 Last time Nondeterministic finite automata (introducing non-determinism into FA) o Definition of NFA o Computation of NFA o Equivalence of DFA and NFA q A language is regular if and only if some NFA recognizes it 2

3 Previously Finite automata o Definition of FA o Computation of FA o Regular language o Regular language and FA o Regular operations o Design an FA Nondeterministic finite automata (introducing non-determinism into FA) o Definition of NFA o Computation of NFA o Equivalence of DFA and NFA 3

4 Previously Finite automata o Definition of FA o Computation of FA o Regular language o Regular language and FA o Regular operations o Design an FA Nondeterministic finite automata (introducing non-determinism into FA) o Definition of NFA o Computation of NFA o Equivalence of DFA and NFA 4

5 Previously Finite automata o Definition of FA o Computation of FA o Regular language o Regular language and FA o Regular operations o Design an FA Nondeterministic finite automata (introducing non-determinism into FA) o Definition of NFA o Computation of NFA o Equivalence of DFA and NFA Regular Expressions (RE) 5

6 Outline for today Regular expressions (REs) o Regular operations and FA o RE examples o Equivalence of FA and REs (next class) Ø Goals: o Learn how to write formal proofs o Understand more about regular operations o Learn to use REs 6

7 Definition Human-aware of Regular Expression Robo.cs Definition - A collection (or set) is closed under an operation if applying this operation to members of the collection returns a member in the collection 7

8 Definition Human-aware of Regular Expression Robo.cs In order to prove equivalence, we must first convince ourselves that RL is closed under regular operations! Definition - A collection (or set) is closed under an operation if applying this operation to members of the collection returns a member in the collection 8

9 Regular operations 9

10 Closed under union 1. Given any RLs A1 and A2 2. Based on the definition, we can construct DFA M1 for A1 and M2 for A2 [ 1. Prove that A1 A2 is also a RL 2. Construct an NFA (DFA) M to simulate both M1 and M2 at the same time and accept if either one accepts o Keep a copy of both M1 and M2; for every step in M, run a step from M1 and then a step in M2; M accepts if either M1 or M2 accepts; otherwise, reject Ø This works now with NFA! " " 10

11 Closed under Human-aware union Robo.cs 1. Given any RLs A1 and A2 2. Based on the definition, we can construct DFA M1 for A1 and M2 for A2 M 1 =(Q 1,, 1,q 0,F 1 ) M 2 =(Q 2,, 2,s 0,F 2 ) 1. Prove that A1 A2 is also a RL 2. Construct an NFA (DFA) M to simulate both M1 and M2 at the same time and accept if either one accepts o Ø [ Keep a copy of both M1 and M2; for every step in M, run a step from M1 and then a step in M2; M accepts if either M1 or M2 accepts; otherwise, reject This works now with NFA! Construct M = (Q, {Q,,,t 0,F},F) to recognize L(M 1 ) [ L(M 2 ) Q = {t 0 } [ Q 1 [ Q 2 is unchanged 8 < { 1 (q, a)} q 2 Q 1 and a 2 (q, a) = { 2 (q, a)} q 2 Q 2 and a 2 : {q 0,s 0 } q = t 0 and a = " q 2 Q, a 2 " t 0 is the start state of M F = F 1 [ F 2 is the set of accept states 11

12 Closed under Human-aware union Robo.cs 1. Given any RLs A1 and A2 2. Based on the definition, we can construct DFA M1 for A1 and M2 for A2 M 1 =(Q 1,, 1,q 0,F 1 ) M 2 =(Q 2,, 2,s 0,F 2 ) 1. Prove that A1 A2 is also a RL 2. Construct an NFA (DFA) M to simulate both M1 and M2 at the same time and accept if either one accepts o Ø [ Keep a copy of both M1 and M2; for every step in M, run a step from M1 and then a step in M2; M accepts if either M1 or M2 accepts; otherwise, reject This works now with NFA! Construct M = (Q, {Q,,,t 0,F},F) to recognize L(M 1 ) [ L(M 2 ) Q = {t 0 } [ Q 1 [ Q 2 is unchanged 8 < { 1 (q, a)} q 2 Q 1 and a 2 (q, a) = { 2 (q, a)} q 2 Q 2 and a 2 : {q 0,s 0 } q = t 0 and a = " q 2 Q, a 2 " t 0 is the start state of M F = F 1 [ F 2 is the set of accept states 12

13 Closed under concatenation 1. Given any RLs A1 and A2 2. Based on the definition, we can construct DFA M1 for A1 and M2 for A2 1. Prove that A1 A2 is also a RL 2. Construct an NFA M to accept A1 A2 " 13

14 Closed under concatenation 1. Given any RLs A1 and A2 2. Based on the definition, we can construct DFA M1 for A1 and M2 for A2 1. Prove that A1 A2 is also a RL 2. Construct an NFA M to accept A1 A2 " 14

15 Closed under concatenation 1. Given any RLs A1 and A2 2. Based on the definition, we can construct DFA M1 for A1 and M2 for A2 1. Prove that A1 A2 is also a RL 2. Construct an NFA M to accept A1 A2 M 1 =(Q 1,, 1,q 0,F 1 ) M 2 =(Q 2,, 2,s 0,F 2 ) Construct M = (Q, {Q,,,t 0,F},F) to recognize L(M 1 ) [ L(M 2 ) Q = Q 1 [ Q 2 is unchanged t 0 = q 0 8 < { 1 (q, a)} q 2 Q 1 and a 2 (q, a) = {s 0 } q 2 F 1 and a = " : { 2 (q, a)} q 2 Q 2 and a 2 q 2 Q, a 2 " F = F 2 15

16 Closed under concatenation 1. Given any RLs A1 and A2 2. Based on the definition, we can construct DFA M1 for A1 and M2 for A2 1. Prove that A1 A2 is also a RL 2. Construct an NFA M to accept A1 A2 M 1 =(Q 1,, 1,q 0,F 1 ) M 2 =(Q 2,, 2,s 0,F 2 ) Construct M = (Q, {Q,,,t 0,F},F) to recognize L(M 1 ) [ L(M 2 ) Q = Q 1 [ Q 2 is unchanged t 0 = q 0 8 < { 1 (q, a)} q 2 Q 1 and a 2 (q, a) = {s 0 } q 2 F 1 and a = " : { 2 (q, a)} q 2 Q 2 and a 2 q 2 Q, a 2 " F = F 2 16

17 Closed under star 1. Given any RL A 2. Based on the definition, we can construct a DFA M for A 1. Prove that A * is also a RL 2. Construct an NFA M to accept A * " 17

18 Closed under star 1. Given any RL A 2. Based on the definition, we can construct a DFA M for A 1. Prove that A * is also a RL 2. Construct an NFA M to accept A * " " 18

19 Closed under star 1. Given any RL A 2. Based on the definition, we can construct a DFA M for A 1. Prove that A * is also a RL 2. Construct an NFA M to accept A * M 1 =(Q 1,, 1,q 0,F 1 ) M 2 =(Q 2,, 2,s 0,F 2 ) Construct M =(Q,,,t 0,F) to recognize A Q = {t 0 } [ Q 1 is unchanged 8 < { 1 (q, a)} q 2 Q 1 and a 2 (q, a) = {q 0 } q 2 F 1 and a = " : {q 0 } q = t 0 and a = " q 2 Q, a 2 " t 0 is the start state F = {t 0 } [ F 1 19

20 Closed under star 1. Given any RL A 2. Based on the definition, we can construct a DFA M for A 1. Prove that A * is also a RL 2. Construct an NFA M to accept A * M 1 =(Q 1,, 1,q 0,F 1 ) M 2 =(Q 2,, 2,s 0,F 2 ) Construct M =(Q,,,t 0,F) to recognize A Q = {t 0 } [ Q 1 is unchanged 8 < { 1 (q, a)} q 2 Q 1 and a 2 (q, a) = {q 0 } q 2 F 1 and a = " : {q 0 } q = t 0 and a = " q 2 Q, a 2 " t 0 is the start state F = {t 0 } [ F 1 20

21 Outline for today Regular expressions (REs) o Regular operations and FA o RE examples o Equivalence of FA and REs (next class) Ø Goals: o Learn how to write formal proofs o Understand more about regular operations o Learn to use REs 21

22 Definition Human-aware of Regular Expression Robo.cs 22

23 Regular Expression Notes about RE Ø A regular expression represents a language q When we use RE R with regular operations, it should be interpreted as L(R) Ø + and * > > [ q Ø In practice: 23

24 Regular Expression More notes o o o o o o o o R + = RR R = R + [ {"} ; = {"} R " = R R ;= ; R [ " = R? R ;is never equal to R? 24

25 RE example w w2 Exampe: L(M) = { {0, 1} * and 001 is a suffix of } w 25

26 RE example w w2 Exampe: L(M) = { {0, 1} * and 001 is a suffix of } w (0 [ 1) 001 or equivalently {0, 1}*001 26

27 RE example 27

28 RE example (aa [ aaa) b or equivalently {aa, aaa}*b 28

29 RE example 29

30 RE example a [ (ba (a [ b)a) + 30

31 Summary Regular expression are defined to be closed under regular operations We have shown in this lecture that o Regular languages are also closely under regular operations o We can express FA using REs Regular Language =? Regular Expressions (RE) 31

32 Outline for today Regular expressions (REs) o Regular operations and FA o RE examples o Equivalence of FA and REs (next class) Ø Goals: o Learn how to write formal proofs o Understand more about regular operations o Learn to use REs Reading assignment for the next class: o Sipser Sec. 1.3 (page Equivalence with FA) Quiz link will be sent out; due date is before the beginning of the next class 32