2017/08/29 Chapter 1.2 in Sipser Ø Announcement:

Transcription

1 Nondeterministic Human-aware Finite Robo.cs Automata 2017/08/29 Chapter 1.2 in Sipser Ø Announcement: q Piazza registration: q First poll will be posted on Piazza soon q Slides for this lecture are here: Lectures/NFA.pdf 1

2 Last time 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 2

3 Limitations Human-aware of FA discussed Robo.cs so far? Limitations of deterministic FA o A single sequence of steps sequential computation What if we need to express parallelism o e.g., threads and processes 3

4 Nondeterministic Human-aware Computation Robo.cs More than one way to perform computation at a step (Q,,,q0,F) For FA, more than one state to transition to in a state! 4

5 Nondeterministic Human-aware Computation Robo.cs 1. Given any RLs A1 and A2 2. Based on the definition, we can construct 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 a machine 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 5

6 Outline for today Nondeterministic finite automata o Definition of NFA o Computation of NFA o Equivalence of DFA and NFA 6

7 Definition of NFA a b " q1 {} {q2} {q3} q2 {q2, q3} {q3} {} q3 {q1} {} {} " = {"} [ P(Q), or the power set of Q : the set of all possible subsets of Q 7

8 Definition of NFA (Q,,,q0,F) Notes about NFA a b " q1 {} {q2} {q3} q2 {q2, q3} {q3} {} q3 {q1} {} {} o o Multiple transitions out of a state on the same symbol Transitions on the empty symbol are allowed nondeterminism o Not required to have a transition on every symbol 8

9 Outline for today Nondeterministic finite automata o Definition of NFA o Computation of NFA o Equivalence of DFA and NFA 9

10 Computation of NFA a b " q1 {} {q2} {q3} q2 {q2, q3} {q3} {} q3 {q1} {} {} 10

11 Computation of NFA reject? discard! 11

12 Computation of NFA discard! 12

13 Computation of NFA 13

14 Computation of NFA discard! 14

15 Computation of NFA Reject 15

16 Computation of NFA a b " q1 {} {q2} {q3} q2 {q2, q3} {q3} {} q3 {q1} {} {} Computation o All computation stops in an non-accept state: reject o A single computation stops in an accept state: accept o 16

17 Computation of NFA q1 reading a discard q3 Start reading b q1 q3 discard reading a q2 q2 q3 reading b discard q3 discard & reject 17

18 Computation of NFA q1 reading a discard q3 Start reading b q1 q3 discard What is the langauge of this NFA? reading a q2 q2 q3 reading b discard q3 discard & reject 18

19 NFA example Exampe: L(M) = { {0, 1} * and 001 is a suffix of } o Pretend that you are the machine reading the string o Identify how much memory you need or what are the states you need to remember? What are the states w w2 w Is this correct now? DFA 19

20 NFA example Exampe: L(M) = { {0, 1} * and 001 is a suffix of } o Pretend that you are the machine reading the string o Identify how much memory you need or what are the states you need to remember? What are the states w w2 w 20

21 NFA example 21

22 Outline for today Nondeterministic finite automata o Definition of NFA o Computation of NFA o Equivalence of DFA and NFA 22

23 Equivalence of DFA and NFA 23

24 Equivalence of DFA and NFA M =(Q 0,, 0,q 0 0,F 0 ) What are the states for the DFA? 24

25 Equivalence of DFA and NFA M =(Q 0,, 0,q0,F 0 0 ) What are the states for the DFA? What is the transition function? 25

26 Equivalence of DFA and NFA M =(Q 0,, 0,q0,F 0 0 ) What are the states for the DFA? What is the transition function? What is the initial state? 26

27 Equivalence of DFA and NFA M =(Q 0,, 0,q0,F 0 0 ) What are the states for the DFA? What is the transition function? What is the initial state? What are the final states? 27

28 Equivalence of DFA and NFA M =(Q 0,, 0,q0,F 0 0 ) What are the states for the DFA? What is the transition function? What is the initial state? What are the final states? 28

29 Equivalence of DFA and NFA M =(Q 0,, 0,q0,F 0 0 ) What are the states for the DFA? What is the transition function? What is the initial state? What are the final states? Remove unreachable states 29

30 Equivalence of DFA and NFA Given an NFA N = a DFA M =(Q 0,, 0,q0,F 0 0 ) to recognize A: o o o o (Q,,,q0,F) Q 0 = P(Q) 8 (R,a)2Q0 0 (R, a) =E( [ that recognizes A, we need to contruct r2r (r, a)) q 0 0 = E({q 0 }) F 0 = {R R 2 Q 0 ^9r(r 2 R ^ r 2 F )} E(R) ={q q can be reached from a state in R going alone zero or more " transitions} 30

31 Equivalence of DFA and NFA Corollary: NFA and DFA are equivalent o Every NFA has an equivalent DFA (just proven) o Every DFA has an equivalent NFA 31

32 Equivalence of DFA and NFA Corollary: NFA and DFA are equivalent o Every NFA has an equivalent DFA (just proven) o Every DFA has an equivalent NFA Corollary: A language is regular is and only if some NFA recognizes it 32

33 Outline for today Nondeterministic finite automata o Definition of NFA o Computation of NFA o Equivalence of DFA and NFA Reading assignment for the next class: o Sipser Sec. 1.1, 1.2 (page 58-66, up to Equivalence with Finite Automata) Quiz link will be sent out; due date is before the beginning of the next class 33