จำนวนสถานะ NUMBER OF STATES

1. จำนวนสถานะ NUMBER OF STATES

2. NUMBER OF STATES ประเด็นที่สนใจ The number of distinct states the finite state machine needs in order to recognize a language is related to the number of distinct strings that must be distinguished from each other.

3. NUMBER OF STATES สามารถแยกความแตกต่างได้ DISTINGUISHABLE นิยาม Let L be a language in *. Two strings x and y in * are distinguishable with respect to L if there is a string z in * so that exactly one of the strings xz and yz is in L. The string z is said to distinguish x and y with respect to L.

4. NUMBER OF STATES สามารถแยกความแตกต่างได้ DISTINGUISHABLE นิยาม Let L be a language in *. Two strings x and y in * are distinguishable with respect to L if L/x ≠ L/y where L/x = { z* | xz  L } L/y = { z* | yz  L }.

5. NUMBER OF STATES สามารถแยกความแตกต่างได้ DISTINGUISHABLE EXAMPLE Let  = { 0, 1 }. Let L be the language associated with (0+1)*10. Two strings x= 01101 and y= 010 in *. Since there is a string z= 0 in * such that xz = 011010 is in L but yz = 0100 is not in L, x and y are distinguishable with respect to L. We may say that x and y are indistinguishable with respect to L if there is no such string z. The strings 0 and 100 are indistinguishable with respect to L.

6. NUMBER OF STATES จำนวนสถานะ NUMBER OF STATES LEMMA Suppose that L  *, and M = (Q,,q0,A,). If x and y are two strings in * for which *(q0,x) = *(q0,y) then x and y are indistinguishable with respective to L. Note: *(q0,x)= qj means that there is a path from q0 to qj with respect to x: *(q0,x) = ((…((q0,x1),x2),…),xj) = qj where x = x1x2…xj.

7. NUMBER OF STATES จำนวนสถานะ NUMBER OF STATES LEMMA Suppose that L  *, and M = (Q,,q0,A,). If x and y are two strings in * for which *(q0,x) = *(q0,y) then x and y are indistinguishable with respective to L. Note: *(q0,x)= qj means that there is a path from q0 to qj with respect to x: *(q0,x) = ((…((q0,x1),x2),…),xj) = qj where x = x1x2…xj. Proof: Let z be any string in *. Consider xz and yz, We have that *(q0,xz) = *(*(q0,x),z) *(q0,yz) = *(*(q0,y),z). Then *(q0,xz) = *(q0,yz). Two strings xz and yz are either both in L or both not in L. Therefore, x and y are indistinguishable with respect to L. QED.

8. NUMBER OF STATES จำนวนสถานะ NUMBER OF STATES THEOREM Suppose that L  *, and for some positive integer n, there are n strings in *, any two of which are distinguishable with respect to L. Then there can be no finite state machine recognizing L with fewer than n states. Proof: Suppose x1,x2,…,xn strings are distinguishable with respect to L. Assume that M is a finite state machine with fewer than n states. By the pigeonhole principle, the state *(q0,x1), *(q0,x2),…*(q0,xn) cannot all be distinct, so for some ij *(q0,xi)=*(q0,xj). Since xi and xj are distinguishable with respect to L, it follows from Lemma that M cannot recognize L. QED.

9. NUMBER OF STATES จำนวนสถานะ NUMBER OF STATES EXAMPLE Let ={0,1}, and let Ln* Ln={x| |x|n and the nth character from the right in x is 1}. For instance, n = 2, all words accepted by L2 is of the form (1+0)*1(0+1). 1 1

10. NUMBER OF STATES จำนวนสถานะ NUMBER OF STATES EXAMPLE Let ={0,1}, and let Ln* Ln={x| |x|n and the nth character from the right in x is 1}. For instance, n = 2, all words accepted by L2 is of the form (1+0)*1(0+1). 1 0 1 0

11. NUMBER OF STATES จำนวนสถานะ NUMBER OF STATES EXAMPLE Let ={0,1}, and let Ln* Ln={x| |x|n and the nth character from the right in x is 1}. For instance, n = 2, all words accepted by L2 is of the form (1+0)*1(0+1). 1 0 0 1 0 0

12. NUMBER OF STATES จำนวนสถานะ NUMBER OF STATES EXAMPLE Let ={0,1}, and let Ln* Ln={x| |x|n and the nth character from the right in x is 1}. For instance, n = 2, all words accepted by L2 is of the form (1+0)*1(0+1). 1 0 0 1 1 0 0 1

13. NUMBER OF STATES จำนวนสถานะ NUMBER OF STATES EXAMPLE Let ={0,1}, and let Ln* Ln={x| |x|n and the nth character from the right in x is 1}. For instance, n = 2, all words accepted by L2 is of the form (1+0)*1(0+1). 1 0 0 1 1 1 0 0 1 1

14. NUMBER OF STATES จำนวนสถานะ NUMBER OF STATES EXAMPLE (1+0)*1(0+1) 0 1 01 1 0 0 0 1 10 1 0 00  0 0 1 0 1 1 11 1

15. NUMBER OF STATES จำนวนสถานะ NUMBER OF STATES THEOREM PALINDROME language over the alphabet {0,1} cannot be accepted by any finite automaton. Proof: Any two strings in {0,1}* are distinguishable with respect to PALINDROME language. QED. EXAMPLE For two strings, 010101 and 1011100 Rejected accepted 11101

16. 7 กราฟการส่งผ่าน TRANSITION GRAPHS

17. TRANSITION GRAPHS วัตถุประสงค์ More powerful machine: read one or more letters of input string at a time b b ab q1 q0 a b aab aba q3 q2 b

18. TRANSITION GRAPHS กราฟการส่งผ่าน TRANSITION GRAPHS นิยาม • A transition graph (abbreviated TG) is a 5-tuple (Q, , q0, , A) where • Q means a finite set of states. •  is a finite input alphabet. • q0  Q named Initial state. • A  Q , A is the set of all accepted states. •  is a function from Q* to P(Q), called transition function.

19. TRANSITION GRAPHS กราฟการส่งผ่าน TRANSITION GRAPHS EXAMPLE 0 2 aa b b b aa 3 1 b Language associated with this TG is (aa+b)*bbb.