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ภาษาสม่ำเสมอ REGULAR LANGUAGES

10. ภาษาสม่ำเสมอ REGULAR LANGUAGES. REGULAR LANGUAGES. วัตถุประสงค์. Language defined by regular expression is called a regular languages . All languages are regular ? (YES/NO) Discuss some properties of all regular languages. REGULAR LANGUAGES. ภาษาสม่ำเสมอ. REGULAR LANGUAGES.

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ภาษาสม่ำเสมอ REGULAR LANGUAGES

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  1. 10 ภาษาสม่ำเสมอ REGULAR LANGUAGES

  2. REGULAR LANGUAGES วัตถุประสงค์ Language defined by regular expression is called a regular languages. All languages are regular ? (YES/NO) Discuss some properties of all regular languages.

  3. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES Let L1 and L2 be regular languages. Then L1+L2, L1L2 and L1* are also regular languages. Proof: By definition of regular expression, regular expression is closed under union, concatenate and Kleene star. ทฤษฎีบท What’s about finite automata?

  4. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES Find L1+L2 b a q1 q0 b a a a  q6 b q2 q3  b q4 b a,b q5 a

  5. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES b Find L1L2 a q1 q0 b a a a b q2 q3 b  q4 b a,b q5 a

  6. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES Find L1* b a q1  q0 b qk a a a b q2 q3 b 

  7. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES นิยาม If L is a language over the alphabet . The complement of L denoted by L’ is the language of all strings of characters from  that are not words in L. The complement of L = * - L. Notice that the complement of L’ is L. ( (L’)’=L.) COMPLEMENT

  8. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES Let L be a regular language. Then L’ is also a regular language. (closed under complementation) Proof: By Kleene’s theorem, there is some FA that accepts the language L. By reversing all final states in that FA into non-final states, and all non-final states into final states, the new machine is a FA that accepts L’. By Kleene’s theorem, L’ is a regular language. QED. ทฤษฎีบท

  9. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES EXAMPLE: Given ={a, b}, and L = a(a+b)*. Find L’. a, b q1 a, b a q2 q0 b

  10. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES EXAMPLE: Given ={a, b}, and L = a(a+b)*. Find L’. a, b q1 a, b a q2 q0 b This FA accepts all words started with b, and also . L’ = +b(a+b)*

  11. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES INTERSECTION Let L1 and L2 be regular languages. Then L1L2 is also a regular language. Proof: By DeMorgan’s law for sets, L1L2 = (L1’+L2’)’. So, L1L2 is a regular language. QED. ทฤษฎีบท How to construct a finite automaton associated with the language?

  12. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES L1 = language((a+b)*aa(a+b)*) L2 = language(b*(ab*ab*)*) a, b a a q0 q1 q2 b b a b q4 q3 a b

  13. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES L1 = language((a+b)*aa(a+b)*) L2 = language(b*(ab*ab*)*) a, b a a q0 q1 q2 b b a b q4 q3 a b Find L1’ and L2’

  14. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES b a b a a a 0,3 2,4 1,4 2,3 b b b a a 1,3 b 0,4 a, b a a q0 q1 q2 b b a b q4 q3 a b

  15. REGULAR LANGUAGES ภาษาสม่ำเสมอ REGULAR LANGUAGES SUMMARY Let L1 L2 be regular languages associated with machines M1=(Q1,,q1,A,1) and M2=(Q2,,q2,A,2). Let M =(Q1Q2,,(q1 ,q2),A,) and transition function is defined by the formula ((p,q),a) = (1(p,a),2(q,a)). Then if A={(p,q)|pA1 or qA2}, M accepts the L1L2. if A={(p,q)|pA1 and qA2}, M accepts the L1L2. if A={(p,q)|pA1 and qA2}, M accepts the L1-L2. ทฤษฎีบท

  16. โจทย์น่าคิด จงหาออโตมาตาจำกัด สำหรับภาษา L1L2 L1 0 B ? 1 A 1,0 0 C 0,1 0 L2 1 Y X

  17. 11 REGULARDECIDABLE

  18. REGULAR DECIDABLE คุณสมบัติความสม่ำเสมอ REGULARITY A language is regular (describable by a regular expression) if and only if it can be accepted by a finite automaton. What inherent property of a language identifies it as being regular ?

  19. REGULAR DECIDABLE คุณสมบัติความสม่ำเสมอ REGULARITY PALINDROME is not regular. Since there are infinitely many distinguishable. Remark Language L contains infinitely many “pairwise distinguishable” with respect to L then L cannot be regular.

  20. REGULAR DECIDABLE คุณสมบัติความสม่ำเสมอ REGULARITY What is the relationship between regular and distinguishable ? Define a relation: We will say that two strings areequivalentif they areindistinguishablewith respect to L.

  21. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ Let L be any language in *. The relation IL on *( the indistinguishability relation) is defined as follows: For any two strings x, y in *, xILy if and only if x and y are indistinguishable with respect to L. In other words, xILy if for any z in *, either xz and yz are both in L or xz and yz are both in L’.

  22. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ตัวอย่าง L be a language over =(0, 1}, defined as follows; x  L with length(x) > 0, x does not contain “double characters”. Regular expression of L = (0+)(10)*(1+). For instance, 0101 IL 10101 since they are indistinguishable with respect to L. For any z *, if 0101z is in L, then 10101z is also in L, if 0101z is not in L, then 10101z is not in L.

  23. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ LEMMA For any language L, IL is an equivalence relation on *. Let x, y and z be strings in *. Reflexive: xILx Symmetric: if xILy then yILx Transitivity: if xILy and yILz then xILz.

  24. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ Proof It is obvious for reflexive and symmetric. Let x, y and z be strings in *. Given xILy and yILz, and for any a  *. Suppose that xa is in L, we will show that za is in L. Since xa is in L and xILy, ya is also in L. Since ya is in L and yILz, za is in L. thus xILz. Q.E.D.

  25. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ตัวอย่าง L be a language over =(0, 1}, defined as follows; x  L with length(x) > 0, x does not contain “double characters”. Regular expression of L = (0+)(10)*(1+). [] = {  } [1] = { 1, 01, 101, 0101, …} [0] = { 0, 10, 010, 1010, ... } [00+11] = { 00, 11, 000, 011, 111, … }

  26. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ข้อสังเกตุ If the set of all equivalence classes of IL is finite, then it is possible to construct an DFA recognizing L in terms of the equivalence classes of IL.

  27. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ข้อสังเกตุ Automaton M=(Q, , q0, A, ) Lq = { x   | *(q0, x) = q } for q  Q. Lq is the set of all strings that end in the state q of M. What is the relationship between them ?

  28. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ตัวอย่าง 0 1 C B 0 1 0  0,1 A 0 1 1 E LA = {  } LB = { 0, 010, 01010, 0101010, … } LC = { 01,0101, 010101, 01010101, … } LD = { 1, 101, 10101, 1010101, … } LE = { 10, 1010, 101010, 10101010, … } L = { 00, 11, 000, 001, 100, 111, … } D 0 1

  29. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ หมายเหตุ • If number of classes of IL and Lq are the same, then • two partitions are identical and • FA is the fewest possible states recognizing L. • For strings x in * and a in  • ([x], a) = [xa].

  30. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ LEMMA IL is right invariant with respect to concatenation. For any x, y in *, and any a in , if x IL y, then xa IL ya. Equivalently, if [x] = [y], then [xa] = [ya]. Proof: Let x IL y. Then x and y are indistinguishable with respect to L. For any z in *, if xz is in L, then yz is in L. Consider xaz, for any a in *, if xaz is in L and let z’ = az, then yz’ is also in L. Thus xa IL ya. Q.E.D.

  31. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ทฤษฎีบท • Let L  *, and QL be the set of equivalence classes of the relation IL on *. • If QL is a finite set, then ML = (QL, , q0, AL, ) is a finite automaton accepting L, where • q0 = [] • AL = { q in QL | q  L  } and • : QL    QL is defined by ([x],a) = [xa]. • Furthermore, • ML has the fewest states of any FA accepting L.

  32. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ พิสูจน์ If QL is a finite set, then ML is finite. Now, ML= (QL, , q0, AL, ) recognizes the language L. For any x  *, x  L if and only if *(q0,x)  AL. Let x L. Since x  [x], we have that x  [x]  L  . Since *(q0,x) = ([],x) = [x]  AL. For u, v  *, (([u], v) = [uv]) If *(q0,x)  AL, then [x]  L  . Let y be an element in [x]  L. We have that y  L. x and y are indistinguishable (same class), then x  L.

  33. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ทฤษฎีบท MYHILL-NERODE THEOREM L is a regular if and only if QL is finite. QL is finite, ML is also finite. ([x],y) = [xy] for any strings x and y in *. The partition Lq is finer than the partition IL.

  34. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ตัวอย่าง Let L = { x in {0, 1}* | x ends with 10 }. Consider three strings, , 1 and 10. Ant two of these strings are distinguishable with respect to L. [] = { , 0, 1, 00, 000, 100, 0000, 0100, … } [1] = { 1, 01, 001, 0001, 00001, … } [10] = { 10, 010, 110, 0010, … }.

  35. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ตัวอย่าง • Let L = { x in {0, 1}* | x ends with 10 }. • Consider three strings, , 1 and 10. Ant two of these strings are distinguishable with respect to L. • ML = (QL, {0, 1}, [], {[10]}, ) be the FA, and • ([], 0) = [] ([], 1) = [1] • ([1], 0) = [10] ([1], 1) = [1] • ([10], 0) = [] ([10], 1) = [1].

  36. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ตัวอย่าง Let L = { x in {0, 1}* | x ends with 10 }. Consider three strings, , 1 and 10. Ant two of these strings are distinguishable with respect to L. 1 0 DFA 0 1 [10] [] [1] 1 0

  37. โจทย์น่าคิด กำหนดให้ ภาษา0n1nเมื่อ n  0 { , 01, 0011, 000111, … } จงแสดงให้เห็นว่า ภาษานี้ไม่เป็นภาษาสม่ำเสมอ ?

  38. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ตัวอย่าง L = { 0n1n | n  0 }, show that L is not regular. Consider any strings of the form 0i and 0j with i  i. They are distinguished by the string 1i. They are infinitely many strings of the form 0i and 0j. Then there are infinitely many distinguished strings.

  39. โจทย์น่าคิด กำหนดให้ ภาษาwwเมื่อ w * { , 00, 11, 0000, 0101, … } จงแสดงให้เห็นว่า ภาษานี้ไม่เป็นภาษาสม่ำเสมอ ?

  40. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ตัวอย่าง L = { ww| w {0, 1}* }, show that L is not regular. Consider any strings of the form 0i and 0j with i  i. They are distinguished by the string 1i0i1i. They are infinitely many strings of the form 0i and 0j. Then there are infinitely many distinguished strings.

  41. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ MINIMAL FINITE STATE MACHINE Number of equivalence classes = number of states in ML then ML is a minimal finite state machine. Find a pair (Lp, Lq) such that Lp and Lq are in the same class, group it into one class. By contraposition, we will find a pair (Lp, Lq) that they are in difference classes.

  42. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ MINIMAL FINITE STATE MACHINE LEMMA For p and q in Q, p  q, if and only if There exists z in * so that exactly one of the two states *(p,z) and (q,z) is in A. A pair (p, q) of states for which Lp and Lq are subsets of different equivalence classes, denoted by p  q.

  43. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ MINIMAL FINITE STATE MACHINE LEMMA For p and q in Q, p  q, if and only if there exists z in * so that exactly one of the two states *(p,z) and (q,z) is in A. Proof: Let p  q. Let x is in Lp and y is in Lq, we have that there exists z in *, xz and yz are distinguishable, *(p,z) = *(*(q0,x),z) = *(q0,xz) *(q,z) = *(*(q0,y),z) = *(q0,yz) only one is in L. Suppose that only one of *(p,z) and (q,z) is in A. It means that z distinguishes x in Lp and y in Lq with respect to L. Therefore p  q.

  44. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ MINIMAL FINITE STATE MACHINE Consider (p,q) be a distinguishable pair. Let r and s be two states in Q. For some a in , such that (r,a) = p and (s,a) = q. Since *(r,az) = *(*(r,a)) = *(p,z) *(s,az) = *(*(s,a)) = *(q,z), we can conclude that (r,s) is also be a distinguishable pair.

  45. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ ALGORITHM MINIMAL FINITE STATE MACHINE ตัวอย่าง 0 1 5 1 2 0 0 1 6 1 0 1 4 0 0 1 0 1 3 7 1 (0+1)*(01)

  46. REGULAR DECIDABLE INDISTINGUISHABILITY ความสามารถในการแยกความไม่แตกต่างกันได้ MINIMAL FINITE STATE MACHINE Algorithm State class class class #1 #2 #3 1 1 2 1,2 3 3 4 1,2,4 5 3,5 6 6 7 3,5,7

  47. REGULAR DECIDABLE NONREGULAR ความไม่สม่ำเสมอ A Language that cannot be defined by a regular expression is called a nonregular language.

  48. REGULAR DECIDABLE NONREGULAR ความไม่สม่ำเสมอ บทวิเคราะห์ Given an infinite regular language L Suppose that there exists a FA with n states associated with L. For any word x in L such that |x|  n, • consider a path associated with x, q0q1q2…qn+1 • then, there exists qi = qj where i j.

  49. REGULAR DECIDABLE NONREGULAR ความไม่สม่ำเสมอ บทวิเคราะห์ a2a*a7b10 accepted a2a2a7b10 accepted a a a a a 1 2 3 9 a a a10b10 a a a2(ak)*a6b10 accepted a2a2ka6b10 accepted FA with 9 states

  50. REGULAR DECIDABLE NONREGULAR ความไม่สม่ำเสมอ ทฤษฎีบท Suppose L is a regular language recognized by a finite automaton with n states. For any u  L, with |u| n, there are strings x, y and z so that u = xyz |xy| n y   For any m  0, xymz  L.

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