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تمام اسب ها همرنگ هستند!

تمام اسب ها همرنگ هستند!. Basis: h = 1 We have only one horse. Clearly all horses in the set are the same color. Inductive hypothesis: In any set of up to h horses all the horses have the same color. Inductive step:

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تمام اسب ها همرنگ هستند!

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  1. تمام اسب ها همرنگ هستند! Basis:h = 1 • We have only one horse. Clearly all horses in the set are the same color. Inductive hypothesis: • In any set of up to h horses all the horses have the same color. Inductive step: • Consider a set of h + 1 horses. Discard one horse; by the induction hypothesis all theremaining horses have the same color. Put that horse back and discard a differenthorse; again all the remaining horses have the same color. So all the horses have the same color as the ones that were not discarded either time and so they all have the same color.

  2. یک رابطه تهی روی یک مجموعه ناتهی چه خصوصیاتی از خواص رابطه هم ارزی را دارد؟

  3. آیا می توان تناظر یک به یک برقرار کرد؟ • نقاط یک نیم دایره و یک خط راست • نقاط هر دو پاره خط دلخواه • مجموعه اعداد حقیقی بین 0 و 1 و مجموعه اعداد حقیقی • مجموعه اعداد حقیقی بین 0 و 1 و مجموعه اعداد حقیقی مثبت

  4. شمارش پذیری

  5. سلسله مراتب Chomsky

  6. گرامرها Design a grammar that generates the set of all strings over alphabet {a, b} that are not palindromes. Design a context free grammar for the following language:

  7. گرامرها برای هر یک از زبان های زیر یک گرامر بنویسید: برای مکمل زبان زیر یک گرامر مستقل از متن بنویسید.

  8. عبارات منظم عبارت زیر را به ساده ترین شکل ممکن بنویسید: ({a,aaaa}{aa,aaaaa} )*

  9. آیا جهت وارد شدن ورودی مؤثر است؟

  10. عبارت منظم متناظر با C3چیست؟

  11. تبدیل DFA به عبارت منظم متناظر 1 q0 q1 1 0 0 0 q2 1

  12. تبدیل DFA به عبارت منظم متناظر 1 ε q3 q0 q1 1 0 0 0 q2 1

  13. تبدیل DFA به عبارت منظم متناظر 1 ε q3 q0 q1 1 0 01*0

  14. تبدیل DFA به عبارت منظم متناظر 1 ε q3 q0 q1 1 0 01*0

  15. تبدیل DFA به عبارت منظم متناظر ε q3 q0 1(01*0)*1 0

  16. تبدیل DFA به عبارت منظم متناظر ε q3 q0 1(01*0)*1 0

  17. تبدیل DFA به عبارت منظم متناظر ε q3 q0 0 0 U 1(01*0)*1

  18. تبدیل DFA به عبارت منظم متناظر (0 U 1(01*0)*1)*

  19. مثال Consider the following DFA over {0, 1}: Q = {p, q, r, s}, start state p, accepting state p. With the following transition table: d (p,0) = s, d (p,1) = p d (q,0) = p, d (q,1) = s d (r,0) = r, d (r,1) = q d (s,0) = q, d (s,1) = r Write the regular expression recognized by this DFA.

  20. اثبات منظم بودن – روش constructive • Let L be a regular language. • Show that the following languages are also regular: • L1 = {v | vwL}, • (L1 is the set of all prefixes of the strings in L.) • L2 = {v | uv L} • (L2 is the set of all suffixes of the strings in L.)

  21. ماشین های حالت متناهی • A: • alphabet S, • start state q0, • a single accepting state qf, • transition function d that is such that for all symbols a in S, d (q0, a) = d (qf, a). • Show that if x is a nonempty string in L(A), then xx, xxx, ... xk is also in L (A).

  22. ماشین های حالت متناهی • Give DFAs for the following languages: • The set of all binary strings which do not have 010 as a substring • The complement of {1,10}* • آیا این عبارت درست است: • تنها یک زبان متناهی وجود دارد که توسط یک ماشین حالت متناهی با یک حالت پذیرفته می شود.

  23. Minimal-size DFA • Show that there is a DFA over {0, 1} with n+1 states that recognizes the language (1n)*. • Show that there does not exist a smaller DFA for this language. (smaller = fewer states).

  24. درست یا نادرست؟ • If L is a nonregular language and M is a regular language then their concatenation LM is not regular. • All finite languages L are regular.

  25. اثبات نامنظم بودن – برهان خلف {ai | i is a prime number} { 0i1j | i > j } The set of words that contain twice as many 0’s as 1’s.

  26. تمرین

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