1 / 46

Finite Automata and Non Determinism

Finite Automata and Non Determinism. http://cis.k.hosei.ac.jp/~yukita/. Definition 1.1: Finite Automaton. State Diagram for M 1. 1. 0. 0. q 3. q 1. q 2. 1. 0, 1. Data Representation for M 1. Task 01 DFA. Implement M 1 with your favorite programming language. GUI

tea
Download Presentation

Finite Automata and Non Determinism

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Finite Automata andNon Determinism http://cis.k.hosei.ac.jp/~yukita/

  2. Definition 1.1: Finite Automaton

  3. State Diagram for M1 1 0 0 q3 q1 q2 1 0, 1

  4. Data Representation for M1

  5. Task 01DFA • Implement M1 with your favorite programming language. • GUI • Two buttons for input 0 and 1 • State chart with the current state highlighted

  6. Language of M1

  7. State Diagram for M5 0 q1 1 0, <RESET> 2, <RESET> 0 1 2 q0 q2 2 1, <RESET>

  8. Data Representation for M5

  9. Informal Description of M5 • M5 keeps a running count of the sum of the numerical symbols it reads, modulo 3. • Every time it receives the <RESET> symbol it resets the count to 0. • M5 accepts if the sum is 0, modulo 3.

  10. Definition 1.7: Regular Language • A language is called a regular language if some finite automaton recognizes it.

  11. Example 1.9: A finite automaton E2 • E2 recognizes the regular language of all strings that contain the string 001 as a substring. • 0010, 1001, 001, and 1111110011110 are all accepted, • but 11 and 0000 are not.

  12. Find a set of states of E2 You • haven’t just seen any symbols of the pattern, • have just seen a 0, • have just seen 00 or, • have just seen the entire pattern 001. Assign the states q,q0,q00, and q001 to these possibilities.

  13. Draw a State Diagram for E2 0, 1 1 0 q 0 q0 q00 q001 1 0 1

  14. Regular Operations on Languages

  15. Example 1.11

  16. Theorem 1.12 Closedness for Union

  17. Proof of Theorem 1.12 Proof of Th 1.12

  18. Construction of M Proof of Th 1.12

  19. Correctness You should check the following. • For any string recognized by M1 is recognized by M. • For any string recognized by M2 is recognized by M. • For any string recognized by M is recognized by M1 or M2. Proof of Th 1.12

  20. Theorem 1.13 Closedness for concatenation

  21. Nondeterminism • To prove Theorem 1.13, we need nondeterminism. • Nondeterminism is a generalization of determinism. So, every deterministic automaton is automatically a nondeterministic automaton.

  22. Nondetermistic Finite Automata • A nondeterministic finite automaton can be different from a deterministic one in that • for any input symbol, nondeterministic one can transit to more than one states. • epsilon transition • NFA and DFA stand for nondeterministic finite automaton and deterministic finite automaton, respectively.

  23. NFA N1 0,1 0,1 q3 1 q4 q1 q2 1 0,e

  24. Parallel world and NFA ... ... accept

  25. Example 1.14 NFA N2 0,1 q3 q4 q1 q2 0,1 1 0,1 Let language A consist of all strings over {0,1} containing a 1 in the third position from the end. N2 recognizes A.

  26. A DFA equivalent to N2 0 0 q010 q110 q000 q100 0 0 1 0 1 1 0 0 1 0 1 q011 q111 q001 q101 1 1 1

  27. Example 1.15 NFA N3 0 0 e 0 e 0 0 Let language A consist of all strings 0k , where k is a multiple of 2 or 3. N3 recognizes A.

  28. A DFA equivalent to N3 0, 1 q-1 1 1 1 1 1 1 q4 q5 q0 q１ q２ q3 0 0 0 0 0 0

  29. Example 1.16 NFA N4 q1 a b e a q3 q2 a,b N4 accepts e, a, baba, and baa. N4 does not accept b, nor babba.

  30. Definition 1.17: NFA

  31. 0,1 0,1 1 0,e 1 q3 q4 q1 q2 Example 1.18 NFA N1

  32. In what situation is Non Determinism relevant? • Von Neumann machines are deterministic. • However, there are many cases where machine specification is all we need.

  33. Theorem 1.19 • Every nondeterministic finite automaton has an equivalent deterministic finite automaton. • Def. The two machines are equivalent is they recognize the same language.

  34. Proof of Th. 1.19 Proof of Th 1.19

  35. Incorporate e arrows Proof of Th 1.19

  36. Corollary 1.20 • A language is regular if and only if some nondeterministic finite automaton recognizes it.

  37. Example 1.21 NFA N4 to DFA 1 b a e a 3 2 a,b

  38. Task 02Parallel World • Write a program that simulates N4. • GUI • Three buttons for input 0, 1, and epsion. • State chart that reflect the branching of the world.

  39. Start and Accept states

  40. The state diagram of D a,b a b {2} {1,2} f {1} a,b b b a b a a {2,3} {1,2,3} {3} a {1,3} a b b

  41. Theorem 1.22The class of regular languages is closed under the union operation. N N1 e e N2

  42. Proof of Th. 1.22

  43. N2 N1 Theorem 1.23The class of regular languages is closed under the concatenation operation. N e e

  44. Proof of Th. 1.23

  45. N1 Theorem 1.24The class of regular languages is closed under the star operation. N e e e

  46. Proof of Th. 1.24

More Related