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Example

Example. Find a DTM that computes the language L = { a n b n : n ≥ 0}. High level program: Repeat Erase an a. Pass along the rest of as. Erase a b Pass along the rest of bs . Go to the beginning of the input.

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Example

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  1. Example • Find a DTM that computes the language L = {anbn : n ≥ 0}. High level program: • Repeat • Erase an a. • Pass along the rest of as. • Erase a b • Pass along the rest of bs. • Go to the beginning of the input. • Until either the whole input is erased (accept) or you find unmatched as or bs (reject).

  2. Example • Find a DTM that computes the language L = {anbn : n ≥ 0}. Answer: a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  3. Testing Test the machine for several possible inputs to see if it works as it should. Test inputs: • ε (it should accept) • aaabbb (it should accept) • aaabb (it should reject) • aabbb (it should reject) • aabba (it should reject)

  4. Test input: ε … … a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  5. Test input: ε … … a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  6. Test input: aaabbb … … a a a b b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  7. Test input: aaabbb … … x a a b b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  8. Test input: aaabbb … … x a a b b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  9. Test input: aaabbb … … x a a b b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  10. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  11. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  12. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  13. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  14. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  15. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  16. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  17. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  18. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  19. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  20. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  21. Test input: aaabbb … … x a a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  22. Test input: aaabbb … … x x a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  23. Test input: aaabbb … … x x a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  24. Test input: aaabbb … … x x a x b b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  25. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  26. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  27. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  28. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  29. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  30. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  31. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  32. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  33. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  34. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  35. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  36. Test input: aaabbb … … x x a x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  37. Test input: aaabbb … … x x x x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  38. Test input: aaabbb … … x x x x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  39. Test input: aaabbb … … x x x x x b a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  40. Test input: aaabbb … … x x x x x x a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  41. Test input: aaabbb … … x x x x x x a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  42. Test input: aaabbb … … x x x x x x a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  43. Test input: aaabbb … … x x x x x x a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  44. Test input: aaabbb … … x x x x x x a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  45. Test input: aaabbb … … x x x x x x a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  46. Test input: aaabbb … … x x x x x x a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  47. Test input: aaabbb … … x x x x x x a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  48. Test input: aaabbb … … x x x x x x a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  49. Test input: aaabbb … … x x x x x x a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

  50. Test input: aaabbb … … x x x x x x a → a , R x → x , R x → x, R b → b , R a → x, R b → x, R q0 q1 q2 □ → □ , R □ → □ , L □ → □ , R qf q3 a → a , L b → b , L x → x , L

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