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Time Complexity in Deterministic Turing Machines for Language Decidability

Explore the time complexity of deterministic Turing Machines in deciding languages and determining the maximum time required to decide any string length. Examples of tractable and intractable problems are discussed, such as Hamiltonian Path and Clique Problems. Non-deterministic algorithms and the Satisfiability Problem are also analyzed.

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Time Complexity in Deterministic Turing Machines for Language Decidability

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  1. Time Complexity Costas Busch - LSU

  2. Consider a deterministic Turing Machine which decides a language Costas Busch - LSU

  3. For any string the computation of terminates in a finite amount of transitions Initial state Accept or Reject Costas Busch - LSU

  4. Decision Time = #transitions Initial state Accept or Reject Costas Busch - LSU

  5. Consider now all strings of length = maximum time required to decide any string of length Costas Busch - LSU

  6. TIME STRING LENGTH Max time to decide string of length Costas Busch - LSU

  7. Time Complexity Class: All Languages decidable by a deterministic Turing Machine in time Costas Busch - LSU

  8. Example: This can be decided in time Costas Busch - LSU

  9. Other example problems in the same class Costas Busch - LSU

  10. Examples in class: Costas Busch - LSU

  11. Examples in class: CYK algorithm Matrix multiplication Costas Busch - LSU

  12. Polynomial time algorithms: constant Represents tractable algorithms: for small we can decide the result fast Costas Busch - LSU

  13. It can be shown: Costas Busch - LSU

  14. The Time Complexity Class Represents: • polynomial time algorithms • “tractable” problems Costas Busch - LSU

  15. Class CYK-algorithm Matrix multiplication Costas Busch - LSU

  16. Exponential time algorithms: Represent intractable algorithms: Some problem instances may take centuries to solve Costas Busch - LSU

  17. Example: the Hamiltonian Path Problem s t Question: is there a Hamiltonian path from s to t? Costas Busch - LSU

  18. s t YES! Costas Busch - LSU

  19. A solution: search exhaustively all paths Exponential time Intractable problem Costas Busch - LSU

  20. The clique problem Does there exist a clique of size k? Costas Busch - LSU

  21. The clique problem Does there exist a clique of size k? Costas Busch - LSU

  22. Example: The Satisfiability Problem Boolean expressions in Conjunctive Normal Form: clauses Variables Question: is the expression satisfiable? Costas Busch - LSU

  23. Example: Satisfiable: Costas Busch - LSU

  24. Example: Not satisfiable Costas Busch - LSU

  25. exponential Algorithm: search exhaustively all possible binary values of the variables Costas Busch - LSU

  26. Non-Determinism Language class: A Non-Deterministic Turing Machine decides each string of length in time Costas Busch - LSU

  27. All computations of on string … … depth … accept … reject accept (deepest leaf) reject Costas Busch - LSU

  28. Non-Deterministic Polynomial time algorithms: Costas Busch - LSU

  29. The class Non-Deterministic Polynomial time Costas Busch - LSU

  30. The satisfiability problem Example: Non-Deterministic algorithm: • Guess an assignment of the variables • Check if this is a satisfying assignment Costas Busch - LSU

  31. Time for variables: • Guess an assignment of the variables • Check if this is a satisfying assignment Total time: Costas Busch - LSU

  32. The satisfiability problem is a - Problem Costas Busch - LSU

  33. Observation: Deterministic Polynomial Non-Deterministic Polynomial Costas Busch - LSU

  34. Open Problem: WE DO NOT KNOW THE ANSWER Costas Busch - LSU

  35. Open Problem: Example: Does the Satisfiability problem have a polynomial time deterministic algorithm? WE DO NOT KNOW THE ANSWER Costas Busch - LSU

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