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Understanding Time Complexity and Turing Machines in Computability Theory

This document explores the concept of time complexity as it relates to deterministic Turing machines and the languages they can decide. It defines decision time, illustrates examples of polynomial and exponential time algorithms, and discusses tractable and intractable problems using examples like the Hamiltonian Path Problem and the Satisfiability Problem. The text emphasizes the distinction between polynomial and non-deterministic polynomial time, raising open questions about whether a polynomial time deterministic algorithm exists for NP-complete problems.

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Understanding Time Complexity and Turing Machines in Computability Theory

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  1. Time Complexity Costas Busch - RPI

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

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

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

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

  6. TIME STRING LENGTH Max time to accept a string of length Costas Busch - RPI

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

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

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

  10. Examples in class: Costas Busch - RPI

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

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

  13. It can be shown: Costas Busch - RPI

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

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

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

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

  18. s t YES! Costas Busch - RPI

  19. A solution: search exhaustively all paths L = {<G,s,t>: there is a Hamiltonian path in G from s to t} Exponential time Intractable problem Costas Busch - RPI

  20. The clique problem Does there exist a clique of size 5? Costas Busch - RPI

  21. The clique problem Does there exist a clique of size 5? Costas Busch - RPI

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

  23. Example: Satisfiable: Costas Busch - RPI

  24. Example: Not satisfiable Costas Busch - RPI

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

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

  27. Non-Deterministic Polynomial time algorithms: Costas Busch - RPI

  28. The class Non-Deterministic Polynomial time Costas Busch - RPI

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

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

  31. The satisfiability problem is an - Problem Costas Busch - RPI

  32. Observation: Deterministic Polynomial Non-Deterministic Polynomial Costas Busch - RPI

  33. Open Problem: WE DO NOT KNOW THE ANSWER Costas Busch - RPI

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

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