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ATM, Halting Problem, P vs. NP. Chapter 4, 5 & 7. Russel’s Paradox. http://www.jimloy.com/logic/russell.htm An Index is a book that lists other books in the library Index of all biology text books in the library. Consider the index of all indices, i.e., book that lists other indices.

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ATM, Halting Problem, P vs. NP

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ATM, Halting Problem, P vs. NP

Chapter 4, 5 & 7


Russel’s Paradox

  • http://www.jimloy.com/logic/russell.htm

  • An Index is a book that lists other books in the library

    • Index of all biology text books in the library.

  • Consider the index of all indices, i.e., book that lists other indices.

    • It would contain itself.

    • Called a self-referential index (form of recursion).

  • Consider the index of all non self-referential indices.

    • The book that lists all indices that are non self-referential.

    • Call it the non-recursive index

  • If the non-recursive index lists itself then its recursive and shouldn’t be in the list.

  • But if it doesn’t list itself, then it is non-recursive and it should be listed as one of the non self-referential books.


ATM reduces to HALTTM

  • Proof by Contradiction

    • Assume we have a Turing Machine R that decides HALTTM.

      • In other words, we can design an algorithm that determines of another algorithm (M) will halt on a given input (w).


ATM reduces to HALTTM

  • Proof by Contradiction

    • Assume we have a Turing Machine R that decides HALTTM.

    • Then, use R to construct a new machine S that can decide ATM.

    • We have already proven that is NOT decidable.

    • So, if we can construct a machine S that can decide ATM based on HALTTM then we must conclude that cannot be decidable.


ATM reduces to HALTTM

  • Proof by Contradiction

    • Assume we have a Turing Machine R that decides HALTTM.

    • Then, use R to construct a new machine S that can decide ATM.

    • We have already proven that ATM is NOT decidable.

    • So, if we can construct a machine S that can decide ATM based on HALTTM then we must conclude that cannot be decidable.


Deeply Understanding ATM

  • ATM is just a set of strings where part of the string is the encoding of a Turing Machine (M) and the other part is an input string (w).

    • The string in ATM are the ones were the input w is accepted by the Machine M.

    • The string in the complement of are the ones the input w is rejected by the Machine M.

  • A decider for ATM cannot exist because there are absurd Turing Machines infinite loop or output nothing.

  • Detecting such absurdity is impossible because you are limited by the power of the same machine that allows for such absurdity.

    • Simulating an absurd Turing Machine with another Turing Machine cannot “undo” it absurdity and allow it to always produce an answer (accept or reject).


Chapter 7

Time Complexity


Big-O


Little-o


O(n2)


O(n log2n)


O(n)


Non-determinism


The Cost of Non-determinism


The Class P


Example of a problem in P


PATH is polynomial


The Class NP


Polynomial Verification

  • Some problems (Hamiltonian Path) cannot be solved in Polynomial Time (P), but…

  • If we could somehow obtain a solution (ask an oracle), the solution could be verified in P-time.


Hamiltonian Path: Decider vs. Verifier

  • There is very subtle difference:

  • Hamiltonian Decider

    • Input: Graph

    • Output: Hamiltonian Path

  • Hamiltonian Verifier

    • Input: Graph, Hamiltonian Path

    • Output: Accept/Reject


Hamiltonian Path Algorithm – Non-deterministic


Understand NP in English

  • Problems where the solutions can be verified in P-Time, but cannot be “discovered” in P-time

  • Problems where the solutions can be “discovered” P-Time using a Non-deterministic Turing Machine

  • Note: Every non-deterministic Turing machine can be made deterministic by adding O(kn)


Non-determinism  O(kn)

  • k is usually 2

    • Given a graph with n vertices, there are 2n possible subsets.

      • 2 options for each vertex

      • Either you are in the subset (1) on not (0)

      • Binary state condition

    • Given a list with n value, there are 2n possible subsets


Non-determinism  O(kn)

“Non deterministically select a subset implies 2n operations.”

  • This is a loop through a Turing machine state that has a branching non-deterministic transition

  • For each n vertex regardless of the tape symbol

    • Add (1) the vertex to the subset and continue

    • Don’t add (2) the vertex and continue


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