Turing Machines

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Turing Machines
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Learning Objectives

• Understand what are Turing machines.
• Understand how to use Turing machines to
  recognize sets, compute functions.
• Understand what are computational complexity,
  computability, and decidability.

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Turing Machines

• A Turing machine T (S, I, f, s0) consists of a
  finite set of states S, and alphabet I containing
  the blank symbol B, a partial function f from S x
  I to S x I x R, L, and a starting state s0,
  where R represents right direction, and L
  represents left direction.
• When f(s, x) is defined, it is a unique triple
  (s, x, d)f(x, s) ? (s, x, d).
• Transition rule five-tuple (s, x, s, x, d).

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Turing Machines

• Control unit loop
• Start at leftmost nonblank cell.
• Read current tape symbol x.
• If f(x, s) is defined, then
• Enter state s.
• Write symbol x in current cell, erasing x.
• Move right or left one cell depending on d.
• If f(x,s) is not defined, then Halt.

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• (s0, 0, s0, 0, R)

• (s0, 1, s1, 1, R)

• (s0, B, s3, B, R)

• (s1, 0, s0, 0, R)

• (s1, 1, s2, 0, L)

• (s1, B, s3, B, R)

• (s2, 1, s3, 0, R)

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Turing Machines for Sets

• Turing machines can be built to recognize sets.
• If V is a subset of an alphabet UI, a Turing
  machine T (S, I, f, s0) recognizes a string x
  in V if and only if T, staring in the initial
  position when x is written on the tape, halts in
  a final state.
• The machine can recognize a set iff it recognizes
  the strings in the set
• The machine does not recognize the set if it does
  not halt or does not halt in a final state.
• Examples Turing Machine to recognize the set
  0 U 1. Turing Machine to recognize the
  set (0 U 1) 1.
• Turing Machine to recognize the set (0 U 1) 1
  (0 U 1) .

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Turing Machines for Functions

• Turing machines can be used to compute functions
  computer that calculates partial functions.
• T(x) y - Turing machine that, given string x
  as input, halts with string y on tape.
• For integer functions, we need to define Turing
  machines on k-tuples of integers.
• Unary representation of integers
• Integer n ? 11111 (n1 times), 0 ? 1
• k-tuple (n1, n2, , nk) ? (n11) 1s
  
  (n2 1) 1s
  
  
  (nk 1) 1s
• Example (2, 0, 1, 3) ?

1111111111
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Turing Machines for Functions

• Example Turing machine for adding two
  nonnegative integers
• Different types of Turing machines
• Move at each step, or not at all.
• Operate on multiple tapes.
• Multiple heads.
• Non deterministic multiple (x, s)

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The Church-Turing Thesis

• Church-Turing Thesisfor any problem that can be
  solved with an effective algorithm, there is a
  Turing machine that can solve this problem.
• Effectively solvable that can be solved using a
  computer with a program written in any language,
  eventually using an unlimited amount of memory.

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Computational Complexity, Computability,
Decidability

• Computational complexity can be expressed in
  varied ways, and through a Turing machine.
• A decision problem asks whether statements from a
  particular class of statements are true
  (yes-or-no problems).
• Example is n prime?Recognize the language
  corresponding to the input values for which the
  answer to the problem is yes.
• Decidable / solvable problem one for which there
  is an effective algorithm that decides whether
  instances of this problem are true.
• Otherwise undecidable / unsolvable problem.

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Computational Complexity, Computability,
Decidability

• The halting problem is the decision problem that
  asks whether a Turing machine T eventually halts
  when given an input string x
• The halting problem is an unsolvable decision
  problem no Turing machine exists that, when
  given an encoding of a Turing machine T and its
  input string x as input, can determine whether T
  eventually halts when started with x written on
  its tape.

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Computational Complexity, Computability,
Decidability

• Computability a function that can be computed by
  a Turing machine is called computable.
• Otherwise uncomputable.
• Class P class of polynomial-time problems
  problems that can be solved by a deterministic
  Turing machine in polynomial time in terms of the
  size of its input.
• Class NP class of problems that can be solved
  in polynomial time by nondeterministic Turing
  machine.
• Problems P ? tractable (Ex prime number
  determination).
• Problems NP ? intractable (Ex Hamilton circuit
  in graph determination).
• All problem in P is on NP, but we do not know
  about the reverse order