CS355%20

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CS355 The Mathematics and Theory of Computer
Science

• Reducibility

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Mapping reducibility

• Being able to reduce problem A to problem B by
  using a mapping reducibility means that a
  computable function exists that converts
  instances of problem A to problem B.
• If we have such a conversion function called a
  reduction, we can solve A with a solver for B.
• The reason is that any instance of A can be
  solved by first using the reduction to convert it
  to an instance of B and then applying the solver
  to B.

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Computable function

• A function f S ? S is a computable function if
  some Turing machine M, on every input w, halts
  with just f(w) on its tape.
• All usual arithmetic operations on integers are
  computable functions.

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Mapping reducibility

• Language A is mapping reducible to language B,
  written A m B, if there is a computable function
  f S ? S, where for every w,
• w A ltgt f(w) B.
• The function f is called the reduction of A to B.

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Mapping reducibility
A
B
f
f reducing A to B
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Mapping reducibility

• A mapping reduction of A to B provides a way to
  convert questions about membership testing in A
  to membership testing in B.
• To test whether w A, we use the reduction f to
  map w to f(w) and test whether f(w) B.
• The term mapping reduction comes from the
  function or mapping that provides the means of
  doing the reduction.

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Mapping reducibility

• If one problem is mapping reducible to a second,
  previously solved problem, we can thereby obtain
  a solution to the original problem.
• Let us look at this.

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Reducibility

• If A reduces to B and B is decidable, then A is
  decidable
• To decide A, convert A to B and decide B.
• If A reduces to B and B is undecidable, then B is
  undecidable
• If B is decidable, then by above A is decidable,
  contradiction.

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Turing-recognisable

• If A m B and A is not Turing-recognisable then B
  is not Turing-recognisable.

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Completeness

• suppose we know that there is a problem A not in
  class X. By finding a reduction from A to B we
  can prove that B is not in X either.

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Completeness

• IF A, B are problems, X is a class of problems,
  and R is a class of reductions, then
• If B R A for every B X, we say that A is hard
  for X under R reductions, or, R-hard for X.

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Completeness

• If is also happens that A X, then we say that A
  is complete for X under R reductions or
  R-complete for X.

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Completeness

• Rather than looking at a specific class of
  reductions we will just look at m-reductions
  which we can use to get a more concise
  terminology.

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Completeness

• If A, B are languages and X is a class of
  languages,
• If B mp A for every B X, we say that A is
  X-hard (mphard for X).
• So, if A is X-hard then it is at least as hard as
  every problem in X.

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Completeness

• If A X and A is X-hard, then we say that
• A is X-complete(mp -complete for X).

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Completeness

• If X, Y are two classes of problems (or their
  associated languages) such that X Y, then
• Any language B that is Y-complete does not belong
  to X.

B
Y
X
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Completeness

• A is X-complete
• B is X-hard and Y-complete
• C is X-hard, Y-hard and Z-complete

B
A
C
Y
X
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Completeness

• The meaning of the word complete is not
  intuitive.
• It comes from Godels and Kleenes recursive
  function theory, and is used to denote that
• A solution to any problem in the set can be
  applied to all others in the set.

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Completeness

• Some points to note
• If A is X-complete it can be viewed as being
  typical of the hardest problems found in X.
• Problems which are hard or complete on one
  machine-based family of language X will have the
  same properties on all other machine models.

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Completeness

1. Almost all reductions establishing hardness or
   completeness of specific problems not only
   transform solvable instances into solvable
   instances, but also general solutions into
   general solutions.

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Completeness

1. Many complete problems have the property that one
   can efficiently compute a solution as soon as one
   can prove that a solution exists.
2. Solving one PSPACE-complete problem in polynomial
   time implies that all PSPACE problems are in P.