Problem solving

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Lecture 4

• Topics
• Problem solving
• Subroutine Theme
• REC language class
• The class of solvable problems
• Closure properties

2
Problem Solving

• Solving problems
• In this course, we spend a lot of time proving
  there are unsolvable problems
• In todays class, we will spend time doing the
  opposite solving problems
• Subroutines
• Using an algorithm for one problem to help build
  an algorithm to solve a second problem

3
Problem 1

• Prime Number Problem
• Input Positive integer n
• Yes/No Question Is n a prime number?
• Questions
• What kind of problem is this?
• Construct a solution to this problem

4
Problem 2

• Maximum clique problem
• Input Graph G (V,E)
• list of nodes and edges
• Output Size of maximum clique in G
• Definitions
• V Set of n nodes vi 1 lt i lt n
• E Set of edges (vi, vj)
• Clique subset C of V such that there is an edge
  between all nodes in C

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Example
v1
v2
v3
v6
v4
v5
V v1, v2, v3, v4, v5, v6 E (v1, v2), (v1,
v5), (v1, v6), (v2, v3), (v2, v4), (v3, v4),
(v4, v5), (v5, v6) Question How could we
encode a graph as a string?
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Example 2
v1
v2
v3
v5
v6
v4
V v1, v2, v3, v4, v5, v6 E Max clique
size? Smallest number of edges needed to increase
m. c. size?
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Algorithm 1

• Construct an algorithm for solving the max clique
  problem

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Problem 2

• Decision clique problem
• Input Graph G (V,E), integer k
• Note the extra input object k
• Yes/No question
• Does the maximum clique in G have size at least k?

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Subroutine Theme

• Assume that algorithm A solves the decision
  clique problem
• Construct an algorithm A for solving the max
  clique problem that uses A as a subroutine

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Example from divisor problem

• integer x, max, j
• cin gtgt x
• max 1
• for (j2 jltx j)
• if (A(x,j)) // using A as a procedure
• max j
• return max

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Problem 3

• Connected graph problem
• Input Graph G (V,E)
• list of nodes and edges
• Yes/No Question Is there a path of edges between
  every pair of nodes?
• Construct an algorithm to solve this problem

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REC

• The class of solvable problems

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Programs and Inputs

• Suppose we have a language L
• In order for P to solve the language recognition
  problem associated with L, what should P do on a
  given string x
• P should halt after a finite amount of time
• P should correctly respond yes/no with respect to
  Is x in L?

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REC

• Formal Definition
• A C program P decides language L iff
• 1) Program P halts on all input strings
• 2) Program P correctly labels all strings it
  halts on as in or not in L.
• A language L is recursive or decidable iff there
  exists a C program P which decides L.
• What if there is a Java program P which decides
  L?
• REC the set of recursive languages

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Churchs Thesis

• What is our modified Churchs Thesis?
• Any algorithm can be written as a C program
• Any problem which cannot be solved by a C
  program cannot be solved by any algorithm.
• Importance?
• The definition of REC is robust.
• We could use other models of computation
• Turing machines, Java programs, Lisp Programs,...

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Languages
Question Is REC a proper subset of the set of
all languages/problems?
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Summary

• Solving problems
• Subroutine theme
• REC
• The class of solvable problems/languages
• Churchs Thesis
• Specific model of computation is not important
• Question Are all problems solvable?