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Lecture 4

Lecture 4. Topics Problem solving Subroutine Theme REC language class The class of solvable problems Closure properties. Problem Solving. Solving problems In this course, we spend a lot of time proving there are unsolvable problems

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Lecture 4

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  1. 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 today’s 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 <= i <= n} • E: Set of edges {(vi, vj)} • Clique: subset C of V such that there is an edge between all nodes in C

  5. Example v1 v2 v3 v6 v5 v4 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?

  6. Example 2 v1 v2 v3 v6 v5 v4 V = {v1, v2, v3, v4, v5, v6} E = { Max clique size? Smallest number of edges needed to increase m. c. size?

  7. Algorithm 1 • Construct an algorithm for solving the max clique problem

  8. Problem 2’ • Decision clique problem • Input: Graph G = (V,E), integerk • Note the extra input object k • Yes/No question: • Does the maximum clique in G have size at least k?

  9. 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

  10. Example from divisor problem integer x, max, j; cin >> x; max = 1; for (j=2; j<x; j++) if (A(x,j)) // using A as a procedure max = j; return max;

  11. 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

  12. REC The class of solvable problems

  13. 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?”

  14. REC • Formal Definition • A C++ program Pdecides 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

  15. Church’s Thesis • What is our modified Church’s 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,...

  16. REC Languages Question: Is REC a proper subset of the set of all languages/problems?

  17. Summary • Solving problems • Subroutine theme • REC • The class of solvable problems/languages • Church’s Thesis • Specific model of computation is not important • Question: Are all problems solvable?

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