How Hard Can a Problem Be ?
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1) The document discusses different levels of problem difficulty, from easy problems that can be solved in polynomial time to hard problems that cannot be solved deterministically in polynomial time. 2) It introduces the concept of a Turing machine as a model of computation and the Church-Turing thesis that anything computable can be computed by a Turing machine. 3) Hard problems, known as NP problems, may be solvable in polynomial time if a "perfect guesser" is available but cannot be solved deterministically in polynomial time unless P=NP. 4) Some problems are provably unsolvable, like the halting problem of determining if an arbitrary program will terminate for a given input.
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How Hard Can a Problem Be ?
- 1. How Hard Can a Problem Be ? Ahmad Saeed Alexandria University
- 2. Agenda Hard for whom ? Easy Problems Hard Problems Unsolvable Problems !!
- 3. A Problem Solver (Turing Machine) Follow the instructions to solve the problem Use a pencil, a rubber and a scrap paper Use them to find the answer You now have a : Turing Machine
- 4. Turing Machines Formally Physically A Scratch Tape ( Infinite ) A Moving head Mathematically A set of possible symbols to write on the tape A set of "states" A transition function between states
- 5. Turing Machine Example Determine whether the number of 1's in a sequence is odd (print 0 at the end of the tape) even (print 1 at the end of the tape)
- 6. Church-Turing Thesis Simply: If it's physically computable, it can be computed by a Turing Machine
- 7. "Easy" Problems (P) The class of problems that could be solved in P olynomial Time Examples Bubble Sort O(n 2 ) Matrix Multiplication O(n 3 ) What about a problem of O(n 100 ) ??
- 9. Nondeterminism (Example) Travelling Salesman Problem Find the path with the least cost to visit a set of cities
- 10. Nondeterminism (Example) What if we were really great guessers ??
- 11. "Hard" Problems (NP) N ondeterministic P olynomial Problems that are solved in a polynomial time if you are (or have access to) a perfect guesser Examples 3 CNF (conjunctive normal form) SAT (satisfiability) Scheduling Problem
- 12. Can all problems be solved by a computer ??
- 13. Halting Problem Given a binary representation of a program P, determine whether it enters an infinite loop on input X or not.
- 14. Halting Problem Proof Assume that we have Halt(P,x) which is a program that returns 0 if program P terminates on input x returns 1 it doesn't program doesn't halt Let M(x) be a program that doesn't terminate (loops forever) if Halt(x,x) returns 0 terminates if Halt(x,x) returns 1 What's the results if M(M) ??
- 15. What have we been talking about ?? Different models of computation boil down to Turing Machine N = NP ??? Computers can't do everything but can we ??
- 16. References Computational Complexity: A Modern Approach Textbook by Sanjeev Arora and Boaz Barak, Cambridge University Press. Lecture notes for cs1510 of Prof Kirk Pruhs of the Dept of Computer Science, University of Pittsburg Lecture notes for CSc2010 of Prof Jaman L. Bhola of the Dept of Computer Science, Georgia State University Wikipedia, 2011
- 17. Any Questions ?? (Open Discussion)
- 18. Thank you for hanging around!! Ahmad Saeed www.ahmad-saeed.co.cc [email_address]