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Week 1, Lecture 1

Problems

Problems can be assumed as membership queries in languages.

Decision Problems

  • Output is a single bit (true or false)
  • Each decision problem is characterized by a subset of the set of all possible inputs (that is, subset of, say, {0,1}*) or languages eg: INPUT: String x ; OUTPUT: YES if x in language L, else NO

Example 1: Primality Testing PRIMES = {n | n is prime}

Example 2 : Graph Connectivity GC = {Graph G | G is connected}

Example 3 : Boolean Satisfiability SAT = {CNF Boolean formula f | f is not a fallacy; i.e. f is satisfiable}

Axioms of Computation

  • It takes non-zero time to retrieve data from far-off locations

    • Machines Are Not Omnipresent
  • Only finite information can be stored/retrieved from finite volume

    • Machines Are Not Omniscient
  • A finite length code only exerts finite amount of control

    • Machines Are Not Omnipotent

How to COMPARE various Solutions?

solutions