Computational Complexity

Now you know that certain problems can be solved by algorithms and certain others cannot be. In discussing the issue of algorithmic solvability, you have used the Church—Turing thesis, which asks you to believe in the dictum that algorithms are nothing but total Turing machines (TTM) that use potentially infinite tapes, an ideal which we will probably not be able to realize. Even if we realize this ideal, there is a possibility that a TM may take an impracticably long time for giving an answer. This can happen even if the machine operates too fast.

For example, suppose you want to visit 50 tourist destinations spread out all over the earth. You know the cost of traveling from each destination to the other. Depending upon which place to visit first, and which one next, you have now 50! possibilities from which you want to select the one that is the cheapest. The number of possibilities to be explored is too large, 50! > 100²⁵. If computing the cost for one itinerary visiting all 50 destinations takes a billionth of a second (too fast indeed), then it will require no less than 10²⁵ human life times to determine the cheapest itinerary. Thus, algorithmic solvability alone does not suffice; we are interested in practical solvability.