Chapter 1 - Introduction: Models of Computation

Summary

The basic question in the theory of computing can be formulated in any of the following ways: What is computable? For which problems can we construct effective mechanical procedures that solve every instance of the problem? Which problems possess algorithms for their solutions?

Fundamental developments in mathematical logic during the 1930s showed the existence of unsolvable problems: No algorithm can possibly exist for the solution of the problem. Thus, the existence of such an algorithm is a logical impossibility—its nonexistence has nothing to do with our ignorance. This state of affairs led to the present formulation of the basic question in the theory of computing. Previously, people always tried to construct an algorithm for every precisely formulated problem until (if ever) the correct algorithm was found. The basic question is of definite practical significance: One should not try to construct algorithms for an unsolvable problem. (There are some notorious examples of such attempts in the past.)

A model of computation is necessary for establishing unsolvability. If one wants to show that no algorithm for a specific problem exists, one must have a precise definition of an algorithm. The situation is different in establishing solvability: It suffices to exhibit some particular procedure that is effective in the intuitive sense. (We use the terms algorithm and effective procedure synonymously.