Computability Theory
0. the Informal Concept
Computability theory is the area of mathematics dealing with the concept of an effective procedure—a procedure that can be carried out by following specific rules. For example, one might ask whether there is some effective procedure—some algorithm—that, given a sentence about the positive integers, will decide whether that sentence is true or false. In other words, is the set of true sentences about the positive integers decidable? Or for a much simpler example, the set of prime numbers is certainly a decidable set. That is, there are mechanical procedures, that are taught in the schools, for deciding of any given positive integer whether or not it is a prime number.
More generally, consider a set S, which can be either a set of natural numbers (the natural numbers are 0, 1, 2, … ), or a set of strings of letters from a finite alphabet. (These two situations are entirely interchangeable. A set of natural numbers is much like a set of base-10 numerals, which are strings of digits. And in the other direction, a string of letters can be coded by a natural number in a variety of ways. The best way is, where the alphabet has k symbols, to utilize k-adic notation, which is like base-k numerals except that the k digits represent 1, 2, …, k, without a 0 digit.) One can say that S is a decidable set if there exists an effective procedure that, given any natural number (in the first case) or string of letters (in the second case), will eventually end by supplying the answer: "Yes" if the given object is a member of S and "No" if it is not a member of S.
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