.1 Introduction 1.3 the Halting Problem

In computability and complexity theory, all problems are reduced to language recognition, where a language is defined as a set of strings over some alphabet, typically {0, 1}. For example, L1 = {10, 11, 101, · · · } is the language that contains all prime numbers, represented in binary. Then the problem of determining whether or not a number x is a prime is the same as determining whether or not x ∈ L1. As a more complicated example, consider the problem of factoring a number N . The language corresponding to this problem is L2 = {〈x, k〉 : x has a nontrivial factor ≤ k}, where 〈x, k〉 denotes an encoding of (x, k) in binary. Then we can determine the factors of N by repeatedly determining whether 〈N,m〉 ∈ L2, doing a binary search on m. (For example, in the case of N = 35, we check 〈35, 35〉, 〈35, 17〉, 〈35, 8〉, 〈35, 4〉, 〈35, 6〉, and finally 〈35, 5〉 in order to determine that 5 is a factor of 35.) With this formulation of a problem, we show that there exist problems that are not computable.