Background and Motivation
Many occasions arise in mathematics where one wishes to use a finite amount of information to describe an element of a (typically infinite) set. This issue is particularly apparent in computational mathematics. Traditional models of computation (such as the Turing machine) have storage capacity which is unbounded, so it is in principle possible to perform computations with the elements of infinite sets. On the other hand, since the amount of storage space in use at any one time is finite, we need each element to have a finite representation.
For various reasons, it is not always possible or desirable to use a system of unique encodings, that is, one in which every element has a single encoding. When using an encoding system without uniqueness, the question naturally arises of whether there is an algorithm which, given as input two encodings, decides whether they represent the same element. Such an algorithm is called a solution to the word problem for the encoding system.
Read more about this topic: Word Problem (mathematics)
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