Group with solvable conjugacy problem

Symbol-free definition
A group with solvable conjugacy problem is a finitely presented group with a finite presentation having the following property: there is an algorithm that, given any two words in the generators, can, in finite time, test whether the two words represent conjugate elements in the group.

The finite time taken depends on the word, but because the generating set is finite, there are only finitely many words of any length, so we cna obtain an upper bound on the time taken by the algorithm as a function of the length of the word.