I have a set of relators $[r_1, r_2, r_3, r_4]$ of a free group on two generators $a$ and $b$, where $\begin{cases} r_1 = a^{2p} \text{ (where $p$ is some positive integer) } \\ r_2 = b^4 \\ t_3 = (ab)^2,\\ r_4 = (ab^{-1})^2 \end{cases}$
how one can express words like $w = a^{-2}ba^2b^{-1}$ that lie in the group generated by the $r_i$ as words formed by the $r_i$. What I tried was to take a $r$ that is an $r_i$ up to a certain power or up to a cyclic rotation of its generators (in this example $ba$ or $ba^{-1}$) and make the product $wr$ such that there are less generators in the original word. I don't know if this method always works as I'm afraid that the resulting words will become bigger and bigger. Is there an specific method that always gives a result?
This is in general a hard problem. You might want to read up on Dehn's algorithm (which is a bit like what you hope for, but requires special circumstances):
https://en.wikipedia.org/wiki/Small_cancellation_theory