Dehn's Word Problem $DWP(G, S)$ asks whether a word in $(S \cup S^{-1})^*$ is the identity in $G$ for a Group $G$ with a finite generating set $S$ of $G$.
I'm new to word problems and one of the first questions that came to my mind is whether we can solve $DWP(G, T)$ given that we can solve $DWP(S, G)$ where $S$ and $T$ are two finite generating sets.
From what I have seen in the literature (e.g. here, Lemma 2.13) we can reduce $DWP(S, G)$ to $DWP(S, T)$, but the proofs rely on having homomorphism $$h: (S \cup S^{-1})^∗ \to (T \cup T^{−1})^∗ $$
But if all we have is an algorithm for $DWP(G,S)$, and not $h$, is there really anything we can say about $DWP(G, T)$? What if $S$ is a subset of $T$?
First write each element $t_i\in T$ as a word in $(S\cup S^{-1})^\ast$: $$t_i=s^{(i)}_{1}\cdots s^{(i)}_{n_i};\quad s^{(i)}_j\in S\cup S^{-1}$$
Now, given two words $w=w(t_1,\dots,t_k)$ and $w'=w'(t_1,\dots,t_k)$ in $(T\cup T^{-1})^\ast$; we want to test whether $\overline{w}=\overline{w'}$ in $G$(here I use $\overline{x}$ to denote the element of $G$ represented by the word $x$).
Substituting the expressions for the $t_i$ in terms of $S\cup S^{-1}$ into $w$ and $w'$ gives new words representing the same elements as $\overline{w}$ and $\overline{w'}$, call these $u$ and $u'$ respectively. Now $\overline{w}=\overline{w'}$ if and only if $\overline{u}=\overline{u'}$, which we can check using our solution to $DWP(G,S)$.
This is the same proof as given in Lemma 2.13, I have just written out the homomorphism $h\colon (S\cup S^{-1})^\ast\to(T\cup T^{-1})^\ast$ explicitly by rewriting the $t_i$'s in terms of the elements of $S$.