Is there an algorithmic way to check if a power-conjugate presentation (for a finite polycyclic group) is consistent?
Background: A finite solvable group $G$ has a subnormal series
$$ G=G_0 \triangleright G_1\triangleright \dots G_n=1$$
with each $G_{k-1}/G_k$ cyclic of prime order $p_k$. A power-conjugate presentation is, informally, a description of $G$ in terms of this series. Specifically, for each $k$, one picks an element $a_k$ of $G_{k-1}$ whose image in the quotient $G_{k-1}/G_k$ is a generator; then $a_k^{p_k}\in G_k$ and as $G_k$ is normal in $G_{k-1}$, $a_k$ acts on $G_k$ by conjugation. One can construct $G_{k-1}$ from $G_k$ via these two pieces of data: the element of $G_k$ equal to $a_k^{p_k}$, and the conjugation action of $a_k$ on $G_k$. A power-conjugate presentation specifies exactly this data for each $a_k$ and thus has the form $$\langle a_1,\dots,a_n\mid \{a_k^{p_k} = \mu_k\}_{1\leq k\leq n}, \{a_j^{a_k} = \mu_{kj}\}_{1\leq k<j\leq n}\rangle $$ where the $\mu_k$ and the $\mu_{kj}$ are words on $a_{k+1},\dots,a_n$ (specifying elements of $G_k$).
Any group presentation having this form is said to be a power-conjugate presentation (whether or not it was arrived at as described above, beginning with some already-existing group $G$). It is said to be consistent if each element of the group presented has a unique representation in the form $a_1^{e_1}\dots a_n^{e_n}$ with $0\leq e_k<p_k$ for each $k$.
Clearly if I begin with a group $G$ and a subnormal series $G=G_0\triangleright \dots \triangleright G_n=1$ as above, and build the power-conjugate presentation as described above, by picking a set of $a_k$'s whose images generate $G_{k-1}/G_k$ and then specifying the values $a_k^{p_k}\in G_k$ and the conjugation action of $a_k$ on $G_k$, the presentation I arrive at will be consistent, as each element $x$ of $G$ has a unique representation as $a_1^{e_1}\dots a_n^{e_n}$ with $0\leq e_k<p_k$ for each $k$. (First look at the image of $x$ in $G_0/G_1$ to determine $e_1$; then $a_1^{-e_1}x$ will be in $G_1$ so look at its image in $G_1/G_2$ to determine $e_2$; etc.)
On the other hand, if I start trying to build a presentation outside of any reference to a preexisting group, just specifying the $\mu_k, \mu_{kj}$ as arbitrary words on $a_{k+1},\dots,a_n$, I may arrive at a presentation that is not consistent.
My question is: is there an algorithmic way to recognize a consistent presentation?
A followup question: if there is, are the computations reasonable to do by hand in small cases?
Yes, this is a special case of confluence testing in the Knuth-Bendix completion process. It is described for polycyclic presentations (including for infinite groups) in Section 12.4 of "Handbook of Computational Group Theory" by Holt, Eick and O'Brien.
For a polycyclic series of length $n$ there are about $n^3/6$ consistency conditions to check - most of these consist of checking the associative law between three generators, but there are some others as well, so it could be done by hand for small $n$, although you are less likely to make mistakes if you use a computer!