 Group encoding of a modular machine

A modular machine can be encoded by HNN extensions of $K={\mathbb{Z}}^2\ast ⟨t⟩$. The kernel of $K\to {\mathbb{Z}}^2$ has a free basis of a group $t(r,s)=x^{-r}{y}^{-s}t{x}^r{y}^s$. Associated-subgroup maps send these basis elements according to the two arithmetic transition types. In the resulting group, membership of $t(r,s)$ in the subgroup generated by $t$ and the instruction stable letters is equivalent to $(r,s)\in H_0(\mathcal{M})$. One more HNN extension centralizes that finitely generated subgroup, converting membership to equality and hence to the word problem for a group.