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We prove that if $A$ is a computable Hopfian finitely presented structure,then $A$ has a computable $d$-$\Sigma_2$ Scott sentence if and only if the weakWhitehead problem for $A$ is decidable. We use this to infer that everyhyperbolic group as well as any polycyclic-by-finite group has a computable$d$-$\Sigma_2$ Scott sentence, thus covering two main classes of finitelypresented groups. Our proof also implies that every weakly Hopfian finitelypresented group is strongly defined by its $\exists^+$-types, a question whicharose in a different context.

Computable Scott sentences and the weak Whitehead problem for finitely presented groups