Recursive $p$-adic numbers and elementary theories of finitely generated pro-$p$-groups

The authors propose a method of effective presentation of finitely generated pro-$p$-groups, and use it to study the elementary theories of such groups. They prove that elementarily equivalent finitely generated pro-$p$-groups are isomorphic. The main result is the following criterion: the elementary theory of a finitely generated nilpotent pro-$p$-group $G$ is decidable if and only if $G$ is effectively presented.
Bibliography: 18 titles.