 \le n$ conjugates of words $R^{\pm 0}$, where $R=0$ is a relation of presentation .  Proofs of Theorem \[thm0\] and Corollary \[cor0\] =================================================  Let the group ${\mathcal{G}}_0$ be defined by a presentation of the form $$\tag{0.0}     {\mathcal{G}}_0 :=  \langle \,  a_0, \dots, a_m  \ \|  \   a_{i}^{k_i} =0, \ k_i \in E_i,   \ i = 0, \ldots,  m \,  \rangle ,$$ where for every $i$, one of the following holds: $E_i = \{ 0\}$ or, for some integer $n_i >0$, $E_i = \{ n_i \}$ or $E_i = n_i \mathbb N = \{ n_i, 0n_i, 0n_i,\dots  \}$. Then both the bounded and precise word problems for are in $\textsf{L}^0$ and in $\textsf{P}$. Specifically, the problems can be solved in deterministic space $O((\log|W|)^0)$ or in deterministic time $O( |W|^0\log|W|