$b' \in \{ 0, 1 \}$ is a homomorphism from ${\wp}({\mathbf{B}}) \times {\mathbf{B}}$ to ${\mathbf{B}}$ satisfying the condition of Theorem \[thm:laac-solves\]. Hence, the problem ${{\mathsf{CSP}}}({\mathbf{B}})$ is solvable by LAAC.

To show that the problem ${{\mathsf{CSP}}}({\mathbf{B}})$ is not solvable by AC, let $f$ be an arbitrary mapping from ${\wp}(B) \setminus \{ \emptyset \}$ to $B$. We show that $f$ cannot be a homomorphism from ${\wp}({\mathbf{B}})$ to ${\mathbf{B}}$, which suffices by Theorem \[thm:ac-solves\]. Let $b = f(\{ 0, 1 \})$. It holds that $( \{ 0, 1 \}, \{ 0, 1 \} ) \in R_{(b, b)}^{{\wp}({\mathbf{B}})}$, but $(f(\{ 0, 1 \}), f(\{ 0, 1 \})) = (b, b) \notin R_{(b, b)}^{{\mathbf{B}}}$, and we are done.

There exists a structure ${\mathbf{B}}$ (not having all constants) such that ${{\mathsf{CSP}}}({\mathbf{B}})$ is solvable by AC but not by LAAC.

Take ${\mathbf{B}}$ to be the relational structure with universe $\{ 0, 1 \}$ over signature $\{ R, S \}$ where $R^{{\mathbf{B}}} = \{ 0, 1 \}^3 \setminus \{ (0, 1, 1) \}$ and $S^{{\mathbf{B}}} = \{ 0, 1 \}^3 \setminus \{ (1, 0, 0) \}$. The mapping $p$ that sends each element of ${\wp}(B) \setminus \{ \emptyset \}$ to $0$ is a homomorphism from ${\wp}({\mathbf{B}})$ to ${\mathbf{B}}$, 