We study the position of the computable setting in the "common theory of locality" developed in arXiv:2106.02066 and arXiv:2204.09329 for local problems on $Δ$-regular trees, $Δ\in ω$. We show that such a problem admits a computable solution on every highly computable $Δ$-regular forest if and only if it admits a Baire measurable solution on every Borel $Δ$-regular forest. We also show that if such a problem admits a computable solution on every computable maximum degree $Δ$ forest then it admits a continuous solution on every maximum degree $Δ$ Borel graph with appropriate topological hypotheses, though the converse does not hold.