  We study complexity classes of local problems on regular trees from the
perspective of distributed local algorithms and descriptive combinatorics. We
show that, surprisingly, some deterministic local complexity classes from the
hierarchy of distributed computing exactly coincide with well studied classes
of problems in descriptive combinatorics. Namely, we show that a local problem
admits a continuous solution if and only if it admits a local algorithm with
local complexity $O(\log^* n)$, and a Baire measurable solution if and only if
it admits a local algorithm with local complexity $O(\log n)$.
