Given a graph $G$ we provide dynamic programming algorithms for many locally checkable vertex subset and vertex partitioning problems. Their runtime is polynomial in the number of equivalence classes of problem-specific equivalence relations on subsets of vertices, defined on a given decomposition tree of $G$. Using these algorithms all these problems become solvable in polynomial time for many well-known graph classes like interval graphs and permutation graphs (Belmonte and Vatshelle (2013)). Given a decomposition of boolean-width $k$ we show that the algorithms will have runtime $O(n^{4}2^{O(k^{2})})$, providing the first large class of problems solvable in fixed-parameter single-exponential time in boolean-width.