In this chapter, we study three algorithmic problems involving computation trees: the optimization, solvability, and satisfiability problems. The solvability problem is concerned with recognizing computation trees that solve problems. The satisfiability problem is concerned with recognizing sentences that are true in at least one structure from a given set of structures. We study how the decidability of the optimization problem depends on the decidability of solvability and satisfiability problems. We also consider various examples with both decidable and undecidable solvability and satisfiability problems.