The group PL([0,1]) of homeomorphisms of [0,1] that are built piecewise from linear functions includes many important subgroups, particularly the group known as Thompson’s group F that appears in research in many areas including logic, group theory, topology, operator theory, etc. For finitely generated “computable” subgroups of PL_+([0,1]) (including Thompson’s F), we use topological properties of the generating homeomorphisms to build algorithms that solve a variety of problems. In this talk I will give an overview of a few fundamental computational questions in group theory, and I will describe (in pictures and words) algorithmic solutions for computable subgroups of PL_+([0,1]). This is joint work with Collin Bleak and Tara Brough.