Consider the P-complete problem Horn which asks whether a given set of Horn clauses is (un)satisfiable. To solve it one keeps a dynamic set of atoms that are forced to be true. Using the clauses one then adds atoms to this set until saturation is reached. It is easy to see that this dynamic set will in general more than constant size even if we allow to discard already proved atoms. Given that we need logarithmic space to store a single atom on a Turing machine tape this seems like a strong intuitive argument for the hypothesis that logarithmic space is different from polynomial time. We thus tried to find formal models of computation in which this intuitive argument can be made rigorous. Thus, we study computational models that can be simulated in logarithmic space and encompass logspace algorithms which manipulate a constant size of objects that require logarithmic space individually such as pointers or graph nodes. The hope is then to be able to show that such models are provably unable to solve P-complete problems. We report in this survey article on our partial results towards this goal as well as the state-of-the-art in general.