Bounded Variable Logic, Parameterized Logarithmic Space, and Savitch's Theorem

We study the parameterized space complexity of model-checking first-order logic with a bounded number of variables. By restricting the number of the quantifier alternations we obtain problems complete for a natural hierarchy between parameterized logarithmic space and FPT. We call this hierarchy the tree hierarchy, provide a machine characterization, and link it to the recently introduced classes PATH and TREE. We show that the lowest class PATH collapses to parameterized logarithmic space only if Savitch’s theorem can be improved. Finally, we settle the complexity with respect to the tree-hierarchy of finding short undirected paths and small undirected trees.