The parameterized space complexity of model-checking bounded variable first-order logic

The parameterized model-checking problem for a class of first-order sentences (queries) asks to decide whether a given sentence from the class holds true in a given relational structure (database); the parameter is the length of the sentence. In 1995 Vardi observed a polynomial time algorithm deciding the model-checking problem for queries with a bounded number of variables. We study its parameterized space complexity. For each bound on the quantifier alternation rank the problem becomes complete for the corresponding level of what we call the tree hierarchy, a hierarchy of parameterized complexity classes defined via space bounded alternating machines between parameterized logarithmic space and fixedparameter tractable time. We observe that a parameterized logarithmic space model-checker for existential bounded variable queries would allow to improve Savitch’s classical simulation of nondeterministic logarithmic space in deterministic space O(log). Further, we define a highly space efficient model-checker for queries with a bounded number of variables and bounded quantifier alternation rank. We study its optimality under the assumption that Savitch’s theorem is optimal.