Two-Variable Logic for Hierarchically Partitioned and Ordered Data

We study Two-Variable First-Order Logic, FO2, under semantic constraints that model hierarchically structured data. Our first logic extends FO2 with a linear order < and a chain of increasingly coarser equivalence relations E_1 ⊆ E_2 ⊆ ... . We show that its finite satisfiability problem is NExpTime-complete. We also demonstrate that a weaker variant of this logic without the linear order enjoys the exponential model property. Our second logic extends FO2 with a chain of nested total preorders ⪯_1 ⊆⪯_2 ⊆ ... . We prove that its finite satisfiability problem is also NExpTime-complete.However, we show that the complexity increases to ExpSpace-complete once access to the successor relations of the preorders is allowed. Our last result is the undecidability of FO2 with two independent chains of nested equivalence relations.