Satisfiability of the Two-Variable Fragment of First-Order Logic over Trees

TL;DR

This paper analyzes the computational complexity of the satisfiability problem for the two-variable fragment of first-order logic over finite trees, revealing EXPSPACE-completeness and complexity reductions for certain restricted cases.

Contribution

It establishes the complexity classifications of the satisfiability problem for various fragments of first-order logic over finite trees, including singular trees and restricted predicates.

Findings

Full logic is EXPSPACE-complete over finite singular trees.

Binary predicate descendant reduces complexity to NEXPTIME.

Guarded logic with one binary predicate is PSPACE-complete over finite singular trees.

Abstract

We consider the satisfiability problem for the two-variable fragment of first-order logic over finite unranked trees. We work with signatures consisting of some unary predicates and the binary navigational predicates child, right sibling, and their respective transitive closures. We prove that the satisfiability problem for the logic containing all these predicates is EXPSPACE-complete. Further, we consider the restriction of the class of structures to singular trees, i.e., we assume that at every node precisely one unary predicate holds. We observe that the full logic and even for unordered trees remain EXPSPACE-complete over finite singular trees, but the complexity decreases for some weaker logics. Namely, the logic with one binary predicate, descendant is NEXPTIME-complete, and its guarded version is PSPACE-complete over finite singular trees, even though both these logics are EXPSPACE-complete over arbitrary finite trees.