On the satisfiability problem for fragments of the two-variable logic with one transitive relation

TL;DR

This paper investigates the satisfiability problem for a fragment of two-variable first-order logic with one transitive relation, showing decidability in certain cases but leaving the full problem open.

Contribution

It introduces a novel model construction technique based on the infinite Ramsey theorem for a specific fragment, clarifying the decidability status of related logic fragments.

Findings

Decidability in 2-NExpTime for guarded existential formulas

The technique does not extend to full two-variable logic with transitivity

The full satisfiability problem remains open and unresolved

Abstract

We study the satisfiability problem for the two-variable first-order logic over structures with one transitive relation. % We show that the problem is decidable in 2-NExpTime for the fragment consisting of formulas where existential quantifiers are guarded by transitive atoms. As this fragment enjoys neither the finite model nor the tree model property, to show decidability we introduce novel model construction technique based on the infinite Ramsey theorem. We also point out why the technique is not sufficient to obtain decidability for the full two-variable logic with one transitive relation, hence contrary to our previous claim, [FO$^2$ with one transitive relation is decidable, STACS 2013: 317-328], the status of the latter problem remains open.