The Triguarded Fragment

2026-08-03

AI Summary

This work addresses the challenge of achieving decidable satisfiability in first-order logic while preserving high expressive power. It introduces the Triadic Guarded Fragment (TGF), which elegantly combines the Guarded Fragment (GF) and the two-variable fragment (FO²) by imposing guarded quantification only on subformulas with three or more free variables. TGF is the first extension of GF that retains both the advantages of GF and FO² while remaining decidable. The paper establishes tight complexity bounds for TGF under various constraints: satisfiability is N2ExpTime-complete when equality is restricted, drops to NExpTime-complete when predicate arities are fixed, and exhibits NP-complete data complexity. Furthermore, TGF enjoys a tight doubly exponential finite model property.

Abstract

A prominent research question in computational logic is how to restrict first-order predicate logic (FO) in such a way that the satisfiability problem becomes decidable. Among others, past efforts have identified two prominent decidable FO fragments of high expressivity: the guarded fragment (GF), and the two-variable fragment (FO2). These fragments are of high interest and crucial importance as they provide significant insights into decidability and expressiveness of other prominent (computational) logics like Modal Logics (MLs)} and various Description Logics (DLs)}, which play a central role in Verification, Knowledge Representation, and other areas. In this article, we show that GF and FO2 can be combined into a new fragment that subsumes both, while maintaining decidability of the satisfiability problem. This fragment, called the triguarded fragment (denoted TGF), is obtained by relaxing the standard definition of GF by requiring guardedness of quantification only for subformulae with three or more free variables. We show that, when restricting the use of equality, satisfiability in TGF is N2ExpTime-complete, dropping to NExpTime-complete when the maximum predicate arity is fixed (a natural assumption in the context of MLs and DLs). We further establish that the problem is NP-complete in terms of data complexity, which is again in line with data complexity results for basic expressive DLs. We observe that many natural extensions of TGF, including the liberal use of equality, lead to undecidability. We also establish that TGF has the finite model property (providing a tight doubly exponential bound on the model size), whence finite satisfiability coincides with satisfiability.

Problem

• first-order logic
• decidability
• satisfiability
• guarded fragment
• two-variable fragment

Innovation

• Triguarded Fragment
• Decidable Fragment
• Guarded Fragment
• Two-Variable Logic
• Finite Model Property