On Homogeneous Model of Fluted Languages

TL;DR

This paper investigates the fluted fragment of first-order logic, demonstrating that satisfiable sentences have homogeneous models, which simplifies analysis and establishes the decidability and complexity of satisfiability problems, including for extensions with counting quantifiers.

Contribution

It introduces the concept of homogeneous models for the fluted fragment, enabling new decidability results and complexity classifications for satisfiability, especially with counting quantifiers.

Findings

Finite satisfiability is TOWER-complete.

Two-variable case reduces complexity to NEXPTime.

Extensions with counting quantifiers increase complexity to Sigma1_1.

Abstract

We study the fluted fragment of first-order logic which is often viewed as a multi-variable non-guarded extension to various systems of description logics lacking role-inverses. In this paper we show that satisfiable fluted sentences (even under reasonable extensions) admit special kinds of ``nice'' models which we call globally/locally homogeneous. Homogeneous models allow us to simplify methods for analysing fluted logics with counting quantifiers and establish a novel result for the decidability of the (finite) satisfiability problem for the fluted fragment with periodic counting. More specifically, we will show that the (finite) satisfiability problem for the language is ${\rm T{\small OWER}}$-complete. If only two variable are used, computational complexity drops to ${\rm NE{\small XP}T{\small IME}}$-completeness. We supplement our findings by showing that generalisations of fluted logics, such as the adjacent fragment, have finite and general satisfiability problems which are, respectively, $\Pi^0_1$- and $\Sigma^0_1$-complete. Additionally, satisfiability becomes $\Sigma^1_1$-complete if periodic counting quantifiers are permitted.