# A Fine-Grained Hierarchy of Hard Problems in the Separated Fragment

## TL;DR

This paper introduces a detailed complexity analysis of the separated fragment (SF) in logic, establishing a hierarchy based on the degree of variable interaction and proving the precise computational complexity for each level.

## Contribution

It defines the degree of interaction as a new measure and proves the $k$-NEXPTIME-completeness of SF-satisfiability for each degree level, refining understanding of SF's complexity.

## Key findings

- SF-satisfiability is $k$-NEXPTIME-complete for each degree $k$.
- SF-satisfiability is non-elementary in the general case.
- Introduces the degree of interaction as a novel complexity measure.

## Abstract

Recently, the separated fragment (SF) has been introduced and proved to be decidable. Its defining principle is that universally and existentially quantified variables may not occur together in atoms. The known upper bound on the time required to decide SF's satisfiability problem is formulated in terms of quantifier alternations: Given an SF sentence $\exists \vec{z} \forall \vec{x}_1 \exists \vec{y}_1 \ldots \forall \vec{x}_n \exists \vec{y}_n . \psi$ in which $\psi$ is quantifier free, satisfiability can be decided in nondeterministic $n$-fold exponential time. In the present paper, we conduct a more fine-grained analysis of the complexity of SF-satisfiability. We derive an upper and a lower bound in terms of the degree of interaction of existential variables (short: degree)}---a novel measure of how many separate existential quantifier blocks in a sentence are connected via joint occurrences of variables in atoms. Our main result is the $k$-NEXPTIME-completeness of the satisfiability problem for the set $SF_{\leq k}$ of all SF sentences that have degree $k$ or smaller. Consequently, we show that SF-satisfiability is non-elementary in general, since SF is defined without restrictions on the degree. Beyond trivial lower bounds, nothing has been known about the hardness of SF-satisfiability so far.