# The Keys to Decidable HyperLTL Satisfiability: Small Models or Very Simple Formulas

## TL;DR

This paper explores the boundaries of decidability in HyperLTL satisfiability by examining small model sizes and simple formulas, providing new decidability results and complexity bounds.

## Contribution

It identifies decidability conditions for HyperLTL satisfiability based on model size restrictions and formula simplicity, extending understanding of its computational limits.

## Key findings

- Decidability achieved for models of bounded size
- Decidability for formulas with nesting depth one
- Tight complexity bounds established for various cases

## Abstract

HyperLTL, the extension of Linear Temporal Logic by trace quantifiers, is a uniform framework for expressing information flow policies by relating multiple traces of a security-critical system. HyperLTL has been successfully applied to express fundamental security policies like noninterference and observational determinism, but has also found applications beyond security, e.g., distributed protocols and coding theory. However, HyperLTL satisfiability is undecidable as soon as there are existential quantifiers in the scope of a universal one. To overcome this severe limitation to applicability, we investigate here restricted variants of the satisfiability problem to pinpoint the decidability border. First, we restrict the space of admissible models and show decidability when restricting the search space to models of bounded size or to finitely representable ones. Second, we consider formulas with restricted nesting of temporal operators and show that nesting depth one yields decidability for a slightly larger class of quantifier prefixes. We provide tight complexity bounds in almost all cases.