On Freeze LTL with Ordered Attributes

TL;DR

This paper extends Freeze LTL with attribute dependency relations, creating a hierarchy of decidable fragments with complex non-primitive recursive satisfiability problems.

Contribution

It introduces a novel extension of Freeze LTL allowing attribute dependency relations, establishing a hierarchy of decidable fragments with high complexity.

Findings

Decidability depends on the type of attribute relations used.

Tree-like relations lead to a hierarchy of complexity classes.

Satisfiability is complete for the class ${\bf F}_{\epsilon_0}$.

Abstract

This paper is concerned with Freeze LTL, a temporal logic on data words with registers. In a (multi-attributed) data word each position carries a letter from a finite alphabet and assigns a data value to a fixed, finite set of attributes. The satisfiability problem of Freeze LTL is undecidable if more than one register is available or tuples of data values can be stored and compared arbitrarily. Starting from the decidable one-register fragment we propose an extension that allows for specifying a dependency relation on attributes. This restricts in a flexible way how collections of attribute values can be stored and compared. This conceptual dimension is orthogonal to the number of registers or the available temporal operators. The extension is strict. Admitting arbitrary dependency relations satisfiability becomes undecidable. Tree-like relations, however, induce a family of decidable fragments escalating the ordinal-indexed hierarchy of fast-growing complexity classes, a recently introduced framework for non-primitive recursive complexities. This results in completeness for the class ${\bf F}_{\epsilon_0}$. We employ nested counter systems and show that they relate to the hierarchy in terms of the nesting depth.